PHYS 5002: Homework 9

Perturbation of a driven system

A system with degenerate eigenstates that is subjected to a time-independent perturbation is being driven at its resonance frequency \((\omega=0)\). Consider the following example.
 
A Hydrogen atom prepared in its ground state is prepared for \(t\le 0\) with spin-up along the \(z\)-axis. At time \(t=0\), a constant \(\mathbf{B}\)-field is turned on which points in an arbitrary direction \((\theta,\phi)\). Neglect fine structure (and \(\mathbf{A}^2\) terms). Compute the probability that the atom will be found in the ground state with spin-down as a function of time.
 
Do the problem first with first-order perturbation theory and then solve it exactly. Discuss the accuracy of the perturbation theory.

This type of problem is called a quench problem, because we have suddenly changed the Hamiltonian at \(t=0\) to another Hamiltonian.

Nailed down spin-half particle

A “nailed-down” spin-\(\frac{1}{2}\) particle is acted on by a constant magnetic field in the \(z\)-direction and by an oscillatory field in the \(xy\)-plane. \[\hat{H}=\hat{H}_0+\hat{V}(t), \ \hat{H}_0=\hbar\Omega_0\hat{S}_z, \ \hat{V}(t)=\hbar\Omega_1(\hat{S}_x\cos{\omega t}+\hat{S}_y\sin{\omega t})\] a.) At \(t=0\) the particle is in the spin-up state along the \(z\)-axis. What is the probability that it will be found up at time \(t\)? (Solve the problem exactly)
 
b.) Use time-dependent perturbation theory to second-order to compute the probability.
 
c.) Compare the perturbation theory to the exact result expanded to second-order. Comment on the accuracy of the perturbation theory.

Time-ordered product gymnastics

Consider the time-dependent harmonic oscillator: \[\hat{H}(t)=\hat{H}_0+\hat{V}(t)\] where \[\hat{H}_0=\hbar\omega\left(\hat{a}^\dagger\hat{a}+\frac{1}{2}\right), \ \hat{V}(t)=C\left(e^{i\Omega t}\hat{a}^\dagger+e^{-i\Omega t}\hat{a}\right)\] a.) Compute \(\hat{U}_S(t,0)\) from the interaction representation formula \[\hat{U}_S(t,0)=e^{-\frac{i}{\hbar}\hat{H}_0t}Te^{-\frac{i}{\hbar}\int_0^tdt'\hat{V}_I(t')}\] to second order in \(\hat{V}\).
 
b.) Compute \(\hat{U}_S(t,0)\) from the formal solution of Schrödinger’s equation \[\hat{U}_S(t,0)=Te^{-\frac{i}{\hbar}\int_0^tdt'\hat{H}(t')}\] to second order in \(\hat{H}\).
 
c.) Compare the two results and comment on the differences.