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1. Position representation of the harmonic oscillator wave functions. (a) Using that the position representation of...
3. The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an...
3. The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Consider an electron trapped by a one-dimensional harmonic potential V(x)=-5 mo?x” (where m is the electron mass, o is a constant angular frequency). In this case, the Schrödinger equation takes the following form, **...
Consider a linear harmonic oscillator whose Hamiltonian is given by 1? д2 Н 2m дд? 2...
Consider a linear harmonic oscillator whose Hamiltonian is given by 1? д2 Н 2m дд? 2 hw(n1/2) with eigenvalues En n 0,1,2,... Please (1) derive its density matrix in momentum representation, and (2) evaluate the mean energy (H with results obtained in last question
Consider a linear harmonic oscillator whose Hamiltonian is given by 1? д2 Н 2m дд? 2 hw(n1/2) with eigenvalues En n 0,1,2,... Please (1) derive its density matrix in momentum representation, and (2) evaluate the mean...
The three-dimensional harmonic oscillator Cartesian wave functions that you found in Prob. 4.46 a...
The three-dimensional harmonic oscillator Cartesian wave functions that you found in Prob. 4.46 are simultaneous eigenfunctions of H and parity (i.e., r →-r), but they are not also simultaneous eigenfunctions of L' and Lz. However, we know that it's possible to construct eigenfunctions of H for the 3D harmonic oscillator that are also eigenfunctions of L, Lz, and parity. Combine the Gaussian factors that appear in your Prob. 4.46 eigenfunctions into a function of r that is independent of θ...
6 The Fermionic Oscillator Suppose that we constructed a harmonic oscillator Hamiltonian H in terms of...
6 The Fermionic Oscillator Suppose that we constructed a harmonic oscillator Hamiltonian H in terms of raising and lowering operators a+,a in the usual way, such that but now whereaa obey the anticommutation relationn (Be careful! The a+,a are operators, rather than numbers.) (a) Suppose I give you a wavefunction that solves the time-independent Schrödinger equation, i.e. such that HUn-EUn-hw (n + ) ψη. Is a+Un also a solution to the time-independent Schrödinger equation If so, what is its energy...
Potential energy function, V(x) = (1/2)mw2x2 Assuming the time-independent Schrödinger equation, show that the following wave...
Potential energy function,
V(x) = (1/2)mw2x2
Assuming the time-independent Schrödinger equation, show that the following wave functions are solutions describing the one-dimensional harmonic behaviour of a particle of mass m, where ?2-h/v/mK, and where co and ci are constants. Calculate the energies of the particle when it is in wave-functions ?0(x) and V1 (z) What is the general expression for the allowed energies En, corresponding to wave- functions Un(x), of this one-dimensional quantum oscillator? 6 the states corresponding to the...
2. Two noninteracting particles, each of mass m, are in the 1-D harmonic oscillator potential Describe...
2. Two noninteracting particles, each of mass m, are in the 1-D harmonic oscillator potential Describe their ground state and the next two excited states, that is, their corresponding (i) wave functions, (i) energy eigenvalues, and (ii) degeneracies if two particles are (a) distinguishable particles, (b) the identical bosons, and (c) the identical fermions. 3. Suppose one particle is in the ground state, and the other is in the first excited state for tw particles described in Prob. 2. Calculate...
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy ,...
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy , and ф are the usual spherical coordinates. (a) In the form given above, why is it clear that the potential energy function V) is (b) For this problem, it will be more convenient to express this spherically-symmetric where r , spherically symmetric? A brief answer is sufficient. potential energy in Cartesian coordinates x, y, and z as physically the same potential energy as the...
For a harmonic oscillator confirm by explicit evaluation of the integral that the two wave functions...
For a harmonic oscillator confirm by explicit evaluation of the integral that the two wave functions of level 1 and 2, psi 1 and psi 2, are orthogonal (the variable x obeys: -∞ < x < +∞). Hint: Use the Hermite polynomials
3. Anharmonicity (6 marks] Consider the three-dimensional isotropic harmonic oscillator 2 1 242 р...
3. Anharmonicity (6 marks] Consider the three-dimensional isotropic harmonic oscillator 2 1 242 рґ which has energy eigenvalues En-hu(n+3/2), where n- 0,1,2.. (a) Calculate the first-order shift in the ground-state energy of the harmonic oscillator due to the addition of an anharmonic term C24 to the potential, where C> 0. (b) Calculate instead the first-order shifts in the energies of the n - 1 ercited states due to the addition of the anharmonic term C (c) For the lowest energy...
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian...
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian 12 12 m2 H = -2m d. 2+ 2 Here m and w are the mass and frequency, respectively. Consider a time-dependent wave function of the form <(x,t) = C'exp (-a(x – 9(t)+ ik(t)z +io(t)), where a and C are positive constants, and g(t), k(t), and o(t) are real functions of time t. 1. Express C in terms of a. [2 marks] 2. By...
3. The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Consider an electron trapped by a one-dimensional harmonic potential V(x)=-5 mo?x” (where m is the electron mass, o is a constant angular frequency). In this case, the Schrödinger equation takes the following form, **...
Consider a linear harmonic oscillator whose Hamiltonian is given by 1? д2 Н 2m дд? 2 hw(n1/2) with eigenvalues En n 0,1,2,... Please (1) derive its density matrix in momentum representation, and (2) evaluate the mean energy (H with results obtained in last question
Consider a linear harmonic oscillator whose Hamiltonian is given by 1? д2 Н 2m дд? 2 hw(n1/2) with eigenvalues En n 0,1,2,... Please (1) derive its density matrix in momentum representation, and (2) evaluate the mean...
The three-dimensional harmonic oscillator Cartesian wave functions that you found in Prob. 4.46 are simultaneous eigenfunctions of H and parity (i.e., r →-r), but they are not also simultaneous eigenfunctions of L' and Lz. However, we know that it's possible to construct eigenfunctions of H for the 3D harmonic oscillator that are also eigenfunctions of L, Lz, and parity. Combine the Gaussian factors that appear in your Prob. 4.46 eigenfunctions into a function of r that is independent of θ...
6 The Fermionic Oscillator Suppose that we constructed a harmonic oscillator Hamiltonian H in terms of raising and lowering operators a+,a in the usual way, such that but now whereaa obey the anticommutation relationn (Be careful! The a+,a are operators, rather than numbers.) (a) Suppose I give you a wavefunction that solves the time-independent Schrödinger equation, i.e. such that HUn-EUn-hw (n + ) ψη. Is a+Un also a solution to the time-independent Schrödinger equation If so, what is its energy...
Potential energy function,
V(x) = (1/2)mw2x2
Assuming the time-independent Schrödinger equation, show that the following wave functions are solutions describing the one-dimensional harmonic behaviour of a particle of mass m, where ?2-h/v/mK, and where co and ci are constants. Calculate the energies of the particle when it is in wave-functions ?0(x) and V1 (z) What is the general expression for the allowed energies En, corresponding to wave- functions Un(x), of this one-dimensional quantum oscillator? 6 the states corresponding to the...
2. Two noninteracting particles, each of mass m, are in the 1-D harmonic oscillator potential Describe their ground state and the next two excited states, that is, their corresponding (i) wave functions, (i) energy eigenvalues, and (ii) degeneracies if two particles are (a) distinguishable particles, (b) the identical bosons, and (c) the identical fermions. 3. Suppose one particle is in the ground state, and the other is in the first excited state for tw particles described in Prob. 2. Calculate...
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy , and ф are the usual spherical coordinates. (a) In the form given above, why is it clear that the potential energy function V) is (b) For this problem, it will be more convenient to express this spherically-symmetric where r , spherically symmetric? A brief answer is sufficient. potential energy in Cartesian coordinates x, y, and z as physically the same potential energy as the...
3. Anharmonicity (6 marks] Consider the three-dimensional isotropic harmonic oscillator 2 1 242 рґ which has energy eigenvalues En-hu(n+3/2), where n- 0,1,2.. (a) Calculate the first-order shift in the ground-state energy of the harmonic oscillator due to the addition of an anharmonic term C24 to the potential, where C> 0. (b) Calculate instead the first-order shifts in the energies of the n - 1 ercited states due to the addition of the anharmonic term C (c) For the lowest energy...
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian 12 12 m2 H = -2m d. 2+ 2 Here m and w are the mass and frequency, respectively. Consider a time-dependent wave function of the form <(x,t) = C'exp (-a(x – 9(t)+ ik(t)z +io(t)), where a and C are positive constants, and g(t), k(t), and o(t) are real functions of time t. 1. Express C in terms of a. [2 marks] 2. By...
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