![n|n) where N-ata. Use the result of [a, at] For harmonic oscillator system, with NIn) to prove 2. atIn) n+n1) and aln) Vn|n-1) (10%)](http://img.homeworklib.com/questions/83c8a5a0-7015-11ea-8b22-5380060b79ec.png?x-oss-process=image/resize,w_560)
Homework Answers
a is step down (lowering)
operator or aligation operator.
a+ is step up (rising ) operator or creation operator.
Add Answer to:
n|n) where N-ata. Use the result of [a, at] For harmonic oscillator system, with NIn) to...
Recall that for the Harmonic oscillator: vmk h Where k 2/2 is the wavenumber and m is the particle mass. =n2R-2k (n-2k...
Recall that for the Harmonic oscillator: vmk h Where k 2/2 is the wavenumber and m is the particle mass. =n2R-2k (n-2k)!k HT ka-2k where a integers; the coresponding wavefuction is Where k and |n are = The first three solutions (lecture 3) for the quantum harmonic oscillator are: n 0,k 0 Ho(1 = hwo/2 = ' Eo 25 E, — Зho/2 Н. (€) n 1, k n 2, k 0,1 E2 5hao/2 H2(42 2, Plot for a 1 ev...
1. Quantum harmonic oscillator (a) Derive formula for standard deviation of position measurement ...
1. Quantum harmonic oscillator (a) Derive formula for standard deviation of position measurement on a particle prepared in the ground state of harmonic oscillator. The formula will depend on h, m andw (b) Estimate order of magnitude of the standard deviation in (a) for the LIGO mirror of mass 10 kg and w 1 Hz. (c) A coherent state lo) is defined to be the eigenstate of the lowering operator with eigenvalue a, i.e. à lo)a) Write la) as where...
Question 3 Consider the one-dimensional harmonic oscillator, and denote its properly normalised e...
ONLY (e) (f) NEEDED THANK YOU :)
Question 3 Consider the one-dimensional harmonic oscillator, and denote its properly normalised energy eigenstates by { | n〉, n = 0, 1, 2, 3, . . .). Define the state where α is a complex number, and C is a normalisation constant. (a) Use a Campbell-Baker-Hausdorff relation (or otherwise) to show that In other words, | α > is an eigenstate of the (non-Hermitian) lowering operator with (complex) eigenvalue α. (b) During lectures...
As a result of a sudden perturbation of the harmonic oscillator originally in the ground state, t...
As a result of a sudden perturbation of the harmonic oscillator originally in the ground state, the restoring force coefficient k in its potential energy U(a) (1/2)k2 changes to k' ak, a>0. Find the proba- bility to find the new oscillator in an excited state.
As a result of a sudden perturbation of the harmonic oscillator originally in the ground state, the restoring force coefficient k in its potential energy U(a) (1/2)k2 changes to k' ak, a>0. Find the proba-...
Calculate the probability that a harmonic oscillator n=(n+1/2)his a state with n odd number if the...
Calculate the probability that a harmonic oscillator
n=(n+1/2)his a state with n odd number if the
oscillator is in contact with a heat bath at temperature
T.
Calculate the probability that a harmonic oscillator w is a state with n odd number if the oscillator is in contact with a heat bath at temperature T. epsilon n=(n+1/2)h
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...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x,...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x, t) = (x)e-1st/ where and E, are the harmonic oscillator's stationary states and their corresponding energies. (a) Show that the expectation value of position is (hint: use the results of Problem 4): (v) = A cos (wt - ) where the real constants A and o are given by: 1 2 Ae-id-1 " Entichtin Interpret this result, comparing it with the motion of a...
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...
Problem 5. (30 points) Consider a Harmonic oscillator with H that H=(ata + 1 / 2)ho,...
Problem 5. (30 points) Consider a Harmonic oscillator with H that H=(ata + 1 / 2)ho, where a=dma)X + i (a) (4 points) Show P, and a x 2h 2h 2moh P. Show also 2moh that [a, a]-l. (b) (6 points) Starting from the commuters la, HJand la', A), where H-H(h) show that the eigenvalues of Hare e,=(n+1/2) for n-0, 1,2, Show also that alm)-nln-l), and a l). (( points) Find the normalized ground state wavefunction by projecting alo)-0 on...
Problem 1 (Harmonic Oscillators) A mass-damper-spring system is a simple harmonic oscillator whose dynamics is governed...
Problem 1 (Harmonic Oscillators) A mass-damper-spring system is a simple harmonic oscillator whose dynamics is governed by the equation of motion where m is the mass, c is the damping coefficient of the damper, k is the stiffness of the spring, F is the net force applied on the mass, and x is the displacement of the mass from its equilibrium point. In this problem, we focus on a mass-damper-spring system with m = 1 kg, c-4 kg/s, k-3 N/m,...
Recall that for the Harmonic oscillator: vmk h Where k 2/2 is the wavenumber and m is the particle mass. =n2R-2k (n-2k)!k HT ka-2k where a integers; the coresponding wavefuction is Where k and |n are = The first three solutions (lecture 3) for the quantum harmonic oscillator are: n 0,k 0 Ho(1 = hwo/2 = ' Eo 25 E, — Зho/2 Н. (€) n 1, k n 2, k 0,1 E2 5hao/2 H2(42 2, Plot for a 1 ev...
1. Quantum harmonic oscillator (a) Derive formula for standard deviation of position measurement on a particle prepared in the ground state of harmonic oscillator. The formula will depend on h, m andw (b) Estimate order of magnitude of the standard deviation in (a) for the LIGO mirror of mass 10 kg and w 1 Hz. (c) A coherent state lo) is defined to be the eigenstate of the lowering operator with eigenvalue a, i.e. à lo)a) Write la) as where...
ONLY (e) (f) NEEDED THANK YOU :)
Question 3 Consider the one-dimensional harmonic oscillator, and denote its properly normalised energy eigenstates by { | n〉, n = 0, 1, 2, 3, . . .). Define the state where α is a complex number, and C is a normalisation constant. (a) Use a Campbell-Baker-Hausdorff relation (or otherwise) to show that In other words, | α > is an eigenstate of the (non-Hermitian) lowering operator with (complex) eigenvalue α. (b) During lectures...
As a result of a sudden perturbation of the harmonic oscillator originally in the ground state, the restoring force coefficient k in its potential energy U(a) (1/2)k2 changes to k' ak, a>0. Find the proba- bility to find the new oscillator in an excited state.
As a result of a sudden perturbation of the harmonic oscillator originally in the ground state, the restoring force coefficient k in its potential energy U(a) (1/2)k2 changes to k' ak, a>0. Find the proba-...
Calculate the probability that a harmonic oscillator
n=(n+1/2)his a state with n odd number if the
oscillator is in contact with a heat bath at temperature
T.
Calculate the probability that a harmonic oscillator w is a state with n odd number if the oscillator is in contact with a heat bath at temperature T. epsilon n=(n+1/2)h
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...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x, t) = (x)e-1st/ where and E, are the harmonic oscillator's stationary states and their corresponding energies. (a) Show that the expectation value of position is (hint: use the results of Problem 4): (v) = A cos (wt - ) where the real constants A and o are given by: 1 2 Ae-id-1 " Entichtin Interpret this result, comparing it with the motion of a...
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...
Problem 5. (30 points) Consider a Harmonic oscillator with H that H=(ata + 1 / 2)ho, where a=dma)X + i (a) (4 points) Show P, and a x 2h 2h 2moh P. Show also 2moh that [a, a]-l. (b) (6 points) Starting from the commuters la, HJand la', A), where H-H(h) show that the eigenvalues of Hare e,=(n+1/2) for n-0, 1,2, Show also that alm)-nln-l), and a l). (( points) Find the normalized ground state wavefunction by projecting alo)-0 on...
Problem 1 (Harmonic Oscillators) A mass-damper-spring system is a simple harmonic oscillator whose dynamics is governed by the equation of motion where m is the mass, c is the damping coefficient of the damper, k is the stiffness of the spring, F is the net force applied on the mass, and x is the displacement of the mass from its equilibrium point. In this problem, we focus on a mass-damper-spring system with m = 1 kg, c-4 kg/s, k-3 N/m,...
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