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Electrical Perturbation (bonus problem, 50 pts) An electron moving in a one-dimensional harmonic potential of frequency...
Consider a one-dimensional (1D) harmonic oscillator problem, where the perturbation V causes a modification of the...
Consider a one-dimensional (1D) harmonic oscillator problem, where the perturbation V causes a modification of the oscillator frequency: 2 K22 H = H. +V, (1) 2 K2 V = - K +K > 0. Of course, this problem (1), (2) is trivially solved exactly yielding the oscillator solutions with a new frequency. Show that corrections to the nth energy level as calculated within the perturbation theory indeed reproduce the exact result, restricting yourselves to terms up to the second order...
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...
19. Suppose that an electron in a one-dimensional harmonic-oscillator potential muo2 is subjected to an oscillating...
19. Suppose that an electron in a one-dimensional harmonic-oscillator potential muo2 is subjected to an oscillating electric field o) cos wt in the x direction (a) If the electron is initially in the ground state, what is the proba- bility that the electron will be in the nth excited state at time t? (b) I , perturbation theory will fail at some time t. What is the critical time?
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...
Suppose a particle is in a one-dimensional harmonic oscillator potential. Suppose that a perturbation is added...
Suppose a particle is in a one-dimensional harmonic oscillator
potential. Suppose that
a perturbation is added at time t = 0 of the form . Assume that at time t = 0 the
particle
is in the ground state. Use first order perturbation theory to find
the probability that at some
time t1 > 0 the particle is in the first excited state of the
harmonic oscillator.
H' = ext.
Assume that the Harmonic Oscillator potential is being perturbed by an additional term that is quadratic...
Assume that the Harmonic Oscillator potential is being perturbed by an additional term that is quadratic inx: Hy = an mw?x?; l«l< 1 Calculate the energy to the first non-zero correction using the Perturbation approach. Use ladder operatorst How does this result compare with the exact one?
1. Consider a charged particle bound in the harmonic oscillator potential V(x) = mw x2. A...
1. Consider a charged particle bound in the harmonic oscillator potential V(x) = mw x2. A weak electric field is applied to the system such that the potential energy, U(X), now has an extra term: V(x) = -qEx. We write the full Hamiltonian as H = Ho +V(x) where Ho = Px +mw x2 V(x) = –qEx. (a) Write down the unperturbed energies, EO. (b) Find the first-order correction to E . (c) Calculate the second-order correction to E ....
please solve with explanations 3. (20 pts) A particle of mass m and charge q is in a one dimensional harmonic oscillato...
please solve with explanations
3. (20 pts) A particle of mass m and charge q is in a one dimensional harmonic oscillator potential ()1ma'. A time dependent uniform electric field E, ()E, os eris 2 applied in the x direction. The particle is in the harmonic oscillator ground state at time a) What is the time dependent perturbation Hamiltonian H'(t) - the potential enegy of the charge in this electric field? b) Find the amplitude ci(t) of finding the particle...
4. Let us revisit the shifted harmonic oscillator from problem set 5, but this time through...
4. Let us revisit the shifted harmonic oscillator from problem set 5, but this time through the lens of perturbation theory. The Hamiltonian of the oscillator is given by * 2m + mw?f? + cî, and, as solved for previously, it has eigenenergies of En = hwan + mwra and eigenstates of (0) = N,,,a1 + role of (rc)*/2, where Do = 42 and a=(mw/h) (a) By treating the term cî as a perturbation, show that the first-order correction to...
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 »...
Consider a one-dimensional (1D) harmonic oscillator problem, where the perturbation V causes a modification of the oscillator frequency: 2 K22 H = H. +V, (1) 2 K2 V = - K +K > 0. Of course, this problem (1), (2) is trivially solved exactly yielding the oscillator solutions with a new frequency. Show that corrections to the nth energy level as calculated within the perturbation theory indeed reproduce the exact result, restricting yourselves to terms up to the second order...
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...
19. Suppose that an electron in a one-dimensional harmonic-oscillator potential muo2 is subjected to an oscillating electric field o) cos wt in the x direction (a) If the electron is initially in the ground state, what is the proba- bility that the electron will be in the nth excited state at time t? (b) I , perturbation theory will fail at some time t. What is the critical time?
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...
Suppose a particle is in a one-dimensional harmonic oscillator
potential. Suppose that
a perturbation is added at time t = 0 of the form . Assume that at time t = 0 the
particle
is in the ground state. Use first order perturbation theory to find
the probability that at some
time t1 > 0 the particle is in the first excited state of the
harmonic oscillator.
H' = ext.
Assume that the Harmonic Oscillator potential is being perturbed by an additional term that is quadratic inx: Hy = an mw?x?; l«l< 1 Calculate the energy to the first non-zero correction using the Perturbation approach. Use ladder operatorst How does this result compare with the exact one?
1. Consider a charged particle bound in the harmonic oscillator potential V(x) = mw x2. A weak electric field is applied to the system such that the potential energy, U(X), now has an extra term: V(x) = -qEx. We write the full Hamiltonian as H = Ho +V(x) where Ho = Px +mw x2 V(x) = –qEx. (a) Write down the unperturbed energies, EO. (b) Find the first-order correction to E . (c) Calculate the second-order correction to E ....
please solve with explanations
3. (20 pts) A particle of mass m and charge q is in a one dimensional harmonic oscillator potential ()1ma'. A time dependent uniform electric field E, ()E, os eris 2 applied in the x direction. The particle is in the harmonic oscillator ground state at time a) What is the time dependent perturbation Hamiltonian H'(t) - the potential enegy of the charge in this electric field? b) Find the amplitude ci(t) of finding the particle...
4. Let us revisit the shifted harmonic oscillator from problem set 5, but this time through the lens of perturbation theory. The Hamiltonian of the oscillator is given by * 2m + mw?f? + cî, and, as solved for previously, it has eigenenergies of En = hwan + mwra and eigenstates of (0) = N,,,a1 + role of (rc)*/2, where Do = 42 and a=(mw/h) (a) By treating the term cî as a perturbation, show that the first-order correction to...
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 »...
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