![The Hamiltonian of a harmonic oscillator is: 2m 2 Show that: (1) [ł, î] =ihp () [f.h]=-ihmoʻx](http://img.homeworklib.com/questions/90cbfb90-4580-11eb-8b96-d9d256812806.png?x-oss-process=image/resize,w_560)
Homework Answers
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The Hamiltonian of a harmonic oscillator is: 2m 2 Show that: (1) [ł, î] =ihp ()...
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...
4. (30 points) Harmonic oscillator with perturbation Recall the Hamiltonian of an harmonic oscillator in 1D:...
4. (30 points) Harmonic oscillator with perturbation Recall the Hamiltonian of an harmonic oscillator in 1D: p21 ÃO = + mwf?, where m is the mass of the particle and w is the angular frequency. Now, let us perturb the oscillator with a quadratic potential. The perturbation is given by Î' = zgmw?h?, where g is a dimensionless constant and g <1. (a) Write down the eigen-energies of the unperturbed Hamiltonian. (b) In Lecture 3, we introduced the lowering (or...
Consider a harmonic oscillator with Hamiltonian given by ?=(p^2/2m)+(1/2)X^2 = (a+)(a-)+(1/2) The current system state is...
Consider a harmonic oscillator with Hamiltonian given by ?=(p^2/2m)+(1/2)X^2 = (a+)(a-)+(1/2) The current system state is the superposition of the lowest and next-to-lowest energy eigenstates that gives the most negative possible value for the average position, use raising and lowering operators to derive the average momentum for this state. then, simplify using ħ = ? = 1
10. A harmonic oscillator with the Hamiltonian H t 2m dr? mooʻr is now subject to...
10. A harmonic oscillator with the Hamiltonian H t 2m dr? mooʻr is now subject to a 2 weak perturbation: H-ix. You are asked to solve the ground state of the new Hamiltonian - À + in two ways. (a) Solve by using the time-independent perturbation theory. Find the lowest non- vanishing order correction to the energy of the ground state. And find the lowest non vanishing order correction to the wavefunction of the ground state. (b) Find the wavefunction...
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...
The wave function for a harmonic oscillator in its first excited state is Consider the harmonic o...
The wave function for a harmonic oscillator in its first excited state is Consider the harmonic oscillator with Hamiltonian and let H --hk and r-cr d. Evaluate E0) for the first excited state using perturbation theory 2m dx 2
The wave function for a harmonic oscillator in its first excited state is Consider the harmonic oscillator with Hamiltonian and let H --hk and r-cr d. Evaluate E0) for the first excited state using perturbation theory 2m dx 2
1. Suppose I have a harmonic oscillator with a small quartic perturbation: 2 рґ 2m 2...
1. Suppose I have a harmonic oscillator with a small quartic perturbation: 2 рґ 2m 2 What are the first-order and second-order corrections to the nth energy levels of the unperturbed harmonic oscillator?
1. The two dimensional Harmonic Oscillator has the Hamiltonian n, n'>denotes the state In> of the...
Please solve with the
explanations of notations
1. The two dimensional Harmonic Oscillator has the Hamiltonian n, n'>denotes the state In> of the x-oscillator and In'> of the y-oscillator. This system is perturbed with the potential energy: Hi-Kix y. The perturbation removes the The perturbation removes the degeneracy of the states | 1,0> and |0,1> a) In first order perturbation theory find the two nondegenerate eigenstates of the full b) Find the corresponding energy eigenvalues. На Hamiltonian as normalized linear...
7 Harmonic oscillator in "energy space" Consider the harmonic oscillator in "energy space", i.e., in terms...
7 Harmonic oscillator in "energy space" Consider the harmonic oscillator in "energy space", i.e., in terms of the basis of eigenvectors n) of the harmonic oscillator Hamiltonian, with Hn) -hwn1/2)]n). We computed these in terms of wavefunctions in position space, ie. pn(x)-(zln), but we can also work purely in terms of the abstract energy eigenvectors in Dirac notation. PS9.pdf 1. You computed the matrix elements 〈nleln) on an earlier problem set. Now find (nn) for general n,n' 2. Find the...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian of a particle of a simple Harmonic oscillator potential in one dimension can be expressed in term of the creation and annihilation operators àt and à, respectively, as: or with In >, n = 0,1,..) are the nth eigenstates of the above Hamiltonian. Part A A.1. Show that the energy levels of a simple harmonic oscillator are E,' Aw (nti), n=0, 12, A.2. Calculate...
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...
4. (30 points) Harmonic oscillator with perturbation Recall the Hamiltonian of an harmonic oscillator in 1D: p21 ÃO = + mwf?, where m is the mass of the particle and w is the angular frequency. Now, let us perturb the oscillator with a quadratic potential. The perturbation is given by Î' = zgmw?h?, where g is a dimensionless constant and g <1. (a) Write down the eigen-energies of the unperturbed Hamiltonian. (b) In Lecture 3, we introduced the lowering (or...
10. A harmonic oscillator with the Hamiltonian H t 2m dr? mooʻr is now subject to a 2 weak perturbation: H-ix. You are asked to solve the ground state of the new Hamiltonian - À + in two ways. (a) Solve by using the time-independent perturbation theory. Find the lowest non- vanishing order correction to the energy of the ground state. And find the lowest non vanishing order correction to the wavefunction of the ground state. (b) Find the wavefunction...
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...
The wave function for a harmonic oscillator in its first excited state is Consider the harmonic oscillator with Hamiltonian and let H --hk and r-cr d. Evaluate E0) for the first excited state using perturbation theory 2m dx 2
The wave function for a harmonic oscillator in its first excited state is Consider the harmonic oscillator with Hamiltonian and let H --hk and r-cr d. Evaluate E0) for the first excited state using perturbation theory 2m dx 2
1. Suppose I have a harmonic oscillator with a small quartic perturbation: 2 рґ 2m 2 What are the first-order and second-order corrections to the nth energy levels of the unperturbed harmonic oscillator?
Please solve with the
explanations of notations
1. The two dimensional Harmonic Oscillator has the Hamiltonian n, n'>denotes the state In> of the x-oscillator and In'> of the y-oscillator. This system is perturbed with the potential energy: Hi-Kix y. The perturbation removes the The perturbation removes the degeneracy of the states | 1,0> and |0,1> a) In first order perturbation theory find the two nondegenerate eigenstates of the full b) Find the corresponding energy eigenvalues. На Hamiltonian as normalized linear...
7 Harmonic oscillator in "energy space" Consider the harmonic oscillator in "energy space", i.e., in terms of the basis of eigenvectors n) of the harmonic oscillator Hamiltonian, with Hn) -hwn1/2)]n). We computed these in terms of wavefunctions in position space, ie. pn(x)-(zln), but we can also work purely in terms of the abstract energy eigenvectors in Dirac notation. PS9.pdf 1. You computed the matrix elements 〈nleln) on an earlier problem set. Now find (nn) for general n,n' 2. Find the...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian of a particle of a simple Harmonic oscillator potential in one dimension can be expressed in term of the creation and annihilation operators àt and à, respectively, as: or with In >, n = 0,1,..) are the nth eigenstates of the above Hamiltonian. Part A A.1. Show that the energy levels of a simple harmonic oscillator are E,' Aw (nti), n=0, 12, A.2. Calculate...
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