#energy eigenvalue

[Click here for a PDF of this post with nicer formatting]

Q: [1] pr 4.11

(a) Given a time reversal invariant Hamiltonian, show that for any energy eigenket

\begin{equation}\label{eqn:totallyAsymmetricPotential:20} \expectation{\BL} = 0. \end{equation}

(b) If the wave function of such a state is expanded as

\begin{equation}\label{eqn:totallyAsymmetricPotential:40} \sum_{l,m} F_{l m} Y_{l m}(\theta,…

Time reversal behavior of solutions to crystal spin Hamiltonian

[Click here for a PDF of this post with nicer formatting]

Q: [1] pr 4.12

Solve the spin 1 Hamiltonian \begin{equation}\label{eqn:crystalSpinHamiltonianTimeReversal:20} H = A S_z^2 + B(S_x^2 – S_y^2). \end{equation}

Is this Hamiltonian invariant under time reversal?

How do the eigenkets change under time reversal?

A:

In spinMatrices.nb the matrix representation of the Hamiltonian is found to be \be…

Energy estimate for an absolute value potential

[Click here for a PDF of this post with nicer formatting]

Here’s a simple problem, a lot like the problem set 6 variational calculation.

Q: [1] 5.21

Estimate the lowest eigenvalue \( \lambda \) of the differential equation

\begin{equation}\label{eqn:absolutePotentialVariation:20} \frac{d^2}{dx^2}\psi + \lr{ \lambda – \Abs{x} } \psi = 0. \end{equation}

Using \( \alpha \) variation with the trial…

A symmetric real Hamiltonian

[Click here for a PDF of this post with nicer formatting]

Question: A symmetric real Hamiltonian ([1] pr. 2.9)

Find the time evolution for the state \( \ket{a’} \) for a Hamiltian of the form

\begin{equation}\label{eqn:symmetricHamiltonianEvolution:20} H = \delta \lr{ \ket{a’}\bra{a’} + \ket{a”}\bra{a”} } \end{equation}

Answer

This Hamiltonian has the matrix representation

\begin{equation}\label{…