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.
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.
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.