[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, \phi),

\end{equation}

what are the phase restrictions on \( F_{lm} \)?

### A: part (a)

For a time reversal invariant Hamiltonian \( H \) we have

\begin{equation}\label{eqn:totallyAsymmetricPotential:60}

H \Theta = \Theta H.

\end{equation}

If \( \ket{\psi} \) is an energy eigenstate with eigenvalue \( E \), we have

\begin{equation}\label{eqn:totallyAsymmetricPotential:80}

\begin{aligned}

H \Theta \ket{\psi}

&= \Theta H \ket{\psi} \\

&= \lambda \Theta \ket{\psi},

\end{aligned}

\end{equation}

so \( \Theta \ket{\psi} \) is also an eigenvalue of \( H \), so can only differ from \( \ket{\psi} \) by a phase factor. That is

\begin{equation}\label{eqn:totallyAsymmetricPotential:100}

\begin{aligned}

\ket{\psi’}

&=

\Theta \ket{\psi} \\

&= e^{i\delta} \ket{\psi}.

\end{aligned}

\end{equation}

Now consider the expectation of \( \BL \) with respect to a time reversed state

\begin{equation}\label{eqn:totallyAsymmetricPotential:120}

\begin{aligned}

\bra{ \psi’} \BL \ket{\psi’}

&=

\bra{ \psi} \Theta^{-1} \BL \Theta \ket{\psi} \\

&=

\bra{ \psi} (-\BL) \ket{\psi},

\end{aligned}

\end{equation}

however, we also have

\begin{equation}\label{eqn:totallyAsymmetricPotential:140}

\begin{aligned}

\bra{ \psi’} \BL \ket{\psi’}

&=

\lr{ \bra{ \psi} e^{-i\delta} } \BL \lr{ e^{i\delta} \ket{\psi} } \\

&=

\bra{\psi} \BL \ket{\psi},

\end{aligned}

\end{equation}

so we have \( \bra{\psi} \BL \ket{\psi} = -\bra{\psi} \BL \ket{\psi} \) which is only possible if \( \expectation{\BL} = \bra{\psi} \BL \ket{\psi} = 0\).

### A: part (b)

Consider the expansion of the wave function of a time reversed energy eigenstate

\begin{equation}\label{eqn:totallyAsymmetricPotential:160}

\begin{aligned}

\bra{\Bx} \Theta \ket{\psi}

&=

\bra{\Bx} e^{i\delta} \ket{\psi} \\

&=

e^{i\delta} \braket{\Bx}{\psi},

\end{aligned}

\end{equation}

and then consider the same state expanded in the position basis

\begin{equation}\label{eqn:totallyAsymmetricPotential:180}

\begin{aligned}

\bra{\Bx} \Theta \ket{\psi}

&=

\bra{\Bx} \Theta \int d^3 \Bx’ \lr{ \ket{\Bx’}\bra{\Bx’} } \ket{\psi} \\

&=

\bra{\Bx} \Theta \int d^3 \Bx’ \lr{ \braket{\Bx’}{\psi} } \ket{\Bx’} \\

&=

\bra{\Bx} \int d^3 \Bx’ \lr{ \braket{\Bx’}{\psi} }^\conj \Theta \ket{\Bx’} \\

&=

\bra{\Bx} \int d^3 \Bx’ \lr{ \braket{\Bx’}{\psi} }^\conj \ket{\Bx’} \\

&=

\int d^3 \Bx’ \lr{ \braket{\Bx’}{\psi} }^\conj \braket{\Bx}{\Bx’} \\

&=

\int d^3 \Bx’ \braket{\psi}{\Bx’} \delta(\Bx- \Bx’) \\

&=

\braket{\psi}{\Bx}.

\end{aligned}

\end{equation}

This demonstrates a relationship between the wave function and its complex conjugate

\begin{equation}\label{eqn:totallyAsymmetricPotential:200}

\braket{\Bx}{\psi} = e^{-i\delta} \braket{\psi}{\Bx}.

\end{equation}

Now expand the wave function in the spherical harmonic basis

\begin{equation}\label{eqn:totallyAsymmetricPotential:220}

\begin{aligned}

\braket{\Bx}{\psi}

&=

\int d\Omega \braket{\Bx}{\ncap}\braket{\ncap}{\psi} \\

&=

\sum_{lm} F_{lm}(r) Y_{lm}(\theta, \phi) \\

&=

e^{-i\delta}

\lr{

\sum_{lm} F_{lm}(r) Y_{lm}(\theta, \phi) }^\conj \\

&=

e^{-i\delta}

\sum_{lm} \lr{ F_{lm}(r)}^\conj Y_{lm}^\conj(\theta, \phi) \\

&=

e^{-i\delta}

\sum_{lm} \lr{ F_{lm}(r)}^\conj (-1)^m Y_{l,-m}(\theta, \phi) \\

&=

e^{-i\delta}

\sum_{lm} \lr{ F_{l,-m}(r)}^\conj (-1)^m Y_{l,m}(\theta, \phi),

\end{aligned}

\end{equation}

so the \( F_{lm} \) functions are constrained by

\begin{equation}\label{eqn:totallyAsymmetricPotential:240}

F_{lm}(r) = e^{-i\delta} \lr{ F_{l,-m}(r)}^\conj (-1)^m.

\end{equation}

# References

[1] Jun John Sakurai and Jim J Napolitano. *Modern quantum mechanics*. Pearson Higher Ed, 2014.

Ragu Kattinakere8 years agowhat do you use for math typesetting?

peeterjoot8 years agoLatex is used for both the blog posts as well as the associated pdfs. See https://peeterjoot.com/latex/ for an overview and example of the markup methods.