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3. (a) Consider a 1-dim harmonic oscillator in its ground state (0) of the unperturbed Hamiltonian at t--0o. Let a pert...
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-...
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...
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...
A particle of charge q and mass m is bound in the ground state of a one-dimensional harmonic osci...
A particle of charge q and mass m is bound in the ground state of a one-dimensional harmonic oscillator potential with frequency oo. At time t-0 a weak spatially uniform electric field (E) is turned on, so that the perturbation to the Hamiltonian can be described as R'(t) =-q Exe-t/t for t> 0. Using first order, time-dependent perturbation theory, calculate the following probabilities: (a) the particle is detected in the first excited state after a very long time (t »...
QUANTUM MECHANICS Problem 4 Consider a one-dimensional charged harmonic oscillator. Let the coordinate be, charge be...
QUANTUM MECHANICS
Problem 4 Consider a one-dimensional charged harmonic oscillator. Let the coordinate be, charge be q, mass be m, and the frequency of the oscillator be u. (a) 79 rat t =-oo, the oscillator is in the ground state 10). A uniform electric field E along x axis is applied betweentoo andtoo with the time dependence of E being given by E(t) ー(t/ア Neglect the induced magnetic field. Find the probability that the oscillator goes to the nth excited...
1. A particle, initially (t -> 0) in the ground state of an infinite, 1D potential...
1. A particle, initially (t -> 0) in the ground state of an infinite, 1D potential box with walls at r 0 and = a, is subjected at time t = 0 to a time-dependent perturbation V (r, t) et/7, with eo a small real number a) Calculate to first order the probability of finding the particle in an excited state for t 0. Consider all final states. Are all possible transitions allowed? b) Examine the time dependence of the...
[4] Consider a harmonic oscillator of mass m and angular frequency ω. At time t-0, the state of this oscillator is given by y(о) со фо) + с ф.) where the states I 0) .) represent the ground state...
[4] Consider a harmonic oscillator of mass m and angular frequency ω. At time t-0, the state of this oscillator is given by y(о) со фо) + с ф.) where the states I 0) .) represent the ground state and first excited state respectively. (a) Write the normalization condition for lv(o) and determine the mean value (H) of the energy in terms of co and ci. (b) With the additional requirement (H)-ho. calculate eoand o,p.
[4] Consider a harmonic oscillator...
1. Consider a spin-0 particle of mass m and charge q moving in a symmetric three-dimensional harm...
1. Consider a spin-0 particle of mass m and charge q moving in a symmetric three-dimensional harmonic oscillator potential with natural frequency W.Att-0 an external magnetic field is turned on which is uniform in space but oscillates with temporal frequency W as follows. E(t)-Bo sin(at) At time t>0, the perturbation is turned off. Assuming that the system starts off at t-0 in the ground state, apply time-dependent perturbation theory to estimate the probability that the system ends up in an...
5. A particle in the harmonic oscillator potential has the initial wave function Psi(x, 0) = A[\psi_{0}(x) + \psi_{1}(x)] for some constant A. Here to and ₁ are the normalized ground state and the first excited state wavefunctions of the harmonic oscill
5. A particle in the harmonic oscillator potential has the initial wave function Psi(x, 0) = A[\psi_{0}(x) + \psi_{1}(x)] for some constant A. Here to and ₁ are the normalized ground state and the first excited state wavefunctions of the harmonic oscillator, respectively. (a) Normalize (r, 0). (b) Find the wavefunction (r, t) at a later time t and hence evaluate (x, t) 2. Leave your answers involving expressions in to and ₁. c) sing the following normalized expression of...
A particle with mass m is in a one-dimensional simple harmonic oscillator potential. At time t = 0 it is described by the state where lo and l) are normalised energy eigenfunctions corresponding to e...
A particle with mass m is in a one-dimensional simple harmonic oscillator potential. At time t = 0 it is described by the state where lo and l) are normalised energy eigenfunctions corresponding to energies E and Ey and b and c are real constants. (a) Find b and c so that (x) is as large as possible. b) Write down the wavefunction of this particle at a time t later c)Caleulate (x) for the particle at time t (d)...
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-...
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...
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...
A particle of charge q and mass m is bound in the ground state of a one-dimensional harmonic oscillator potential with frequency oo. At time t-0 a weak spatially uniform electric field (E) is turned on, so that the perturbation to the Hamiltonian can be described as R'(t) =-q Exe-t/t for t> 0. Using first order, time-dependent perturbation theory, calculate the following probabilities: (a) the particle is detected in the first excited state after a very long time (t »...
QUANTUM MECHANICS
Problem 4 Consider a one-dimensional charged harmonic oscillator. Let the coordinate be, charge be q, mass be m, and the frequency of the oscillator be u. (a) 79 rat t =-oo, the oscillator is in the ground state 10). A uniform electric field E along x axis is applied betweentoo andtoo with the time dependence of E being given by E(t) ー(t/ア Neglect the induced magnetic field. Find the probability that the oscillator goes to the nth excited...
1. A particle, initially (t -> 0) in the ground state of an infinite, 1D potential box with walls at r 0 and = a, is subjected at time t = 0 to a time-dependent perturbation V (r, t) et/7, with eo a small real number a) Calculate to first order the probability of finding the particle in an excited state for t 0. Consider all final states. Are all possible transitions allowed? b) Examine the time dependence of the...
[4] Consider a harmonic oscillator of mass m and angular frequency ω. At time t-0, the state of this oscillator is given by y(о) со фо) + с ф.) where the states I 0) .) represent the ground state and first excited state respectively. (a) Write the normalization condition for lv(o) and determine the mean value (H) of the energy in terms of co and ci. (b) With the additional requirement (H)-ho. calculate eoand o,p.
[4] Consider a harmonic oscillator...
1. Consider a spin-0 particle of mass m and charge q moving in a symmetric three-dimensional harmonic oscillator potential with natural frequency W.Att-0 an external magnetic field is turned on which is uniform in space but oscillates with temporal frequency W as follows. E(t)-Bo sin(at) At time t>0, the perturbation is turned off. Assuming that the system starts off at t-0 in the ground state, apply time-dependent perturbation theory to estimate the probability that the system ends up in an...
A particle with mass m is in a one-dimensional simple harmonic oscillator potential. At time t = 0 it is described by the state where lo and l) are normalised energy eigenfunctions corresponding to energies E and Ey and b and c are real constants. (a) Find b and c so that (x) is as large as possible. b) Write down the wavefunction of this particle at a time t later c)Caleulate (x) for the particle at time t (d)...
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