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Exercise 6.5 Consider a spinless particle of mass m which is confined to move under the influence of a three-dimensional potential Î(x, y, z) = :| 0 +∞ for 0 < x

Exercise 6.5 Consider a spinless particle of mass m which is confined to move under the influence of a three-dimensional potential Î(x, y, z) = :| 0 +∞ for 0 < x

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(e) If the state vector of the nth excited state is [n) = |nx)|ny) or (x|nx) (y|ny) = ynx (x)Yny (y), 386 (xy\n) = calculate the expectation value of the operator  = 4 + ² in the state [n) as a function of the quantum numbers nx and ny. Exercise 6.2 A particle of mass m moves in the xy plane in the potential V(x, y) = ={ (a) Write down the time-independent Schrödinger equation for this particle and reduce it to a set of familiar one-dimensional equations. (b) Find the normalized eigenfunctions and the eigenenergies. Exercise 6.3 A particle of mass m moves in the xy plane in a two-dimensional rectangular well 0 ≤ y ≤ b, V(x, y) = 6.6. EXERCISES 1 CHAPTER 6. THREE-DIMENSIONAL PROBLEMS By reducing the time-independent Schrödinger equation to a set of more familiar one-dimensional equations, find the normalized wave functions and the energy levels of this particle. V(x, y, z) = mo²y for all y and 0 < x < a, +∞o, elsewhere. Exercise 6.4 Consider an anisotropic three-dimensional harmonic oscillator potential V(x, y, z) = = {m{o} x² + o²} y² + o² 2²). (a) Evaluate the energy levels in terms of wx, wy, and wz. (b) Calculate [Ĥ, ο]. Do you expect the wave functions to be eigenfunctions of 1²? { 0, (c) Find the three lowest levels for the case wx = wy = 2wz/3, and determine the degener- acy of each level. +∞, Exercise 6.5 Consider a spinless particle of mass m which is confined to move under the influence of a three-dimensional potential V(x, y, z) = =1 0 +∞ 0 < x <a, elsewhere. (a) Find the expression for the energy levels En、nyn, and their corresponding wave func- tions. (b) If a = 2b find the energies of the five lowest states and their degeneracies. Exercise 6.6 A particle of mass m moves in the three-dimensional potential 1 V₁(x, y) = — mos² (x² + y²), nw² 2 for 0 < x <a, 0 < y <a, 0 < z <b, elsewhere. mo²² for 0 < x < a, 0 ≤ y ≤ a, and z> 0 +∞, elsewhere. (a) Write down the time-independent Schrödinger equation for this particle and reduce it to a set of familiar one-dimensional equations; then find the normalized wave function Ynxnyn₂(x, y, z). (b) Find the allowed eigenenergies of this particle and show that they can be written as: En¸nyn₂ = En¸ny + Enz. (c) Find the four lowest energy levels in the xy plane (i.e., Enxny) and their corresponding deg racies. Exercise 6.7 A particle of mass m moves in the potential V(x, y, z) = V₁(x, y) + V₂(z) where V₂(z) = { 0, +∞o, 387 0 ≤ z≤ a, elsewhere. (a) Calculate the energy levels and the wave function of this particle. (b) Let us now turn off V₂(z) (i.e., m is subject only to V₁ (x, y)). Calculate the degeneracy gn of the nth energy level (note that n = nx + ny).
Transcribed Image Text:(e) If the state vector of the nth excited state is [n) = |nx)|ny) or (x|nx) (y|ny) = ynx (x)Yny (y), 386 (xy\n) = calculate the expectation value of the operator  = 4 + ² in the state [n) as a function of the quantum numbers nx and ny. Exercise 6.2 A particle of mass m moves in the xy plane in the potential V(x, y) = ={ (a) Write down the time-independent Schrödinger equation for this particle and reduce it to a set of familiar one-dimensional equations. (b) Find the normalized eigenfunctions and the eigenenergies. Exercise 6.3 A particle of mass m moves in the xy plane in a two-dimensional rectangular well 0 ≤ y ≤ b, V(x, y) = 6.6. EXERCISES 1 CHAPTER 6. THREE-DIMENSIONAL PROBLEMS By reducing the time-independent Schrödinger equation to a set of more familiar one-dimensional equations, find the normalized wave functions and the energy levels of this particle. V(x, y, z) = mo²y for all y and 0 < x < a, +∞o, elsewhere. Exercise 6.4 Consider an anisotropic three-dimensional harmonic oscillator potential V(x, y, z) = = {m{o} x² + o²} y² + o² 2²). (a) Evaluate the energy levels in terms of wx, wy, and wz. (b) Calculate [Ĥ, ο]. Do you expect the wave functions to be eigenfunctions of 1²? { 0, (c) Find the three lowest levels for the case wx = wy = 2wz/3, and determine the degener- acy of each level. +∞, Exercise 6.5 Consider a spinless particle of mass m which is confined to move under the influence of a three-dimensional potential V(x, y, z) = =1 0 +∞ 0 < x <a, elsewhere. (a) Find the expression for the energy levels En、nyn, and their corresponding wave func- tions. (b) If a = 2b find the energies of the five lowest states and their degeneracies. Exercise 6.6 A particle of mass m moves in the three-dimensional potential 1 V₁(x, y) = — mos² (x² + y²), nw² 2 for 0 < x <a, 0 < y <a, 0 < z <b, elsewhere. mo²² for 0 < x < a, 0 ≤ y ≤ a, and z> 0 +∞, elsewhere. (a) Write down the time-independent Schrödinger equation for this particle and reduce it to a set of familiar one-dimensional equations; then find the normalized wave function Ynxnyn₂(x, y, z). (b) Find the allowed eigenenergies of this particle and show that they can be written as: En¸nyn₂ = En¸ny + Enz. (c) Find the four lowest energy levels in the xy plane (i.e., Enxny) and their corresponding deg racies. Exercise 6.7 A particle of mass m moves in the potential V(x, y, z) = V₁(x, y) + V₂(z) where V₂(z) = { 0, +∞o, 387 0 ≤ z≤ a, elsewhere. (a) Calculate the energy levels and the wave function of this particle. (b) Let us now turn off V₂(z) (i.e., m is subject only to V₁ (x, y)). Calculate the degeneracy gn of the nth energy level (note that n = nx + ny).
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