Time-dependent perturbation theory

Formal

Problem:

A system makes transitions between eigenstates of H0 under the action of the time dependent Hamiltonian H0 + εU0sinωt, εU0 << H0.  Assume that at t = 0 the system is in the state |Φi>.  Find an expression for the probability of transition from |Φi> to |Φf>, where |Φi> and |Φf> are eigenstates of H0 with eigenvalues Ei and Ef.  Show that this probability is small unless Ef - Ei ~ ±ħω.
[This shows that a charged particle in an oscillating electric field with angular frequency ω will exchange energy with the field only in multiples of E = ħω.]

Solution:

Problem:

A system makes transitions between eigenstates of H0 under the action of the time dependent Hamiltonian H0 + Wcosωt, W << H0.  Assume that at t = 0 the system is in the state |ψi>.
(a)  Find an expression for the probability of transition from |ψi> to |ψf>, where |ψi> and |ψf> are eigenstates of H0 with eigenvalues Ei and Ef.
(b)  Find an expression for the probability of transition from |ψi> to |ψf>, for a constant perturbation W.

Solution:

Solution:


Infinite well

Problem:

Consider a one-dimensional, infinitely deep well.  Let U(x) = 0, for 0 < x < a, and U(x) = ∞ everywhere else.  The eigenstates of H0 = p2/(2m) + U(x) are Φn(x) = (2/a)½sin(nπx/a) with eigenvalues En = n2π2ħ2/(2ma2).
Assume that at t = 0 you put a coin on the bottom of the well.
W(x) = W0, a/4 < x < 3a/4 for t > 0, W(x) = 0 everywhere else, H = H0 + W.
If at t = 0 the system is in the state Φ3(x), what is the probability of finding it in Φ1(x) at time t?

Solution:

Problem:

Consider a particle in the ground state of a two-dimensional infinite square well.  The width of the well is L.  The ground state is not degenerate, but the excited energy levels are degenerate. 
A perturbation H' = A x sinωt with A = h2/(100 m L3) is turned on at t = 0.  Find the probability that at time t the particle will in the first excited state, for each of the degenerate states.

Solution:

Problem:

A particle of mass m and electric charge e, moving in one dimension, is confined to an interval of length a and is subject to a uniform electric field E. 
Initially it is in the eigenstate of kinetic energy with eigenvalue Ek = k2π2ħ2/(2ma2), where k is an integer. 
Find, to first order in E2, the probability that after a time t its kinetic energy will be found to be El where k ≠ l.

Solution:


Harmonic oscillator

Problem:

A one-dimensional harmonic oscillator is in its ground state for t < 0.
For  t > 0 it is subjected to a time-dependent but spatially uniform force in the x-direction, 
F = F0exp(-t/τ).
(a)  Using time-dependent perturbation theory to first order, obtain the probability of finding the oscillator in its first excited state for t > 0.  Show that as t approaches infinity (with τ finite), the limit of your expression is independent of time.  Is this result reasonable, or surprising?
(b)  Can the oscillator be found in higher excited states (for t > 0 )?

Solution:

Problem:

Consider a one-dimensional oscillator in its ground state.
The unperturbed Hamiltonian is H0 = p2/(2m) + ½mω02x2.
At t = 0 the Hamiltonian becomes H = H0 + H1, where H1 = ½mω12x2cosft, ω1 << ω0.
Calculate the transition probability to the second excited state.
Can there be a transition to any other excited state?

Note:
The stationary states of H0 = p2/(2m) + mω02x2/2 are un(x) = NnHn(αx)exp(-α2x2/2)
with Nn = [α/(π½2nn!)]½, α = (mω0/ħ)½  and Hn(αx) a Hermite polynomial.
Recursion relation: Hn+1(αx) = 2αxHn(αx) - 2nHn-1(αx).
H0(αx) = 1,   H1(αx) = 2αx,   H2(αx) = 4(αx)2 - 2.

Solution:


Rotator

Problem:

Consider a symmetrical top, spinning about is symmetry axis with non-zero angular momentum J.  The moment of inertia of the top is I and its Hamiltonian is given by H = J2/2I.  Assume the top is perturbed by an external magnetic field B, with interaction H' = -μ∙B, where μ = GJμB/ħ is the magnetic moment of the top, G > 0, and μB is the Bohr magneton.  Assume that the top has an integer total angular momentum quantum number.
(a)  What is the energy of the top when B = 0?
(b)  What are the ground state energy and the energy of the first excited state of the spinning top when B = B0k, B0 > 0?
(c)  Assume that B becomes time dependent,
B(t) = B0k, t < 0,  B(t) = B0k + i ΔBexp(-λt), t > 0.
where ΔB << B0 and λ > 0. 
If the top is in its first excited state for t < 0, find the probability that the top is in the ground state of part (b) for t > 0.

Solution:


Atoms

Problem:

Assume an atom is interacting with a monochromatic (visible light) EM plane wave
E
(r,t) = E0k cos(ky - ωt).
Assume the wave functions Φnlm(r) of the atomic electrons are characterized by the quantum numbers n, l, and m.  Derive the selection rules for electric dipole transitions.

Solution:

Problem:

The electron in a hydrogen atom is in a 3d state.  Neglect the fine structure.
(a)  To what state or states (i.e. 1s etc.) can it go by radiating a photon in an allowed transition?
(b)  What is the degeneracy of the electron (include spin, but ignore spin-orbit interaction) in a 3d state?

Solution: