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7.E: Quantum Mechanics (Exercises)

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Conceptual Questions

7.1 wave functions.

1. What is the physical unit of a wave function, \(\displaystyle Ψ(x,t)\)? What is the physical unit of the square of this wave function?

2. Can the magnitude of a wave function \(\displaystyle (Ψ∗(x,t)Ψ(x,t))\) be a negative number? Explain.

3. What kind of physical quantity does a wave function of an electron represent?

4. What is the physical meaning of a wave function of a particle?

5. What is the meaning of the expression “expectation value?” Explain.

7.2 The Heisenberg Uncertainty Principle

6. If the formalism of quantum mechanics is ‘more exact’ than that of classical mechanics, why don’t we use quantum mechanics to describe the motion of a leaping frog? Explain.

7. Can the de Broglie wavelength of a particle be known precisely? Can the position of a particle be known precisely?

8. Can we measure the energy of a free localized particle with complete precision?

9. Can we measure both the position and momentum of a particle with complete precision?

7.3 The Schrӧdinger Equation

10. What is the difference between a wave function \(\displaystyle ψ(x,y,z)\) and a wave function \(\displaystyle Ψ(x,y,z,t)\) for the same particle?

11. If a quantum particle is in a stationary state, does it mean that it does not move?

12. Explain the difference between time-dependent and -independent Schrӧdinger’s equations.

13. Suppose a wave function is discontinuous at some point. Can this function represent a quantum state of some physical particle? Why? Why not?

7.4 The Quantum Particle in a Box

14. Using the quantum particle in a box model, describe how the possible energies of the particle are related to the size of the box.

15. Is it possible that when we measure the energy of a quantum particle in a box, the measurement may return a smaller value than the ground state energy? What is the highest value of the energy that we can measure for this particle?

16. For a quantum particle in a box, the first excited state (\(\displaystyle Ψ_2\)) has zero value at the midpoint position in the box, so that the probability density of finding a particle at this point is exactly zero. Explain what is wrong with the following reasoning: “If the probability of finding a quantum particle at the midpoint is zero, the particle is never at this point, right? How does it come then that the particle can cross this point on its way from the left side to the right side of the box?

7.5 The Quantum Harmonic Oscillator

17. Is it possible to measure energy of \(\displaystyle 0.75ℏω\) for a quantum harmonic oscillator? Why? Why not? Explain.

18. Explain the connection between Planck’s hypothesis of energy quanta and the energies of the quantum harmonic oscillator.

19. If a classical harmonic oscillator can be at rest, why can the quantum harmonic oscillator never be at rest? Does this violate Bohr’s

correspondence principle?

20. Use an example of a quantum particle in a box or a quantum oscillator to explain the physical meaning of Bohr’s correspondence principle.

21. Can we simultaneously measure position and energy of a quantum oscillator? Why? Why not?

7.6 The Quantum Tunneling of Particles through Potential Barriers

22. When an electron and a proton of the same kinetic energy encounter a potential barrier of the same height and width, which one of them will

tunnel through the barrier more easily? Why?

23. What decreases the tunneling probability most: doubling the barrier width or halving the kinetic energy of the incident particle?

24. Explain the difference between a box-potential and a potential of a quantum dot.

25. Can a quantum particle ‘escape’ from an infinite potential well like that in a box? Why? Why not?

26. A tunnel diode and a resonant-tunneling diode both utilize the same physics principle of quantum tunneling. In what important way are they different?

27. Compute \(\displaystyle |Ψ(x,t)|^2\) for the function \(\displaystyle Ψ(x,t)=ψ(x)sinωt\), where \(\displaystyle ω\) is a real constant.

28. Given the complex-valued function \(\displaystyle f(x,y)=(x−iy)/(x+iy)\), calculate \(\displaystyle |f(x,y)|^2\).

29. Which one of the following functions, and why, qualifies to be a wave function of a particle that can move along the entire real axis?

(a) \(\displaystyle ψ(x)=Ae^{−x^2}\);

(b) \(\displaystyle ψ(x)=Ae^{−x};\)

(c) \(\displaystyle ψ(x)=Atanx\);

(d) \(\displaystyle ψ(x)=A(sinx)/x\);

(e) \(\displaystyle ψ(x)=Ae^{−|x|}\).

30. A particle with mass m moving along the x -axis and its quantum state is represented by the following wave function: \(\displaystyle Ψ(x,t)=\begin{cases}0&x<0\\Axe^{−αx}e^{−iEt/ℏ}&,x≥0\end{cases}\), where \(\displaystyle α=2.0×10^{10}m^{−1}\).

(a) Find the normalization constant.

(b) Find the probability that the particle can be found on the interval \(\displaystyle 0≤x≤L\).

(c) Find the expectation value of position.

(d) Find the expectation value of kinetic energy.

31. A wave function of a particle with mass m is given by \(\displaystyle ψ(x)=\begin{cases}Acosαx&−\frac{π}{2α}≤x≤+\frac{π}{2α}\\0&otherwise\end{cases}\), where \(\displaystyle α=1.00×10^{10}/m\).

(b) Find the probability that the particle can be found on the interval \(\displaystyle 0≤x≤0.5×10^{−10}m\).

(c) Find the particle’s average position.

(d) Find its average momentum.

(e) Find its average kinetic energy \(\displaystyle −0.5×10^{−10}m≤x≤+0.5×10^{−10}m\).

32. A velocity measurement of an \(\displaystyle α\)-particle has been performed with a precision of 0.02 mm/s. What is the minimum uncertainty in its position?

33. A gas of helium atoms at 273 K is in a cubical container with 25.0 cm on a side.

(a) What is the minimum uncertainty in momentum components of helium atoms?

(b) What is the minimum uncertainty in velocity components?

(c) Find the ratio of the uncertainties in

(b) to the mean speed of an atom in each direction.

34. If the uncertainty in the \(\displaystyle y\)-component of a proton’s position is 2.0 pm, find the minimum uncertainty in the simultaneous measurement of the proton’s \(\displaystyle y\)-component of velocity. What is the minimum uncertainty in the simultaneous measurement of the proton’s xx-component of velocity?

35. Some unstable elementary particle has a rest energy of 80.41 GeV and an uncertainty in rest energy of 2.06 GeV. Estimate the lifetime of this particle.

36. An atom in a metastable state has a lifetime of 5.2 ms. Find the minimum uncertainty in the measurement of energy of the excited state.

37. Measurements indicate that an atom remains in an excited state for an average time of 50.0 ns before making a transition to the ground state with the simultaneous emission of a 2.1-eV photon.

(a) Estimate the uncertainty in the frequency of the photon.

(b) What fraction of the photon’s average frequency is this?

38. Suppose an electron is confined to a region of length 0.1 nm (of the order of the size of a hydrogen atom).

(a) What is the minimum uncertainty of its momentum?

(b) What would the uncertainty in momentum be if the confined length region doubled to 0.2 nm?

39. Combine Equation \(7.4.1\) and Equation \(7.4.2\) to show \(\displaystyle k^2=\frac{ω^2}{c^2}\).

40. Show that \(\displaystyle Ψ(x,t)=Ae^{i(kx−ωt)}\) is a valid solution to Schrӧdinger’s time-dependent equation.

41. Show that \(\displaystyle Ψ(x,t)=Asin(kx−ωt)\) and \(\displaystyle Ψ(x,t)=Acos(kx−ωt)\) do not obey Schrӧdinger’s time-dependent equation.

42. Show that when \(\displaystyle Ψ_1(x,t)\) and \(\displaystyle Ψ_2(x,t)\) are solutions to the time-dependent Schrӧdinger equation and A,B are numbers, then a function \(\displaystyle Ψ(x,t)\) that is a superposition of these functions is also a solution: \(\displaystyle Ψ(x,t)=AΨ_1(x,t)+BΨ_1(x,t)\).

43. A particle with mass m is described by the following wave function: \(\displaystyle ψ(x)=Acoskx+Bsinkx\), where A, B, and k are constants. Assuming that the particle is free, show that this function is the solution of the stationary Schrӧdinger equation for this particle and find the energy that the particle has in this state.

44. Find the expectation value of the kinetic energy for the particle in the state, \(\displaystyle Ψ(x,t)=Ae^{i(kx−ωt)}\). What conclusion can you draw from your solution?

45. Find the expectation value of the square of the momentum squared for the particle in the state, \(\displaystyle Ψ(x,t)=Ae^{i(kx−ωt)}\). What conclusion can you draw from your solution?

46. A free proton has a wave function given by \(\displaystyle Ψ(x,t)=Ae^{i(5.02×10^{11}x−8.00×10^{15}t)}\). The coefficient of x is inverse meters (\(\displaystyle m^{−1}\)) and the coefficient on t is inverse seconds (\(\displaystyle s^{−1}\)). Find its momentum and energy.

47. Assume that an electron in an atom can be treated as if it were confined to a box of width \(\displaystyle 2.0Å\). What is the ground state energy of the electron? Compare your result to the ground state kinetic energy of the hydrogen atom in the Bohr’s model of the hydrogen atom.

48. Assume that a proton in a nucleus can be treated as if it were confined to a one-dimensional box of width 10.0 fm.

(a) What are the energies of the proton when it is in the states corresponding to \(\displaystyle n=1, n=2,\) and \(\displaystyle n=3\)?

(b) What are the energies of the photons emitted when the proton makes the transitions from the first and second excited states to the ground state?

49. An electron confined to a box has the ground state energy of 2.5 eV. What is the width of the box?

50. What is the ground state energy (in eV) of a proton confined to a one-dimensional box the size of the uranium nucleus that has a radius of approximately 15.0 fm?

51. What is the ground state energy (in eV) of an αα-particle confined to a one-dimensional box the size of the uranium nucleus that has a radius of approximately 15.0 fm?

52. To excite an electron in a one-dimensional box from its first excited state to its third excited state requires 20.0 eV. What is the width of the box?

53. An electron confined to a box of width 0.15 nm by infinite potential energy barriers emits a photon when it makes a transition from the first excited state to the ground state. Find the wavelength of the emitted photon.

54. If the energy of the first excited state of the electron in the box is 25.0 eV, what is the width of the box?

55. Suppose an electron confined to a box emits photons. The longest wavelength that is registered is 500.0 nm. What is the width of the box?

56. Hydrogen \(\displaystyle H_2\) molecules are kept at 300.0 K in a cubical container with a side length of 20.0 cm. Assume that you can treat the molecules as though they were moving in a one-dimensional box.

(a) Find the ground state energy of the hydrogen molecule in the container.

(b) Assume that the molecule has a thermal energy given by \(\displaystyle k_BT/2\) and find the corresponding quantum number n of the quantum state that would correspond to this thermal energy.

57. An electron is confined to a box of width 0.25 nm.

(a) Draw an energy-level diagram representing the first five states of the electron.

(b) Calculate the wavelengths of the emitted photons when the electron makes transitions between the fourth and the second excited states, between the second excited state and the ground state, and between the third and the second excited states.

58. An electron in a box is in the ground state with energy 2.0 eV.

(a) Find the width of the box.

(b) How much energy is needed to excite the electron to its first excited state?

(c) If the electron makes a transition from an excited state to the ground state with the simultaneous emission of 30.0-eV photon, find the quantum number of the excited state?

59.  Show that the two lowest energy states of the simple harmonic oscillator, \( ψ_0(x) \) and \( ψ_1(x) \) from \[\psi_n (x) = N_n e^{-\beta^2 x^2/2} H_n (\beta x) \nonumber \] with \(n = 0,1,2,3, ...\) satisfy the relatant time-independent Schrӧdinger equation \[-\dfrac{\hbar}{2m} \dfrac{d^2 \psi(x)}{dx^2} + \dfrac{1}{2}m\omega^2 x^2 \psi(x) = E\psi (x). \nonumber \]

60. If the ground state energy of a simple harmonic oscillator is 1.25 eV, what is the frequency of its motion?

61. When a quantum harmonic oscillator makes a transition from the \(\displaystyle (n+1)\) state to the n state and emits a 450-nm photon, what is its frequency?

62. Vibrations of the hydrogen molecule \(\displaystyle H_2\) can be modeled as a simple harmonic oscillator with the spring constant \(\displaystyle k=1.13×10^3N/m\) and mass \(\displaystyle m=1.67×10^{−27}kg\).

(a) What is the vibrational frequency of this molecule?

(b) What are the energy and the wavelength of the emitted photon when the molecule makes transition between its third and second excited states?

63. A particle with mass 0.030 kg oscillates back-and-forth on a spring with frequency 4.0 Hz. At the equilibrium position, it has a speed of 0.60 m/s. If the particle is in a state of definite energy, find its energy quantum number.

64. Find the expectation value ⟨\(\displaystyle x^2\)⟩ of the square of the position for a quantum harmonic oscillator in the ground state. Note: \(\displaystyle ∫^{+∞}_{−∞}dxx^2e ^{−ax^2}=\sqrt{π}(2a^{3/2})^{−1}\).

65. Determine the expectation value of the potential energy for a quantum harmonic oscillator in the ground state. Use this to calculate the expectation value of the kinetic energy.

66. Verify that \(\displaystyle ψ_1(x)\) given by Equation 7.57 is a solution of Schrӧdinger’s equation for the quantum harmonic oscillator.

67. Estimate the ground state energy of the quantum harmonic oscillator by Heisenberg’s uncertainty principle. Start by assuming that the product of the uncertainties \(\displaystyle Δx\) and \(\displaystyle Δp\) is at its minimum. Write \(\displaystyle Δp\) in terms of \(\displaystyle Δx\) and assume that for the ground state \(\displaystyle x≈Δx\) and \(\displaystyle p≈Δp\), then write the ground state energy in terms of x . Finally, find the value of x that minimizes the energy and find the minimum of the energy.

68. A mass of 0.250 kg oscillates on a spring with the force constant 110 N/m. Calculate the ground energy level and the separation between the adjacent energy levels. Express the results in joules and in electron-volts. Are quantum effects important?

69. Show that the wave function in

(a) Equation 7.68 satisfies Equation 7.61, and

(b) Equation 7.69 satisfies Equation 7.63.

70. A 6.0-eV electron impacts on a barrier with height 11.0 eV. Find the probability of the electron to tunnel through the barrier if the barrier width is

(a) 0.80 nm and

(b) 0.40 nm.

71. A 5.0-eV electron impacts on a barrier of with 0.60 nm. Find the probability of the electron to tunnel through the barrier if the barrier height is

(a) 7.0 eV;

(b) 9.0 eV; and

(c) 13.0 eV.

72. A 12.0-eV electron encounters a barrier of height 15.0 eV. If the probability of the electron tunneling through the barrier is 2.5 %, find its width.

73. A quantum particle with initial kinetic energy 32.0 eV encounters a square barrier with height 41.0 eV and width 0.25 nm. Find probability that the particle tunnels through this barrier if the particle is

(a) an electron and,

(b) a proton.

74. A simple model of a radioactive nuclear decay assumes that \(\displaystyle α\)-particles are trapped inside a well of nuclear potential that walls are the barriers of a finite width 2.0 fm and height 30.0 MeV. Find the tunneling probability across the potential barrier of the wall for αα-particles having kinetic energy

(a) 29.0 MeV and

(b) 20.0 MeV. The mass of the \(\displaystyle α\)-particle is \(\displaystyle m=6.64×10^{−27}kg\).

75. A muon, a quantum particle with a mass approximately 200 times that of an electron, is incident on a potential barrier of height 10.0 eV. The kinetic energy of the impacting muon is 5.5 eV and only about 0.10% of the squared amplitude of its incoming wave function filters through the barrier. What is the barrier’s width?

76. A grain of sand with mass 1.0 mg and kinetic energy 1.0 J is incident on a potential energy barrier with height 1.000001 J and width 2500 nm. How many grains of sand have to fall on this barrier before, on the average, one passes through?

Additional Problems

77. Show that if the uncertainty in the position of a particle is on the order of its de Broglie’s wavelength, then the uncertainty in its momentum is on the order of the value of its momentum.

78. The mass of a \(\displaystyle ρ\)-meson is measured to be \(\displaystyle 770MeV/c^2\) with an uncertainty of \(\displaystyle 100MeV/c^2\). Estimate the lifetime of this meson.

79. A particle of mass m is confined to a box of width L . If the particle is in the first excited state, what are the probabilities of finding the particle in a region of width 0.020 L around the given point x :

(a) \(\displaystyle x=0.25L\);

(b) \(\displaystyle x=0.40L\);

(c) \(\displaystyle x=0.75L\); and

(d) \(\displaystyle x=0.90L\).

80. A particle in a box [0; L ] is in the third excited state. What are its most probable positions?

81. A 0.20-kg billiard ball bounces back and forth without losing its energy between the cushions of a 1.5 m long table

(a) If the ball is in its ground state, how many years does it need to get from one cushion to the other? You may compare this time interval to the age of the universe.

(b) How much energy is required to make the ball go from its ground state to its first excited state? Compare it with the kinetic energy of the ball moving at 2.0 m/s.

82. Find the expectation value of the position squared when the particle in the box is in its third excited state and the length of the box is L .

83. Consider an infinite square well with wall boundaries \(\displaystyle x=0\) and \(\displaystyle x=L\). Show that the function \(\displaystyle ψ(x)=Asinkx\) is the solution to the stationary Schrӧdinger equation for the particle in a box only if \(\displaystyle k=\sqrt{2mE}/ℏ\). Explain why this is an acceptable wave function only if k is an integer multiple of \(\displaystyle π/L\).

84. Consider an infinite square well with wall boundaries \(\displaystyle x=0\) and \(\displaystyle x=L\). Explain why the function \(\displaystyle ψ(x)=Acoskx\) is not a solution to the stationary Schrӧdinger equation for the particle in a box.

85. Atoms in a crystal lattice vibrate in simple harmonic motion. Assuming a lattice atom has a mass of \(\displaystyle 9.4×10^{−26}kg\), what is the force constant of the lattice if a lattice atom makes a transition from the ground state to first excited state when it absorbs a \(\displaystyle 525-µm\) photon?

86. A diatomic molecule behaves like a quantum harmonic oscillator with the force constant 12.0 N/m and mass \(\displaystyle 5.60×10^{−26}kg\).

(a) What is the wavelength of the emitted photon when the molecule makes the transition from the third excited state to the second excited state?

(b) Find the ground state energy of vibrations for this diatomic molecule.

87. An electron with kinetic energy 2.0 MeV encounters a potential energy barrier of height 16.0 MeV and width 2.00 nm. What is the probability that the electron emerges on the other side of the barrier?

88. A beam of mono-energetic protons with energy 2.0 MeV falls on a potential energy barrier of height 20.0 MeV and of width 1.5 fm. What percentage of the beam is transmitted through the barrier?

Challenge Problems

89. An electron in a long, organic molecule used in a dye laser behaves approximately like a quantum particle in a box with width 4.18 nm. Find the emitted photon when the electron makes a transition from the first excited state to the ground state and from the second excited state to the first excited state.

90. In STM, an elevation of the tip above the surface being scanned can be determined with a great precision, because the tunneling-electron current between surface atoms and the atoms of the tip is extremely sensitive to the variation of the separation gap between them from point to point along the surface. Assuming that the tunneling-electron current is in direct proportion to the tunneling probability and that the tunneling probability is to a good approximation expressed by the exponential function \(\displaystyle e^{−2βL}\) with \(\displaystyle β=10.0/nm\), determine the ratio of the tunneling current when the tip is 0.500 nm above the surface to the current when the tip is 0.515 nm above the surface.

91. If STM is to detect surface features with local heights of about 0.00200 nm, what percent change in tunneling-electron current must the STM electronics be able to detect? Assume that the tunneling-electron current has characteristics given in the preceding problem.

92. Use Heisenberg’s uncertainty principle to estimate the ground state energy of a particle oscillating on an spring with angular frequency, \(\displaystyle ω=\sqrt{k/m}\), where k is the spring constant and m is the mass.

93. Suppose an infinite square well extends from \(\displaystyle −L/2\) to \(\displaystyle +L/2\). Solve the time-independent Schrӧdinger’s equation to find the allowed energies and stationary states of a particle with mass m that is confined to this well. Then show that these solutions can be obtained by making the coordinate transformation \(\displaystyle x'=x−L/2\) for the solutions obtained for the well extending between 0 and L .

94. A particle of mass m confined to a box of width L is in its first excited state \(\displaystyle ψ_2(x)\).

(a) Find its average position (which is the expectation value of the position).

(b) Where is the particle most likely to be found?

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Exploring Quantum Mechanics: A Collection of 700+ Solved Problems for Students, Lecturers, and Researchers

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Noah Graham; Exploring Quantum Mechanics: A Collection of 700+ Solved Problems for Students, Lecturers, and Researchers . Physics Today 1 June 2014; 67 (6): 51–52. https://doi.org/10.1063/PT.3.2423

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Exploring Quantum Mechanics: A Collection of 700+ Solved Problems for Students, Lecturers, and Researchers , Victor   Galitski , Boris   Karnakov , Vladimir   Kogan , and Victor   Galitski  Jr , Oxford U. Press , 2013. $165.00 (912 pp.). ISBN 978-0-19-923271-0

I like to start my upper-level undergraduate quantum mechanics course with a quote from physicist David Griffiths: “I do not believe one can intelligently discuss what quantum mechanics means until one has a firm sense of what quantum mechanics does .” Exploring Quantum Mechanics: A Collection of 700+ Solved Problems for Students, Lecturers, and Researchers by Victor Galitski, Boris Karnakov, Vladimir Kogan, and Victor Galitski Jr provides a wide range of opportunities to learn what quantum mechanics does through an impressive collection of solved problems.

The book originates from a smaller work assembled by Galitski and Kogan in the mid 1950s; that work was then expanded, 20 years later, by Galitski in collaboration with Karnakov. Alongside a highly productive research career, Galitski created the earlier work while struggling against political oppression under Joseph Stalin and the expanded version while fighting cancer of a more literal form. Three decades after the publication of the first Russian edition in 1981, his grandson, Galitski Jr, took on the long process of editing, translating, and expanding the problems. The result is a gem of old-world craftsmanship, well worth a place alongside the other classic texts of quantum mechanics in any physicist’s library.

The problems cover topics in quantum mechanics at considerable depth, across a wide range of difficulty and sophistication. Some are simple exercises, many are suitable for advanced undergraduates, and the majority are suitable for the graduate level or beyond. Although I can’t claim to have checked every problem and solution, the ones I sampled showed care and attention to detail. In a few cases I might have expanded on what the book provides in preparing a problem set or solution set for a class, but those problems tended to be the more straightforward ones; if such brevity allowed for more of the book’s 912 pages to be devoted to detailed discussions of the more complex and subtle problems, it’s a choice I wholeheartedly applaud.

Each chapter begins with a brief summary of key concepts and formulas, which serves as a useful reference for the subsequent problems and solutions. The presentation closely parallels a standard full-year graduate quantum mechanics course and provides a comprehensive range of problems for each topic. There is a particularly extensive selection of problems in atomic and nuclear physics, often connecting closely to experimental measurements. I was most impressed, however, by the sheer inventiveness and creativity required to formulate a wide range of problems that illuminate the many subtle facets of quantum mechanics but for which the calculations involved nonetheless remain tractable.

As Galitski Jr points out in the preface, this sort of thorough, detailed collection is a product of “people living and working in completely different times, and they were quite different from us, today’s scientists: with their attention spans undiminished by constant exposure to email, internet, and television, and with their minds free of petty worries about citation counts, indices, and rankings, they were able to devote 100% of their attention to science and take the time to focus on difficult problems that really mattered.”

Ironically, where today’s technology may have the most to offer to education is in managing large-scale collections of specialized knowledge of the sort found in this book. Although I’m doubtful about the value of the internet in replacing the teacher, I think it has a lot to offer toward upgrading the textbook, and I am frequently struck (especially while making up problem sets and exams) by how valuable it would be to assemble a large-scale online database of carefully crafted problems and solutions. Instructors around the world could contribute those cherished problems each of us has developed, typographical errors could be eliminated by crowdsourcing, and problems could be efficiently indexed by difficulty level and subject. All it would take to get such a project off the ground would be to assemble an initial critical mass of solved problems. Perhaps a member of the next generation will come along to take up that challenge.

Noah Graham is an associate professor of physics at Middlebury College in Vermont. He regularly teaches upper-level undergraduate quantum mechanics and his research applies scattering theory and computational methods to calculations of Casimir forces and the stability of coherent field configurations in classical and quantum field theory.

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Exploring Quantum Mechanics: A Collection of 700+ Solved Problems for Students, Lecturers, and Researchers

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A series of seminal technological revolutions has led to a new generation of electronic devices miniaturized to such tiny scales that the strange laws of quantum physics come into play. There is no doubt that, unlike scientists and engineers of the past, technology leaders of the future will have to rely on quantum mechanics in their everyday work. This makes teaching and learning the subject of paramount importance for further progress. Mastering quantum physics is a very non-trivial task and its deep understanding can only be achieved through working out real-life problems and examples. It is notoriously difficult to come up with new quantum-mechanical problems that would be solvable with a pencil and paper, and within a finite amount of time. This book presents some 700+ original problems in quantum mechanics together with detailed solutions, covering nearly 1,000 pages on all aspects of quantum science.

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Home > Books > Quantum Mechanics

Exactly Solvable Problems in Quantum Mechanics

Submitted: 29 April 2020 Reviewed: 03 July 2020 Published: 18 August 2020

DOI: 10.5772/intechopen.93317

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Some of the problems in quantum mechanics can be exactly solved without any approximation. Some of the exactly solvable problems are discussed in this chapter. Broadly there are two main approaches to solve such problems. They are (i) based on the solution of the Schrödinger equation and (ii) based on operators. The normalized eigen function, eigen values, and the physical significance of some of the selected problems are discussed.

• exactly solvable
• Schrödinger equation
• eigen function
• eigen values

Author Information

Lourdhu bruno chandrasekar *.

• Department of Physics, Periyar Maniammai Institute of Science and Technology, Vallam, India

Kanagasabapathi Gnanasekar

• Department of Physics, The American College, India

Marimuthu Karunakaran

• Department of Physics, Alagappa Government Arts College, India

*Address all correspondence to: [email protected]

1. Potential well

The potential well is the region where the particle is confined in a small region. In general, the potential of the confined region is lower than the surroundings ( Figure 1 ) [ 1 , 2 ].

Infinite potential well.

The potential of the system is defined as

The one dimensional Schrödinger equation in Cartesian coordinate is given as

In the infinite potential well, the confined particle is present in the well region (Region-II) for an infinitely long time. So the solution of the Schrödinger equation in the Region-II and Region-III can be omitted for our discussion right now. The Schrödinger equation in the Region-II is written as

The solution of the Eq. (2) is

At x = − L , and at x = L , the wave function vanishes since the potential is infinite. Hence, At x = − L ,

Similarly, at x = L

i. 2 A 2 cos αL = 0 ⟹ cos αL = 0 ⟹ αL = nπ / 2 ⟹ α 2 = n 2 π 2 / 4 L 2 ; n = 1 , 3 , 5 , … … . Since α 2 = 2 mE ℏ 2 , 2 mE ℏ 2 = n 2 π 2 / 4 L 2 , the energy eigen value is found as

The eigen function is Ψ = A 1 cos αx

According to the normalization condition,

Hence the normalized eigen function for n = 1 , 3 , 5 , … … is

ii. 2 A 1 sin αL = 0 ⟹ sin αL = 0 ⟹ αL = nπ / 2 ⟹ α 2 = n 2 π 2 / 4 L 2 ; n = 2 , 4 , 6 , … … . For this case, n = 2 , 4 , 6 , … … , the corresponding energy eigen value is

The eigen function is Ψ = A 2 cos αx and the normalized eigen function is

In Summary, the eigen value is E = n 2 π 2 ℏ 2 / 8 m L 2 for all positive integer values of “n.” The normalized eigen functions are

The minimum energy state can be calculated by setting n = 1 , which corresponds to the ground state. The ground state energy is

The energy difference between the successive states is simply the difference between the energy eigen value of the corresponding state. For example, ∆ E 12 = E 1 ∼ E 2 = 3 E 1 and ∆ E 23 = E 2 ∼ E 3 = 5 E 1 . Hence the energy difference between any two successive states is not the same.

Though the eigen functions for odd and even values of “n” are different, the energy eigen value remains the same.

If the potential well is chosen in the limit 0 < x < 2 L (width of the well is 2 L ), the energy eigen value is the same as given in Eqs.(6) and (8) . But if the limit is chosen as 0 < x < L (width of the well is L ), the for all positive integers of “n,” the eigen function is Ψ = 2 / L 1 / 2 sin nπx / L and the energy eigen function is E = n 2 π 2 ℏ 2 / 2 m L 2 .

2. Step potential

Step potential is a problem that has two different finite potentials [ 3 ]. Classically, the tunneling probability is 1 when the energy of the particle is greater than the height of the barrier. But the result is not true based on wave mechanics ( Figure 2 ).

Step potential.

The potential of the system

The Schrödinger equation in the Region-I and Region-II is given, respectively as,

Case (i): when E < V 0 , the solutions of the Schrödinger equations in the Region-I and Region-II, respectively, are given as

where α 2 = 2 mE ℏ 2 and β 2 = 2 m E − V 0 ℏ 2 . Here, B 2 exp βx represents the exponentially increasing wave along the x-direction. The wave function Ψ 2 must be finite as x → ∞ . This is possible only by setting B 2 = 0 . Hence the eigen function in the Region-II is

According to admissibility conditions of wave functions, at x = 0 , Ψ 1 = Ψ 2 and Ψ 1 ′ = Ψ 2 ′ . It gives us

From these two equations,

The reflection coefficient R is given as

It is interesting to note that all the particles that encounter the step potential are reflected back. This is due to the fact that the width of the step potential is infinite. The number of particles in the process is conserved, which leads that T = 0 , since T + R = 1 .

Case (ii): when E > V 0 , the solutions are given as

where β 2 = 2 m E − V 0 ℏ 2 . As x → ∞ , the wave function Ψ 2 must be finite. Hence

Ψ 2 = A 2 exp iβx by setting B 2 = 0 . According to the boundary conditions at x = 0 ,

From these equations,

The reflection coefficient R and the transmission coefficient T, respectively, are given as

From these easily one can show that

The results again indicate that the total number of particles which encounters the step potential is conserved.

3. Potential barrier

This problem clearly explains the wave-mechanical tunneling [ 3 , 4 ]. The potential of the system is given as ( Figure 3 )

Potential barrier.

In the Region-I, the Schrödinger equation is Ψ ′ ′ + α 2 Ψ = 0 . The wave function in this region is given as

In Region-II, if E < V 0 , the Schrödinger equation is Ψ ′ ′ − β 2 Ψ = 0 . The solution of the equation is given as

The Schrödinger equation in the Region-III is Ψ ′ ′ + α 2 Ψ = 0 . The corresponding solution is Ψ 3 = A 3 exp iαx + B 3 exp − iαx . But in the Region-III, the waves can travel only along positive x-direction and there is no particle coming from the right, B 3 = 0 . Hence

At x = 0 , Ψ 1 = Ψ 2 and Ψ 1 ′ = Ψ 2 ′ . These give us two equations

At x = L , Ψ 2 = Ψ 3 and Ψ 2 ′ = Ψ 3 ′ . These conditions give us another two equations

Solving the equations from (27) to (30) , one can find the coefficients in the equations. The reflection coefficient is R is found as

The transmission coefficient T is found as

When E < V 0 , though the energy of the incident particles is less than the height of the barrier, the particle can tunnel into the barrier region. This is in contrast to the laws of classical physics. This is known as the tunnel effect.

As V 0 → ∞ , the transmission coefficient is zero. Hence the tunneling is not possible only when V 0 → ∞ .

When the length of the barrier is an integral multiple of π / β , there is no reflection from the barrier. This is termed as resonance scattering.

The tunneling probability depends on the height and width of the barrier.

Later, Kronig and Penney extended this idea to explain the motion of a charge carrier in a periodic potential which is nothing but the one-dimensional lattices.

4. Delta potential

The Dirac delta potential is infinitesimally narrow potential only at some point (generally at the origin, for convenience) [ 3 ]. The potential of the system

Here λ is the positive constant, which is the strength of the delta potential. Here, we confine ourselves only to the bound states, hence E < 0 ( Figure 4 ).

Dirac delta potential.

The Schrödinger equation is

The solution of the Schrödinger equation is given as

where β 2 = − 2 mE ℏ 2 . At x = 0 , Ψ 1 = Ψ 2 . So the coefficients A 1 and A 2 are equal. But Ψ 1 ′ ≠ Ψ 2 ′ , since the first derivative causes the discontinuity. The first derivatives are related by the following equation

This gives us

Equating the value of β gives the energy eigen value as

The energy eigen value expression does not have any integer like in the case of the potential well. Hence there is only one bound state which is available for a particular value of “m.”

The eigen function can be evaluated as follows: The eigen function is always continuous. At x = 0 gives us A 1 = A 2 = A . Hence the eigen function is

To normalize Ψ ,

This gives us A = β = mλ ℏ .

5. Linear harmonic oscillator

Simple harmonic oscillator, damped harmonic oscillator, and force harmonic oscillator are the few famous problems in classical physics. But if one looks into the atomic world, the atoms are vibrating even at 0 K. Such atomic oscillations need the tool of quantum physics to understand its nature. In all the previous examples, the potential is constant in any particular region. But in this case, the potential is a function of the position coordinate “x.”

5.1 Schrodinger method

The potential of the linear harmonic oscillator as a function of “x” is given as ( Figure 5 ) [ 4 , 5 , 6 ]:

Potential energy of the linear harmonic oscillator.

The time-independent Schrödinger equation is given as

The potential is not constant since it is a function of “x”; Eq. (40) cannot solve directly as the previous problems. Let

Using the new constant β and the variable α , the Schrödinger equation has the form

The asymptotic Schrödinger equation α → ∞ is given as

The general solution of the equation is exp ± a 2 / 2 . As α → ∞ , exp + a 2 / 2 becomes infinite, hence it cannot be a solution. So the only possible solution is exp − a 2 / 2 . Based on the asymptotic solution, the general solution of Eq. (42) is given as

The normalized eigen function is

The solution given in Eq. (43) is valid if the condition 2 n + 1 − 2 E ℏ ω = 0 holds. This gives the energy eigen value as

The integer n = 0 represents the ground state, n = 1 represents the first excited state, and so on. The ground state energy of the linear harmonic oscillator is E = ℏ ω / 2 . This minimum energy is known as ground state energy.

The ground state normalized eigen function is

The energy difference between any two successive levels is ℏ ω . Hence the energy difference between any two successive levels is constant. But this is not true in the case of real oscillators.

5.2 Operator method

The operator method is also one of the convenient methods to solve the exactly solvable problem as well as approximation methods in quantum mechanics [ 5 ]. The Hamiltonian of the linear harmonic oscillator is given as,

Let us define the operator “a,” lowering operator, in such a way that

and the corresponding Hermitian adjoint, raising operator, is

Here, x p represents the commutation between the operators x and p . x p = iℏ and Eq. (49) becomes

In the same way, one can find the a a + and it is given as

Adding Eqs. (50) and (51) gives us the Hamiltonian in terms of the operators.

Subtracting Eq. (50) from (51) gives, a a + − a + a = 1 . This can be simplified as

The Hamiltonian H acts on any state ∣ n that gives the eigen value E n times the same state ∣ n , that is, H ∣ n = E n ∣ n .

The expectation value of a + a is

Let us consider the ground state ∣ 0 .

Since a ∣ 0 = 0 , 0 a + a 0 = 0 . Thus,

Similarly, the energy of the first excited state is found as follows:

In the same way, E 2 = 5 ℏ ω / 2 , E 3 = 7 ℏ ω / 2 , and so on. Hence, one can generalize the result as

The uncertainties in position and momentum, respectively, are given as

In order to evaluate the uncertainties x 2 , x 2 , p 2 , and p 2 have to be evaluated. From Eqs. (47) and (48) the position and momentum operators are found as

a. The expectation value of ‘x’ is given as,

b. The expectation value of momentum is

From Eq. (58) and (59) , the uncertainty in position and momentum, respectively are given as,

6. Conclusions

The minimum uncertainty state is the ground state. In this state, ∆ x = ℏ 2 mω 1 / 2 and ∆ p = mωℏ 2 1 / 2 .

Hence the minimum uncertainty product is ∆ x . ∆ p = ℏ 2 . Since the other states have higher uncertainty than the ground state, the general uncertainty is ∆ x . ∆ p ≥ ℏ 2 . This is the mathematical representation of Heisenberg’s uncertainty relation.

Since Ψ 0 x corresponds to the low energy state, a Ψ 0 x = 0 . This gives us the ground state eigen function. This can be done as follows:

Integrating the above equation gives,

The normalized eigen function is given as

The other eigen states can be evaluated using the equation, Ψ n x = a + n / n ! Ψ 0 x .

7. Particle in a 3D box

The confinement of a particle in a three-dimensional potential is discussed in this section [ 4 , 6 ]. The potential is defined as ( Figure 6 )

Three-dimensional potential box.

The three dimensional time-independent Schrödinger equation is given as

Let the eigen function Ψ x y z is taken as the product of Ψ x x , Ψ y y and Ψ z z according to the technique of separation of variables. i.e., Ψ x y z = Ψ x x Ψ y y Ψ z z .

Divide the above equation by Ψ x y z gives us

Now we can boldly write E as E x x + E y y + E z z

Now the equation can be separated as follows:

The normalized eigen function Ψ x x is given as

In the same way, Ψ y y and Ψ z z are given as

Hence, the eigen function Ψ x y z is given as

The energy given values are given as

The total energy E is

In a cubical potential box, a = b = c , then the energy eigen value becomes,

The minimum energy that corresponds to the ground state is E 1 = 3 π 2 ℏ 2 2 m a 2 . Here n x = n y = n z = 1 .

Different states with different quantum numbers may have the same energy. This phenomenon is known as degeneracy. For example, the states (i) n x = 2 ; n y = n z = 1 , (ii) n y = 2 ; n x = n z = 1 ; and (iii) n z = 2 ; n x = n y = 1 have the same energy of E = 6 π 2 ℏ 2 m a 2 . So we can say that the energy 6 π 2 ℏ 2 m a 2 has a 3-fold degenerate.

The states (111), (222), (333), (444),…. has no degeneracy.

In this problem, the state may have zero-fold degeneracy, 3-fold degeneracy or 6-fold degeneracy.

• 1. Griffiths DJ. Introduction to Quantum Mechanics. 2nd ed. India: Pearson
• 2. Singh K, Singh SP. Elements of Quantum Mechanics. 1st ed. India: S. Chand & Company Ltd
• 3. Gasiorowicz S. Quantum Mechanics. 3rd ed. India: Wiley
• 4. Schiff LI. Quantum Mechanics. 4th ed. India: McGraw Hill International Editions
• 5. Peleg Y, Pnini R, Zaarur E, Hecht E. Quantum Mechanics. 2nd ed. India: McGraw Hill Editions
• 6. Aruldhas G. Quantum Mechanics. 2nd ed. India: Prentice-Hall

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Quantum Mechanics: Problems with solutions

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Essential Advanced Physics is a series comprising four parts: Classical Mechanics , Classical Electrodynamics , Quantum Mechanics and Statistical Mechanics . Each part consists of two volumes, Lecture Notes and Problems with Solutions, further supplemented by an additional collection of test problems and solutions available to qualifying university instructors. This volume, Quantum Mechanics: Problems with Solutions contains detailed model solutions to the exercise problems formulated in the companion Lecture Notes volume. In many cases, the solutions include result discussions that enhance the lecture material. For the reader's convenience, the problem assignments are reproduced in this volume.

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Introduction.

Konstantin K Likharev

1D wave mechanics

Higher dimensionality effects, bra–ket formalism, some exactly solvable problems, perturbative approaches, open quantum systems, multiparticle systems, elements of relativistic quantum mechanics, back matter, d o i.

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Konstantin K Likharev received his PhD from the Lomonosov Moscow State University, USSR in 1969, and a habilitation degree of Doctor of Sciences from USSR's High Attestation Committee in 1979. From 1969 to 1990 Dr Likharev was a Staff Scientist of Moscow State University. In 1991 he assumed a Professorship at Stony Brook University (Distinguished Professor since 2002, John S. Toll Professor since 2017). During his research career, Dr Likharev worked in the fields of nonlinear classical and dissipative quantum dynamics, and solid-state physics and electronics, notably including superconductor electronics and nanoelectronics—most recently, with applications to neuromorphic networks. Dr Likharev has authored more than 250 original publications, over 80 review papers and book chapters, two monographs and several patents. Dr Likharev is a Fellow of the APS and IEEE.

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Solved Problems in Quantum Mechanics (UNITEXT for Physics) 1st ed. 2019 Edition

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This book presents a large collection of problems in Quantum Mechanics that are solvable within a limited time and using simple mathematics. The problems test both the student’s understanding of each topic and their ability to apply this understanding concretely.

Solutions to the problems are provided in detail, eliminating only the simplest steps. No problem has been included that requires knowledge of mathematical methods not covered in standard courses, such as Fuchsian differential equations.

The book is in particular designed to assist all students who are preparing for written examinations in Quantum Mechanics, but will also be very useful for teachers who have to pose problems to their students in lessons and examinations.

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Leonardo Angelini is an Associate Professor in Theoretical Physics at the University of Bari, Italy. He is author of about 60 publications in international journals in various research fields: approximation methods in quantum mechanics and field theory, modeling in elementary particle physics, phase transitions in statistical mechanics, non-equilibrium phenomena in extensive chaotic systems, and algorithms for data analysis with applications to elementary particle physics and biomedical signals. He has taught a variety of courses in Theoretical Physics, including Field Theory, Statistical Mechanics, Quantum Mechanics, Advanced Mathematical Methods of Physics, Simulation Techniques, and Computational Physics Laboratory. He currently teaches Quantum Mechanics for the bachelor's degree in Physics.

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May 22, 2023

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Quantum Theory’s ‘Measurement Problem’ May Be a Poison Pill for Objective Reality

Solving a notorious quantum quandary could require abandoning some of science’s most cherished assumptions about the physical world

By Anil Ananthaswamy

A core mystery of quantum physics hints that objective reality is illusory—or that the quantum world is even weirder than expected.

Yaroslav Kushta/Getty Images

Imagine a physicist observing a quantum system whose behavior is akin to a coin toss: it could come up heads or tails. They perform the quantum coin toss and see heads. Could they be certain that their result was an objective, absolute and indisputable fact about the world? If the coin was simply the kind we see in our everyday experience, then the outcome of the toss would be the same for everyone: heads all around! But as with most things in quantum physics, the result of a quantum coin toss would be a much more complicated “It depends.” There are theoretically plausible scenarios in which another observer might find that the result of our physicist’s coin toss was tails.

At the heart of this bizarreness is what’s called the measurement problem. Standard quantum mechanics accounts for what happens when you measure a quantum system : essentially, the measurement causes the system’s multiple possible states to randomly “collapse” into one definite state. But this accounting doesn’t define what constitutes a measurement —hence, the measurement problem.

Attempts to avoid the measurement problem—for example, by envisaging a reality in which quantum states don’t collapse at all—have led physicists into strange terrain where measurement outcomes can be subjective. “One major aspect of the measurement problem is this idea ... that observed events are not absolute,” says Nicholas Ormrod of the University of Oxford. This, in short, is why our imagined quantum coin toss could conceivably be heads from one perspective and tails from another.

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But is such an apparently problematic scenario physically plausible or merely an artifact of our incomplete understanding of the quantum world? Grappling with such questions requires a better understanding of theories in which the measurement problem can arise—which is exactly what Ormrod, along with Vilasini Venkatesh of the Swiss Federal Institute of Technology in Zurich and Jonathan Barrett of Oxford, have now achieved. In a recent preprint , the trio proved a theorem that shows why certain theories—such as quantum mechanics—have a measurement problem in the first place and how one might develop alternative theories to sidestep it, thus preserving the “absoluteness” of any observed event. Such theories would, for instance, banish the possibility of a coin toss coming up heads to one observer and tails to another.

But their work also shows that preserving such absoluteness comes at a cost many physicists would deem prohibitive. “It’s a demonstration that there is no pain-free solution to this problem,” Ormrod says. “If we ever can recover absoluteness, then we’re going to have to give up on some physical principle that we really care about.”

Ormrod, Venkatesh and Barrett’s paper “addresses the question of which classes of theories are incompatible with absoluteness of observed events—and whether absoluteness can be maintained in some theories, together with other desirable properties,” says Eric Cavalcanti of Griffith University in Australia. (Cavalcanti, along with physicist Howard Wiseman and their colleagues, defined the term “absoluteness of observed events” in prior work that laid some of the foundations for Ormrod, Venkatesh and Barrett’s study.)

Holding on to absoluteness of observed events, it turns out, could mean that the quantum world is even weirder than we know it to be .

The Heart of the Problem

Gaining a sense of what exactly Ormrod, Venkatesh and Barrett have achieved requires a crash course in the basic arcana of quantum foundations. Let’s start by considering our hypothetical quantum system that can, when observed, come up either heads or tails.

In textbook quantum theory, before collapse, the system is said to be in a superposition of two states, and this quantum state is described by a mathematical construct called a wave function, which evolves in time and space. This evolution is both deterministic and reversible: given an initial wave function, one can predict what it’ll be at some future time, and one can in principle run the evolution backward to recover the prior state. Measuring the wave function, however, causes it to collapse, mathematically speaking, such that the system in our example shows up as either heads or tails.

This collapse-inducing process is the murky source of the measurement problem: it’s an irreversible, one-time-only affair—and no one even knows what defines the process or boundaries of measurement. What amounts to a “measurement” or, for that matter, an “observer”? Do either of these things have physical constraints, such as minimal or maximal sizes? And must they, too, be subject to various slippery quantum effects, or can they be somehow considered immune from such complications? None of these questions have easy, agreed-upon answers—but theorists have no shortage of proffered solutions.

Given the example system, one model that preserves the absoluteness of the observed event—meaning that it’s either heads or tails for all observers—is the Ghirardi-Rimini-Weber theory (GRW). In GRW, quantum systems can exist in a superposition of states until they reach some as-yet-underdetermined size, at which point the superposition spontaneously and randomly collapses, independent of an observer. Whatever the outcome—heads or tails in our example—it shall hold for all observers.

But GRW, which belongs to a broader class of “spontaneous collapse” theories, seemingly runs afoul of a long-cherished physical principle: the preservation of information. Just as a burned book could, in principle, be read by reassembling its pages from its ashes (ignoring the burning book’s initial emission of thermal radiation, for simplicity’s sake), preservation of information implies that a quantum system’s evolution through time should allow its antecedent states to be known. By postulating a random collapse, GRW theory destroys the possibility of knowing what led up to the collapsed state—which, by most accounts, means information about the system prior to its transformation becomes irrecoverably lost. “[GRW] would be a model that gives up information preservation, thereby preserving absoluteness of events,” Venkatesh says.

A counterexample that allows for nonabsoluteness of observed events is the “many worlds” interpretation of quantum mechanics. In this view, our example wave function will branch into multiple contemporaneous realities, such that in one “world,” the system will come up heads, while in another, it’ll be tails. In this conception, there is no collapse. “So the question of what happens is not absolute; it’s relative to a world,” Ormrod says. Of course, in trying to avoid the collapse-induced measurement problem, the many worlds interpretation introduces the mind-numbing branching of wave functions and runaway proliferation of worlds at each and every fork in the quantum road—an unpalatable scenario for many.

Nevertheless, the many worlds interpretation is an example of what are called perspectival theories, wherein the outcome of a measurement depends on the observer’s perspective.

Crucial Aspects of Reality

To prove their theorem without getting mired in any particular theory or interpretation, quantum mechanical or otherwise, Ormrod, Venkatesh and Barrett focused on perspectival theories that obey three important properties. Again, we need some fortitude to grasp the import of these properties and to appreciate the rather profound outcome of the researchers’ proof.

The first property is called Bell nonlocality (B). It was first identified in 1964 by physicist John Bell in an eponymous theorem and has been shown to be an undisputed empirical fact about our physical reality. Let’s say that Alice and Bob each have access to one of a pair of particles, which are described by a single state. Alice and Bob make individual measurements of their respective particles and do this for a number of similarly prepared pairs of particles. Alice chooses her type of measurement freely and independently of Bob, and vice versa. That Alice and Bob choose their measurement settings of their own free will is an important assumption. Then, when they eventually compare notes, the duo will find that their measurement outcomes are correlated in a manner that implies the states of the two particles are inseparable: knowing the state of one tells you about the state of the other. Theories that can explain such correlations are said to be Bell nonlocal.

The second property is the preservation of information (I). Quantum systems that show deterministic and reversible evolution satisfy this condition. But the requirement is more general. Imagine that you are wearing a green sweater today. In an information-preserving theory, it should still be possible, in principle, 10 years hence to retrieve the color of your sweater even if no one saw you wearing it. But “if the world is not information-preserving, then it might be that in 10 years’ time, there’s simply no way to find out what color jumper I was wearing,” Ormrod says.

The third is a property called local dynamics (L). Consider two events in two regions of spacetime. If there exists a frame of reference in which the two events appear simultaneous, then the regions of space are said to be “spacelike separated.” Local dynamics implies that the transformation of a system in one of these regions cannot causally affect the transformation of a system in the other region any faster than the speed of light, and vice versa, where a transformation is any operation that takes a set of input states and produces a set of output states. Each subsystem undergoes its own transformation, and so does the entire system as a whole. If the dynamics are local, the transformation of the full system can be decomposed into transformations of its individual parts: the dynamics are said to be separable. “The local dynamics [constraint] ensures that you are not somehow faking Bell [nonlocality],” Venkatesh says.

In quantum theory, transformations can be decomposed into their constituent parts. “So quantum theory is dynamically separable,” Ormrod says. In contrast, when two particles share a state that’s Bell nonlocal (that is, when two particles are entangled, per quantum theory), the state is said to be inseparable into the individual states of the two particles. If transformations behaved similarly, in that the global transformation could not be described in terms of the transformations of individual subsystems, then the whole system would be dynamically inseparable.

All the pieces are in place to understand the trio’s result. Ormrod, Venkatesh and Barrett’s work comes down to a sophisticated analysis of how such “BIL” theories (those satisfying all three aforementioned properties) handle a deceptively simple thought experiment. Imagine that Alice and Bob, each in their own lab, make a measurement on one of a pair of particles. Both Alice and Bob make one measurement each, and both do the exact same measurement. For example, they might both measure the spin of their particle in the up-down direction.

Viewing Alice and Bob and their labs from the outside are Charlie and Daniela, respectively. In principle, Charlie and Daniela should be able to measure the spin of the same particles, say, in the left-right direction. In an information-preserving theory, this should be possible.

Let’s take the specific example of what might happen in standard quantum theory. Charlie, for example, treats Alice, her lab and the measurement she makes as one system that is subject to deterministic, reversible evolution. Assuming that he has complete control of the overall system, Charlie can reverse the process such that the particle comes back to its original state (like a burned book being reconstituted from its ashes). Daniela does the same with Bob and his lab. Now Charlie and Daniela each make a different measurement on their respective particles in the left-right direction.

Using this scenario, the team proved that the predictions of any BIL theory for the measurement outcomes of the four observers contradict the absoluteness of observed events. In other words, “ all BIL theories have a measurement problem,” Ormrod says.

Choose Your Poison

This leaves physicists at an unpalatable impasse: either accept the nonabsoluteness of observed events or give up one of the assumptions of a BIL theory.

Venkatesh thinks that there’s something compelling about giving up absoluteness of observed events. After all, she says, physics successfully transitioned from a rigid Newtonian framework to a more nuanced and fluid Einsteinian description of reality. “We had to adjust some notions of what we thought was absolute. There was absolute space and time for Newton,” Venkatesh says. But in Albert Einstein’s conception of the universe, space and time are one, and this single spacetime isn’t something absolute but can warp in ways that don’t fit with Newtonian ways of thinking.

On the other hand, a perspectival theory that depends on observers creates its own problems. Most prominently, how can one do science within the confines of a theory where two observers cannot agree on the outcomes of measurements? “It’s not clear that science can work in the way [it’s] supposed to work if we’re not coming up with predictions for observed events that we take to be absolute,” Ormrod says.

So if one were to insist on absoluteness of observed events, then something has to give. It’s not going to be Bell nonlocality or preservation of information: the former is on solid empirical footing, and the latter is considered an important aspect of any theory of reality. The focus shifts to local dynamics—in particular, to dynamical separability.

Dynamical separability is “kind of an assumption of reductionism,” Ormrod says. “You can explain the big stuff in terms of these little pieces.”

Preserving the absoluteness of observed events could imply that such reductionism doesn’t hold: just like a Bell nonlocal state cannot be reduced to some constituent states, it may be that the dynamics of a system are similarly holistic, adding another kind of nonlocality to the universe. Importantly, giving it up doesn’t cause a theory to fall afoul of Einstein’s theories of relativity, much like physicists have argued that Bell nonlocality doesn’t require superluminal or nonlocal causal influences but merely nonseparable states.

“Perhaps the lesson of Bell is that the states of distant particles are inextricably linked, and the lesson of the new ... theorems is that their dynamics are, too,” Ormrod, Venkatesh and Barrett wrote in their paper.

“I like the idea of rejecting dynamical separability a lot, because if it works, then ... we get to have our cake and eat it, [too],” Ormrod says. “We get to continue to believe what we take to be the most fundamental things about the world: the fact that relativity theory is true, and information is preserved, and this kind of thing. But we also get to believe in absoluteness of observed events.”

Jeffrey Bub , a philosopher of physics and a professor emeritus at the University of Maryland, College Park, is willing to swallow some bitter pills if that means living in an objective universe. “I would want to hold on to the absoluteness of observed events,” he says. “It seems, to me, absurd to give this up just because of the measurement problem in quantum mechanics.” To that end, Bub thinks a universe in which dynamics are not separable is not such a bad idea. “I guess I would agree, tentatively, with the authors that [dynamical] nonseparability is the least unpalatable option,” he says.

The problem is that no one yet knows how to construct a theory that rejects dynamical separability—assuming it’s even possible to construct—while holding on to the other properties such as preservation of information and Bell nonlocality.

A More Profound Nonlocality

Griffith University’s Howard Wiseman, who is seen as a founding figure for such theoretical musings, appreciates Ormrod, Venkatesh and Barrett’s effort to prove a theorem that is applicable but not specific to quantum mechanics. “It’s nice that they are pushing in that direction,” he says. “We can say things more generally without referring to quantum mechanics at all.”

He points out that the thought experiment used in the analysis doesn’t require Alice, Bob, Charlie and Daniela to make any choices—they always make the same measurements. As a result, the assumptions used to prove the theorem don’t explicitly include an assumption about freedom of choice because no one is exercising such a choice. Normally, the fewer the assumptions, the stronger the proof, but that might not be the case here, Wiseman says. That’s because the first assumption—that the theory must accommodate Bell nonlocality—requires agents to have free will. Any empirical test of Bell nonlocality involves Alice and Bob choosing of their own free will the types of measurements they make. So if a theory is Bell nonlocal, it implicitly acknowledges the free will of the experimenters. “What I suspect is that they are sneaking in a free choice assumption,” Wiseman says.

This is not to say that the proof is weaker. Rather it would have been stronger if it had not required an assumption of free will. As it happens, free will remains a requirement. Given that, the most profound import of this theorem could be that the universe is nonlocal in an entirely new way. If so, such nonlocality would equal or rival Bell nonlocality, an understanding of which has paved the way for quantum communications and quantum cryptography. It’s anybody’s guess what a new kind of nonlocality—hinted at by dynamical nonseparability—would mean for our understanding of the universe.

In the end, only experiments will point the way toward the correct theory, and quantum physicists can only prepare themselves for any eventuality. “Irrespective of one’s personal view on which [theory] is a better one, all of them have to be explored,” Venkatesh says. “Ultimately, we’ll have to look at the experiments we can perform. It could be one way or the other, and it’s good to be prepared.”

• © 2012

Solved Problems in Quantum and Statistical Mechanics

• Michele Cini 0 ,
• Francesco Fucito 1 ,
• Mauro Sbragaglia 2

Department of Physics, University of Rome, Tor Vergata, Italy Laboratori Nazionali Frascati, INFN, Italy

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Table of contents (12 chapters)

Front matter, theoretical background, summary of quantum and statistical mechanics.

Michele Cini, Francesco Fucito, Mauro Sbragaglia

Quantum Mechanics — Problems

Formalism of quantum mechanics and one dimensional problems, angular momentum and spin, central force field, perturbation theory and wkb method, statistical mechanics — problems, thermodynamics and microcanonical ensemble, canonical ensemble, grand canonical ensemble, kinetic physics, bose-einstein gases, fermi-dirac gases, fluctuations and complements, back matter.

This textbook is the result of many years of teaching quantum and statistical mechanics, drawing on exercises and exam papers used on courses taught by the authors. The subjects of the exercises have been carefully selected to cover all the material which is most needed by students. 

Each exercise is carefully solved in full details, explaining the theory behind the solution with particular care for those issues that students often find difficult, or which are often neglected in other books on the subject. The exercises in this book never require extensive calculations  but tend to be somewhat unusual  and force the solver  to think about the problem starting from first principles, rather than by analogy with some previously solved exercise.

• Approximations
• Fluctuations
• Kinetic theory
• Quantum Mechanics

Department of Physics, University of Rome, Tor Vergata, Italy

Laboratori nazionali frascati, infn, italy.

Book Title : Solved Problems in Quantum and Statistical Mechanics

Authors : Michele Cini, Francesco Fucito, Mauro Sbragaglia

Series Title : UNITEXT

DOI : https://doi.org/10.1007/978-88-470-2315-4

Publisher : Springer Milano

eBook Packages : Physics and Astronomy , Physics and Astronomy (R0)

Copyright Information : Springer-Verlag Italia S.r.l., part of Springer Nature 2012

Softcover ISBN : 978-88-470-2314-7 Published: 30 January 2012

eBook ISBN : 978-88-470-2315-4 Published: 30 March 2012

Series ISSN : 2038-5714

Series E-ISSN : 2532-3318

Edition Number : 1

Number of Pages : VIII, 399

Topics : Quantum Physics , Thermodynamics , Complex Systems , Statistical Physics and Dynamical Systems

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Realizing clean qubits for quantum computers using electrons on helium

Future quantum computers could be based on electrons floating above liquid helium, according to study by a RIKEN physicist and collaborators, appearing in Physical Review Applied .

Today's computers are based on shuttling electrons around in silicon. Electrons in silicon could also form the basis of a completely different breed of computers—quantum computers. Numerous efforts are underway to realize quantum computers using electrons in various solid-state crystals, starting with silicon.

By exploiting the quantum nature of tiny objects, quantum computers promise to revolutionize computing by solving problems that are intractable using the most powerful supercomputers available today.

While efforts to create qubits using electrons in solid-state crystals have achieved significant success, increasing the number of qubits (the quantum equivalent of bits) is challenging because defects and impurities in solid-state crystals create unpredictable electrical potentials, making it difficult to produce many uniform qubits.

One way to overcome this problem would be to use electrons floating in a vacuum as qubits, since vacuum is defect free.

"Solid-state crystals will always have some defects, which means we cannot create a perfect environment for electrons," says Erika Kawakami of the RIKEN Center for Quantum Computing. "That is problematic if we want to create a lot of uniform qubits. And so it's better to have qubits in vacuum."

In 1999, researchers theoretically proposed to realize qubits based on electrons floating on liquid helium for the first time. In this physical system, electrons float in vacuum slightly above the surface of liquid helium. This was a groundbreaking proposal, but it was limited to basic operations of quantum gates because quantum-computer research was still in its infancy.

Now, in a theoretical study, the team has shown how the quantum gates can be realized more concretely using electrons floating above liquid helium .

Central to their proposal is a hybrid qubit involving the vertically quantized charge state and the spin state of a floating electron. The charge state of the electron allows it to be easily manipulated over moderate distances using an electric field , while the spin state can be used to stably store data. The interaction between the spin and charge states of the electron enables data to be transferred between the two electron properties.

"We've proposed how to realize one-qubit and two-qubit gates using electrons on helium and estimated their fidelities," says Kawakami. "We've also specified how we can scale up the number of qubits. That is something new."

Their system uses an array of tiny ferromagnetic pillars to trap electrons above helium. It should be possible to squeeze more than 10 million qubits into an area the size of a postage stamp.

The team now intends to embrace the challenge of implementing their proposal experimentally.

Journal information: Physical Review Applied , arXiv

Provided by RIKEN

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Unlocking the quantum future

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Quantum computing is the next frontier for faster and more powerful computing technologies. It has the potential to better optimize routes for shipping and delivery, speed up battery development for electric vehicles, and more accurately predict trends in financial markets. But to unlock the quantum future, scientists and engineers need to solve outstanding technical challenges while continuing to explore new applications.

One place where they’re working towards this future is the MIT Interdisciplinary Quantum Hackathon , or iQuHACK for short (pronounced “i-quack,” like a duck). Each year, a community of quhackers (quantum hackers) gathers at iQuHACK to work on quantum computing projects using real quantum computers and simulators. This year, the hackathon was held both in-person at MIT and online over three days in February.

Quhackers worked in teams to advance the capability of quantum computers and to investigate promising applications. Collectively, they tackled a wide range of projects, such as running a quantum-powered dating service, building an organ donor matching app, and breaking into quantum vaults. While working, quhackers could consult with scientists and engineers in attendance from sponsoring companies. Many sponsors also received feedback and ideas from quhackers to help improve their quantum platforms.

But organizing iQuHACK 2024 was no easy feat. Co-chairs Alessandro Buzzi and Daniela Zaidenberg led a committee of nine members to hold the largest iQuHACK yet. “It wouldn’t have been possible without them,” Buzzi said. The hackathon hosted 260 in-person quhackers and 1,000 remote quhackers, representing 77 countries in total. More than 20 scientists and engineers from sponsoring companies also attended in person as mentors for quhackers.

Each team of quhackers tackled one of 10 challenges posed by the hackathon’s eight major sponsoring companies. Some challenges asked quhackers to improve computing performance, such as by making quantum algorithms faster and more accurate. Other challenges asked quhackers to explore applying quantum computing to other fields, such as finance and machine learning. The sponsors worked with the iQuHACK committee to craft creative challenges with industry relevance and societal impact. “We wanted people to be able to address an interesting challenge [that has] applications in the real world,” says Zaidenberg.

One team of quhackers looked for potential quantum applications and found one close to home: dating. A team member, Liam Kronman, had previously built dating apps but disliked that matching algorithms for normal classical computers “require [an overly] strict setup.” With these classical algorithms, people must be split into two groups — for example, men and women — and matches can only be made between these groups. But with quantum computers, matching algorithms are more flexible and can consider all possible combinations, enabling the inclusion of multiple genders and gender preferences. 

Kronman and his team members leveraged these quantum algorithms to build a quantum-powered dating platform called MITqute (pronounced “meet cute”). To date, the platform has matched at least 240 people from the iQuHACK and MIT undergrad communities. In a follow-up survey, 13 out of 41 respondents reported having talked with their match, with at least two pairs setting up dates. “I really lucked out with this one,” one respondent wrote. 

Another team of quhackers also based their project on quantum matching algorithms but instead leveraged the algorithms’ power for medical care. The team built a mobile app that matches organ donors to patients, earning them the hackathon’s top social impact award. 

But they almost didn’t go through with their project. “At one point, we were considering scrapping the whole thing because we thought we couldn’t implement the algorithm,” says Alma Alex, one of the developers. After talking with their hackathon mentor for advice, though, the team learned that another group was working on a similar type of project — incidentally, the MITqute team. Knowing that others were tackling the same problem inspired them to persevere.

A sense of community also helped to motivate other quhackers. For one of the challenges , quhackers were tasked with hacking into 13 virtual quantum vaults. Teams could see each other’s progress on each vault in real time on a leaderboard, and this knowledge informed their strategies. When the first vault was successfully hacked by a team, progress from many other teams spiked on that vault and slowed down on others, says Daiwei Zhu, a quantum applications scientist at IonQ and one of the challenge’s two architects.

The vault challenge may appear to be just a fun series of puzzles, but the solutions can be used in quantum computers to improve their efficiency and accuracy. To hack into a vault, quhackers had to first figure out its secret key — an unknown quantum state — using a maximum of 20 probing tests. Then, they had to change the key’s state to a target state. These types of characterizations and modifications of quantum states are “fundamental” for quantum computers to work, says Jason Iaconis, a quantum applications engineer at IonQ and the challenge’s other architect. 

But the best way to characterize and modify states is not yet clear. “Some of the [vaults] we [didn’t] even know how to solve ourselves,” Zhu says. At the end of the hackathon, six vaults had at least one team mostly hack into them. (In the quantum world where gray areas exist, it’s possible to partly hack into a vault.)

The community of scientists and engineers formed at iQuHACK persists beyond the weekend, and many members continue to grow the community outside the hackathon. Inspired quhackers have gone on to start their own quantum computing clubs at their universities. A few years ago, a group of undergraduate quhackers from different universities formed a Quantum Coalition that now hosts their own quantum hackathons. “It’s crazy to see how the hackathon itself spreads and how many people start their own initiatives,” co-chair Zaidenberg says. 

The three-day hackathon opened with a keynote from MIT Professor Will Oliver, which included an overview of basic quantum computing concepts, current challenges, and computing technologies. Following that were industry talks and a panel of six industry and academic quantum experts, including MIT Professor Peter Shor, who is known for developing one of the most famous quantum algorithms. The panelists discussed current challenges, future applications, the importance of collaboration, and the need for ample testing.

Later, sponsors held technical workshops where quhackers could learn the nitty-gritty details of programming on specific quantum platforms. Day one closed out with a talk by research scientist Xinghui Yin on the role of quantum technology at LIGO, the Laser Interferometer Gravitational-Wave Observatory that first detected gravitational waves. The next day, the hackathon’s challenges were announced at 10 a.m., and hacking kicked off at the MIT InnovationHQ. In the afternoon, attendees could also tour MIT quantum computing labs.

Hacking continued overnight at the MIT Museum and ended back at MIT iHQ at 10 a.m. on the final day. Quhackers then presented their projects to panels of judges. Afterward, industry speakers gave lightning talks about each of their company’s latest quantum technologies and future directions. The hackathon ended with a closing ceremony, where sponsors announced the awards for each of the 10 challenges. 

The hackathon was captured in a three-part video by Albert Figurt , a resident artist at MIT. Figurt shot and edited the footage in parallel with the hackathon. Each part represented one day of the hackathon and was released on the subsequent day.

Throughout the weekend, quhackers and sponsors consistently praised the hackathon’s execution and atmosphere. “That was amazing … never felt so much better, one of the best hackathons I did from over 30 hackathons I attended,” Abdullah Kazi, a quhacker, wrote on the iQuHACK Slack.

Ultimately, “[we wanted to] help people to meet each other,” co-chair Buzzi says. “The impact [of iQuHACK] is scientific in some way, but it’s very human at the most important level.”

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Physicists make record-breaking 'quantum vortex' to study the mysteries of black holes

Physicists created a 'quantum vortex,' which flows with 500 times less viscosity than water and could be used to study the space-time warping caused by black holes.

Scientists have created a giant quantum tornado inside a helium superfluid, and they want to use it to probe the enigmatic nature of black holes .

The whirlpool — made from liquid helium cooled to near absolute zero — moves without friction, making it mimic the way rotating black holes warp the space-time that surrounds them.

By studying the vortex, physicists could glean important insight into the behavior of the cosmic monsters. The researchers published their findings March 20 in the journal Nature .

"Using superfluid helium has allowed us to study tiny surface waves in greater detail and accuracy than with our previous experiments in water," lead author Patrik Svancara , a physicist at the University of Nottingham in the U.K., said in a statement . "As the viscosity of superfluid helium is extremely small, we were able to meticulously investigate their interaction with the superfluid tornado and compare the findings with our own theoretical projections."

Related: Supermassive black hole at the heart of the Milky Way is approaching the cosmic speed limit, dragging space-time along with it

The workings of black holes remain a persistent mystery for physicists. The known laws of physics break in the presence of these extreme objects' infinite gravitational pulls. For those looking to combine Einstein's theory of general relativity with quantum mechanics , this means black holes' warping of space-time offers an alluring pull.

In the absence of a cataclysmic space-time rupture on Earth, the team behind the new study looked to a model system that could simulate some of the extreme eddies that exist around black holes. After supercooling liquid helium to a few fractions above absolute zero, they placed it inside a tank with a propeller at the bottom to stir up a vortex inside the fluid.

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Then, by watching how the superfluid (which flows roughly 500 times more easily than water) moved, the researchers observed how thousands of tiny vortices inside it combined into a giant whirlpool.

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"Superfluid helium contains tiny objects called quantum vortices, which tend to spread apart from each other," Svancara said in the statement. "In our set-up, we've managed to confine tens of thousands of these quanta in a compact object resembling a small tornado, achieving a vortex flow with record-breaking strength in the realm of quantum fluids."

By studying the quantum whirlpool, the scientists found convincing similarities to how black holes behave in space. Most notably, they observed a similar black hole phenomenon called ringdown, which is when a newly merged black hole wobbles on its axis.

Now that the simpler parallels have been observed, the researchers will train their experiment on more mysterious aspects of black hole behavior.

This "could eventually lead us to predict how quantum fields behave in curved spacetimes around astrophysical black holes," co-author Silke Weinfurtner , a professor of physics at the University of Nottingham, said in the statement.

Ben Turner is a U.K. based staff writer at Live Science. He covers physics and astronomy, among other topics like tech and climate change. He graduated from University College London with a degree in particle physics before training as a journalist. When he's not writing, Ben enjoys reading literature, playing the guitar and embarrassing himself with chess.

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Quantum Physics

Title: an efficient quantum algorithm for linear system problem in tensor format.

Abstract: Solving linear systems is at the foundation of many algorithms. Recently, quantum linear system algorithms (QLSAs) have attracted great attention since they converge to a solution exponentially faster than classical algorithms in terms of the problem dimension. However, low-complexity circuit implementations of the oracles assumed in these QLSAs constitute the major bottleneck for practical quantum speed-up in solving linear systems. In this work, we focus on the application of QLSAs for linear systems that are expressed as a low rank tensor sums, which arise in solving discretized PDEs. Previous works uses modified Krylov subspace methods to solve such linear systems with a per-iteration complexity being polylogarithmic of the dimension but with no guarantees on the total convergence cost. We propose a quantum algorithm based on the recent advances on adiabatic-inspired QLSA and perform a detailed analysis of the circuit depth of its implementation. We rigorously show that the total complexity of our implementation is polylogarithmic in the dimension, which is comparable to the per-iteration complexity of the classical heuristic methods.

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