time reversal

Plane wave and spinor under time reversal

December 16, 2015 phy1520 No comments , , , , ,

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Q: [1] pr 4.7

  1. (a)
    Find the time reversed form of a spinless plane wave state in three dimensions.

  2. (b)
    For the eigenspinor of \( \Bsigma \cdot \ncap \) expressed in terms of polar and azimuthal angles \( \beta\) and \( \gamma \), show that \( -i \sigma_y \chi^\conj(\ncap) \) has the reversed spin direction.

A: part (a)

The Hamiltonian for a plane wave is

\begin{equation}\label{eqn:timeReversalPlaneWaveAndSpinor:20}
H = \frac{\Bp^2}{2m} = i \PD{t}.
\end{equation}

Under time reversal the momentum side transforms as

\begin{equation}\label{eqn:timeReversalPlaneWaveAndSpinor:40}
\begin{aligned}
\Theta \frac{\Bp^2}{2m} \Theta^{-1}
&=
\frac{\lr{ \Theta \Bp \Theta^{-1}} \cdot \lr{ \Theta \Bp \Theta^{-1}} }{2m} \\
&=
\frac{(-\Bp) \cdot (-\Bp)}{2m} \\
&=
\frac{\Bp^2}{2m}.
\end{aligned}
\end{equation}

The time derivative side of the equation is also time reversal invariant
\begin{equation}\label{eqn:timeReversalPlaneWaveAndSpinor:60}
\begin{aligned}
\Theta i \PD{t}{} \Theta^{-1}
&=
\Theta i \Theta^{-1} \Theta \PD{t}{} \Theta^{-1} \\
&=
-i \PD{(-t)}{} \\
&=
i \PD{t}{}.
\end{aligned}
\end{equation}

Solutions to this equation are linear combinations of

\begin{equation}\label{eqn:timeReversalPlaneWaveAndSpinor:80}
\psi(\Bx, t) = e^{i \Bk \cdot \Bx – i E t/\Hbar},
\end{equation}

where \( \Hbar^2 \Bk^2/2m = E \), the energy of the particle. Under time reversal we have

\begin{equation}\label{eqn:timeReversalPlaneWaveAndSpinor:100}
\begin{aligned}
\psi(\Bx, t)
\rightarrow e^{-i \Bk \cdot \Bx + i E (-t)/\Hbar}
&= \lr{ e^{i \Bk \cdot \Bx – i E (-t)/\Hbar} }^\conj \\
&=
\psi^\conj(\Bx, -t)
\end{aligned}
\end{equation}

A: part (b)

The text uses a requirement for time reversal of spin states to show that the Pauli matrix form of the time reversal operator is

\begin{equation}\label{eqn:timeReversalPlaneWaveAndSpinor:120}
\Theta = -i \sigma_y K,
\end{equation}

where \( K \) is a complex conjugating operator. The form of the spin up state used in that demonstration was

\begin{equation}\label{eqn:timeReversalPlaneWaveAndSpinor:140}
\begin{aligned}
\ket{\ncap ; +}
&= e^{-i S_z \beta/\Hbar} e^{-i S_y \gamma/\Hbar} \ket{+} \\
&= e^{-i \sigma_z \beta/2} e^{-i \sigma_y \gamma/2} \ket{+} \\
&= \lr{ \cos(\beta/2) – i \sigma_z \sin(\beta/2) }
\lr{ \cos(\gamma/2) – i \sigma_y \sin(\gamma/2) } \ket{+} \\
&= \lr{ \cos(\beta/2) – i \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix} \sin(\beta/2) }
\lr{ \cos(\gamma/2) – i \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix} \sin(\gamma/2) } \ket{+} \\
&=
\begin{bmatrix}
e^{-i\beta/2} & 0 \\
0 & e^{i \beta/2}
\end{bmatrix}
\begin{bmatrix}
\cos(\gamma/2) & -\sin(\gamma/2) \\
\sin(\gamma/2) & \cos(\gamma/2)
\end{bmatrix}
\begin{bmatrix}
1 \\
0
\end{bmatrix} \\
&=
\begin{bmatrix}
e^{-i\beta/2} & 0 \\
0 & e^{i \beta/2}
\end{bmatrix}
\begin{bmatrix}
\cos(\gamma/2) \\
\sin(\gamma/2) \\
\end{bmatrix} \\
&=
\begin{bmatrix}
\cos(\gamma/2)
e^{-i\beta/2}
\\
\sin(\gamma/2)
e^{i \beta/2}
\end{bmatrix}.
\end{aligned}
\end{equation}

The state orthogonal to this one is claimed to be

\begin{equation}\label{eqn:timeReversalPlaneWaveAndSpinor:180}
\begin{aligned}
\ket{\ncap ; -}
&= e^{-i S_z \beta/\Hbar} e^{-i S_y (\gamma + \pi)/\Hbar} \ket{+} \\
&= e^{-i \sigma_z \beta/2} e^{-i \sigma_y (\gamma + \pi)/2} \ket{+}.
\end{aligned}
\end{equation}

We have

\begin{equation}\label{eqn:timeReversalPlaneWaveAndSpinor:200}
\begin{aligned}
\cos((\gamma + \pi)/2)
&=
\textrm{Re} e^{i(\gamma + \pi)/2} \\
&=
\textrm{Re} i e^{i\gamma/2} \\
&=
-\sin(\gamma/2),
\end{aligned}
\end{equation}

and
\begin{equation}\label{eqn:timeReversalPlaneWaveAndSpinor:220}
\begin{aligned}
\sin((\gamma + \pi)/2)
&=
\textrm{Im} e^{i(\gamma + \pi)/2} \\
&=
\textrm{Im} i e^{i\gamma/2} \\
&=
\cos(\gamma/2),
\end{aligned}
\end{equation}

so we should have

\begin{equation}\label{eqn:timeReversalPlaneWaveAndSpinor:240}
\ket{\ncap ; -}
=
\begin{bmatrix}
-\sin(\gamma/2)
e^{-i\beta/2}
\\
\cos(\gamma/2)
e^{i \beta/2}
\end{bmatrix}.
\end{equation}

This looks right, but we can sanity check orthogonality

\begin{equation}\label{eqn:timeReversalPlaneWaveAndSpinor:260}
\begin{aligned}
\braket{\ncap ; -}{\ncap ; +}
&=
\begin{bmatrix}
-\sin(\gamma/2)
e^{i\beta/2}
&
\cos(\gamma/2)
e^{-i \beta/2}
\end{bmatrix}
\begin{bmatrix}
\cos(\gamma/2)
e^{-i\beta/2}
\\
\sin(\gamma/2)
e^{i \beta/2}
\end{bmatrix} \\
&=
0,
\end{aligned}
\end{equation}

as expected.

The task at hand appears to be the operation on the column representation of \( \ket{\ncap; +} \) using the Pauli representation of the time reversal operator. That is

\begin{equation}\label{eqn:timeReversalPlaneWaveAndSpinor:160}
\begin{aligned}
\Theta \ket{\ncap ; +}
&=
-i \sigma_y K
\begin{bmatrix}
e^{-i\beta/2} \cos(\gamma/2) \\
e^{i \beta/2} \sin(\gamma/2)
\end{bmatrix} \\
&=
-i \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix}
\begin{bmatrix}
e^{i\beta/2} \cos(\gamma/2) \\
e^{-i \beta/2} \sin(\gamma/2)
\end{bmatrix} \\
&=
\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}
\begin{bmatrix}
e^{i\beta/2} \cos(\gamma/2) \\
e^{-i \beta/2} \sin(\gamma/2)
\end{bmatrix} \\
&=
\begin{bmatrix}
-e^{-i \beta/2} \sin(\gamma/2) \\
e^{i\beta/2} \cos(\gamma/2) \\
\end{bmatrix} \\
&= \ket{\ncap ; -},
\end{aligned}
\end{equation}

which is the result to be demononstrated.

References

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

Time reversal behavior of solutions to crystal spin Hamiltonian

December 15, 2015 phy1520 No comments , , , ,

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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
\begin{equation}\label{eqn:crystalSpinHamiltonianTimeReversal:40}
H =
\Hbar^2
\begin{bmatrix}
A+\frac{B}{2} & 0 & \frac{B}{2} \\
-\frac{i B}{\sqrt{2}} & B & -\frac{i B}{\sqrt{2}} \\
\frac{B}{2} & 0 & A+\frac{B}{2} \\
\end{bmatrix}.
\end{equation}

The eigenvalues are
\begin{equation}\label{eqn:crystalSpinHamiltonianTimeReversal:60}
\setlr{ \Hbar^2 A, \Hbar^2 B, \Hbar^2(A + B)},
\end{equation}

and the respective eigenvalues (unnormalized) are

\begin{equation}\label{eqn:crystalSpinHamiltonianTimeReversal:80}
\setlr{
\begin{bmatrix}
-1 \\
0 \\
1
\end{bmatrix},
\begin{bmatrix}
0 \\
1 \\
0
\end{bmatrix},
\begin{bmatrix}
1 \\
-\frac{i \sqrt{2} B}{A} \\
1 \\
\end{bmatrix}
}.
\end{equation}

Under time reversal, the Hamiltonian is

\begin{equation}\label{eqn:crystalSpinHamiltonianTimeReversal:100}
H \rightarrow A (-S_z)^2 + B ( (-S_x)^2 – (-S_y)^2 ) = H,
\end{equation}

so we expect the eigenkets for this Hamiltonian to vary by at most a phase factor. To check this, first recall that the time reversal action on a spin one state is

\begin{equation}\label{eqn:crystalSpinHamiltonianTimeReversal:120}
\Theta \ket{1, m} = (-1)^m \ket{1, -m},
\end{equation}

or

\begin{equation}\label{eqn:crystalSpinHamiltonianTimeReversal:140}
\begin{aligned}
\Theta \ket{1} &= -\ket{-1} \\
\Theta \ket{0} &= \ket{0} \\
\Theta \ket{-1} &= -\ket{1}.
\end{aligned}
\end{equation}

Let’s write the eigenkets respectively as

\begin{equation}\label{eqn:crystalSpinHamiltonianTimeReversal:160}
\begin{aligned}
\ket{A} &= -\ket{1} + \ket{-1} \\
\ket{B} &= \ket{0} \\
\ket{A+B} &= \ket{1} + \ket{-1} – \frac{i \sqrt{2} B}{A} \ket{0}.
\end{aligned}
\end{equation}

Noting that the time reversal operator maps complex numbers onto their conjugates, the time reversed eigenkets are

\begin{equation}\label{eqn:crystalSpinHamiltonianTimeReversal:180}
\begin{aligned}
\ket{A} &\rightarrow \ket{-1} – \ket{-1} = -\ket{A} \\
\ket{B} &\rightarrow \ket{0} = \ket{B} \\
\ket{A+B} &\rightarrow -\ket{1} – \ket{-1} + \frac{i \sqrt{2} B}{A} \ket{0} = -\ket{A+B}.
\end{aligned}
\end{equation}

Up to a sign, the time reversed states match the unreversed states.

References

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

Third update of aggregate notes for phy1520, Graduate Quantum Mechanics.

November 9, 2015 phy1520 1 comment , , , , , , , , , , , , , , , ,

I’ve posted a third update of my aggregate notes for PHY1520H Graduate Quantum Mechanics, taught by Prof. Arun Paramekanti. In addition to what was noted previously, this contains lecture notes up to lecture 13, my solutions for the third problem set, and some additional worked practice problems.

Most of the content was posted individually in the following locations, but those original documents will not be maintained individually any further.

PHY1520H Graduate Quantum Mechanics. Lecture 13: Time reversal (cont.), and angular momentum. Taught by Prof. Arun Paramekanti

November 7, 2015 phy1520 No comments , , , ,

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Disclaimer

Peeter’s lecture notes from class. These may be incoherent and rough.

These are notes for the UofT course PHY1520, Graduate Quantum Mechanics, taught by Prof. Paramekanti, covering \textchapref{{4}}, \textchapref{{3}} [1] content.

Time reversal (cont.)

Given a time reversed state

\begin{equation}\label{eqn:qmLecture13:20}
\ket{\tilde{\Psi}(t)} = \Theta \ket{\Psi(0)}
\end{equation}

which can alternately be written

\begin{equation}\label{eqn:qmLecture13:40}
\Theta^{-1} \ket{\tilde{\Psi}(t)} = \ket{\Psi(-t)} = e^{i \hat{H} t/\Hbar} \ket{\Psi(0)}
\end{equation}

The left hand side can be expanded as the evolution of the state as found at time \( -t \)

\begin{equation}\label{eqn:qmLecture13:60}
\begin{aligned}
\Theta^{-1} \ket{\tilde{\Psi}(t)}
&=
\Theta^{-1} e^{-i \hat{H} t/\Hbar} \ket{\tilde{\Psi}(-t)} \\
&=
\Theta^{-1} e^{-i \hat{H} t/\Hbar} \Theta \ket{\Psi(0)}.
\end{aligned}
\end{equation}

To first order for a small time increment \( \delta t \), we have

\begin{equation}\label{eqn:qmLecture13:80}
\lr{ 1 + i \frac{\hat{H}}{\Hbar} \delta t } \ket{\Psi(0)} =
\Theta^{-1} \lr{ 1 – i \frac{\hat{H}}{\Hbar} \delta t } \Theta \ket{\Psi(0)},
\end{equation}

or

\begin{equation}\label{eqn:qmLecture13:120}
i \frac{\hat{H}}{\Hbar} \delta t \ket{\Psi(0)}
=
\Theta^{-1} (- i) \frac{\hat{H}}{\Hbar} \delta t \Theta \ket{\Psi(0)}.
\end{equation}

Since this holds for any state \( \ket{\Psi(0)} \), the time reversal operator satisfies

\begin{equation}\label{eqn:qmLecture13:140}
i \hat{H}
=
\Theta^{-1} (- i) \hat{H} \Theta.
\end{equation}

Note that the factors of \( i \) have not been canceled on purpose, since we are allowing for the time reversal operator to not necessarily commute with imaginary numbers.

There are two possible solutions

  • If \( \Theta \) is unitary where \( \Theta i = i \Theta \), then

    \begin{equation}\label{eqn:qmLecture13:160}
    \hat{H}
    =
    -\Theta^{-1} \hat{H} \Theta,
    \end{equation}

    or
    \begin{equation}\label{eqn:qmLecture13:180}
    \Theta \hat{H}
    =
    – \hat{H} \Theta.
    \end{equation}

    Consider the implications of this on energy eigenstates
    \begin{equation}\label{eqn:qmLecture13:200}
    \hat{H} \ket{\Psi_n} = E_n \ket{\Psi_n},
    \end{equation}

    \begin{equation}\label{eqn:qmLecture13:220}
    \Theta \hat{H} \ket{\Psi_n} = E_n \Theta \ket{\Psi_n},
    \end{equation}

    but

    \begin{equation}\label{eqn:qmLecture13:240}
    -\hat{H} \Theta \ket{\Psi_n} = E_n \Theta \ket{\Psi_n},
    \end{equation}

    or

    \begin{equation}\label{eqn:qmLecture13:260}
    \hat{H} \lr{ \Theta \ket{\Psi_n}} = -E_n \lr{ \Theta \ket{\Psi_n} }.
    \end{equation}

    This would mean that \( \lr{ \Theta \ket{\Psi_n}} \) is an eigenket of \( \hat{H} \), but with a negative energy eigenvalue.

  • \( \Theta \) is antiunitary, where \( \Theta i = -i \Theta \).

    This time
    \begin{equation}\label{eqn:qmLecture13:280}
    i \hat{H} = i \Theta^{-1} \hat{H} \Theta,
    \end{equation}

    so

    \begin{equation}\label{eqn:qmLecture13:300}
    \Theta \hat{H} = \hat{H} \Theta.
    \end{equation}

    Acting on an energy eigenket, we’ve got

    \begin{equation}\label{eqn:qmLecture13:1400}
    \Theta \hat{H} \ket{\Psi_n}
    =
    E_n \lr{ \Theta \ket{\Psi_n} },
    \end{equation}

    and
    \begin{equation}\label{eqn:qmLecture13:1420}
    \lr{ \hat{H} \Theta } \ket{\Psi_n}
    =
    \hat{H} \lr{ \Theta \ket{\Psi_n} },
    \end{equation}

    so \( \Theta \ket{\Psi_n} \) is an eigenstate with energy \( E_n \).

What properties do we expect from \( \Theta \)?

We expect
\begin{equation}\label{eqn:qmLecture13:320}
\begin{aligned}
\hat{x} &\rightarrow \hat{x} \\
\hat{p} &\rightarrow -\hat{p} \\
\hat{\BL} &\rightarrow -\hat{\BL}
\end{aligned}
\end{equation}

where we have a sign flip in the time dependent momentum operator (and therefore angular momentum), but not for position. If we have

\begin{equation}\label{eqn:qmLecture13:340}
\Theta^{-1} \hat{x} \Theta = \hat{x},
\end{equation}

if that’s true, then how about the momentum operator in the position basis
\begin{equation}\label{eqn:qmLecture13:360}
\begin{aligned}
\Theta^{-1} \hat{p} \Theta
&=
\Theta^{-1} \lr{ -i \Hbar \PD{x}{} } \Theta \\
&=
\Theta^{-1} \lr{ -i \Hbar } \Theta \PD{x}{} \\
&=
i \Hbar \Theta^{-1} \Theta \PD{x}{} \\
&=
-\hat{p}.
\end{aligned}
\end{equation}

How about the \( x,p \) commutator? For that we have

\begin{equation}\label{eqn:qmLecture13:380}
\begin{aligned}
\Theta^{-1} \antisymmetric{\hat{x}}{\hat{p}} \Theta
&=
\Theta^{-1} \lr{ i \Hbar } \Theta \\
&=
-i \Hbar \Theta^{-1} \Theta \\
&=
– \antisymmetric{\hat{x}}{\hat{p}}.
\end{aligned}
\end{equation}

For the the angular momentum operators

\begin{equation}\label{eqn:qmLecture13:420}
\hat{L}_i = \epsilon_{ijk} \hat{r}_j \hat{p}_k,
\end{equation}

the time reversal operator should flip the sign due to its action on \( \hat{p}_k \).

Time reversal acting on spin 1/2 (Fermions). Attempt I.

Consider two spin states \( \ket{\uparrow}, \ket{\downarrow} \). What should the action of the time reversal operator on such a state be? Let’s (incorrectly) start by supposing that the time reversal operator effects are

\begin{equation}\label{eqn:qmLecture13:440}
\begin{aligned}
\Theta \ket{\uparrow} &= \ket{\downarrow} \\
\Theta \ket{\downarrow} &= \ket{\uparrow}.
\end{aligned}
\end{equation}

Given a general state
so that if

\begin{equation}\label{eqn:qmLecture13:740}
\ket{\Psi} = a \ket{\uparrow} + b \ket{\downarrow},
\end{equation}

the action of the time reversal operator would be

\begin{equation}\label{eqn:qmLecture13:760}
\Theta \ket{\Psi} = a^\conj \ket{\downarrow} + b^\conj \ket{\uparrow}.
\end{equation}

That action is:

\begin{equation}\label{eqn:qmLecture13:460}
\begin{aligned}
a \rightarrow b^\conj \\
b \rightarrow a^\conj
\end{aligned}
\end{equation}

Let’s consider whether or not such an action a spin operator with properties

\begin{equation}\label{eqn:qmLecture13:480}
\antisymmetric{\hat{S}_i}{\hat{S}_j} = i \epsilon_{ijk} \hat{S}_k.
\end{equation}

produce the desired inversion of sign

\begin{equation}\label{eqn:qmLecture13:500}
\Theta^{-1} \hat{S}_i \Theta = – \hat{S}_i.
\end{equation}

The expectations of the spin operators (without any application of time reversal) are

\begin{equation}\label{eqn:qmLecture13:1440}
\begin{aligned}
\bra{\Psi} \hat{S}_x \ket{\Psi}
&=
\frac{\Hbar}{2}
\lr{ a^\conj \bra{\uparrow} + b^\conj \bra{\downarrow} }
\sigma_x
\lr{ a \ket{\uparrow} + b \ket{\downarrow} } \\
&=
\frac{\Hbar}{2}
\lr{ a^\conj \bra{\uparrow} + b^\conj \bra{\downarrow} }
\lr{ a \ket{\downarrow} + b \ket{\uparrow} } \\
&=
\frac{\Hbar}{2}
\lr{ a^\conj b + b^\conj a },
\end{aligned}
\end{equation}

\begin{equation}\label{eqn:qmLecture13:1460}
\begin{aligned}
\bra{\Psi} \hat{S}_y \ket{\Psi}
&=
\frac{\Hbar}{2}
\lr{ a^\conj \bra{\uparrow} + b^\conj \bra{\downarrow} }
\sigma_y
\lr{ a \ket{\uparrow} + b \ket{\downarrow} } \\
&=
\frac{i\Hbar}{2}
\lr{ a^\conj \bra{\uparrow} + b^\conj \bra{\downarrow} }
\lr{ a \ket{\downarrow} – b \ket{\uparrow} } \\
&=
\frac{\Hbar}{2 i} \lr{ a^\conj b – b^\conj a },
\end{aligned}
\end{equation}

\begin{equation}\label{eqn:qmLecture13:1480}
\begin{aligned}
\bra{\Psi} \hat{S}_z \ket{\Psi}
&=
\frac{\Hbar}{2}
\lr{ a^\conj \bra{\uparrow} + b^\conj \bra{\downarrow} }
\sigma_z
\lr{ a \ket{\uparrow} – b \ket{\downarrow} } \\
&=
\frac{\Hbar}{2} \lr{ \Abs{a}^2 – \Abs{b}^2 }
\end{aligned}
\end{equation}

The time reversed actions are

\begin{equation}\label{eqn:qmLecture13:1560}
\begin{aligned}
\bra{\Psi} \Theta^{-1} \hat{S}_x \Theta \ket{\Psi}
&=
\frac{\Hbar}{2}
\lr{ a^\conj \bra{\downarrow} + b^\conj \bra{\uparrow} }
\sigma_x
\lr{ a \ket{\downarrow} + b \ket{\uparrow} } \\
&=
\frac{\Hbar}{2}
\lr{ a^\conj \bra{\downarrow} + b^\conj \bra{\uparrow} }
\lr{ a \ket{\uparrow} + b \ket{\downarrow} } \\
&=
\frac{\Hbar}{2}
\lr{ a^\conj b + b^\conj a },
\end{aligned}
\end{equation}

\begin{equation}\label{eqn:qmLecture13:1580}
\begin{aligned}
\bra{\Psi} \Theta^{-1} \hat{S}_y \Theta \ket{\Psi}
&=
\frac{\Hbar}{2}
\lr{ a^\conj \bra{\downarrow} + b^\conj \bra{\uparrow} }
\sigma_y
\lr{ a \ket{\downarrow} + b \ket{\uparrow} } \\
&=
\frac{i\Hbar}{2}
\lr{ a^\conj \bra{\downarrow} + b^\conj \bra{\uparrow} }
\lr{ -a \ket{\uparrow} + b \ket{\downarrow} } \\
&=
\frac{\Hbar}{2 i} \lr{ -a^\conj b + b^\conj a },
\end{aligned}
\end{equation}

\begin{equation}\label{eqn:qmLecture13:1600}
\begin{aligned}
\bra{\Psi} \Theta^{-1} \hat{S}_z \Theta \ket{\Psi}
&=
\frac{\Hbar}{2}
\lr{ a^\conj \bra{\downarrow} + b^\conj \bra{\uparrow} }
\sigma_z
\lr{ a \ket{\downarrow} + b \ket{\uparrow} } \\
&=
\frac{\Hbar}{2}
\lr{ a^\conj \bra{\downarrow} + b^\conj \bra{\uparrow} }
\lr{ -a \ket{\downarrow} + b \ket{\uparrow} } \\
&=
\frac{\Hbar}{2} \lr{ -\Abs{a}^2 + \Abs{b}^2 }
\end{aligned}
\end{equation}

We see that this is not right, because the sign for the x component has not been flipped.

Spin 1/2 (Fermions). Attempt II.

Again assuming

\begin{equation}\label{eqn:qmLecture13:580}
\ket{\Psi} = a \ket{\uparrow} + b \ket{\downarrow},
\end{equation}

now try the action

\begin{equation}\label{eqn:qmLecture13:780}
\Theta \ket{\Psi} = a^\conj \ket{\downarrow} – b^\conj \ket{\uparrow}.
\end{equation}

This is the action:

\begin{equation}\label{eqn:qmLecture13:600}
\begin{aligned}
a \rightarrow -b^\conj \\
b \rightarrow a^\conj
\end{aligned}
\end{equation}

The correct action of time reversal on the basis states (up to a phase choice) is

\begin{equation}\label{eqn:qmLecture13:630}
\boxed{
\begin{aligned}
\Theta \ket{\uparrow} &= \ket{\downarrow} \\
\Theta \ket{\downarrow} &= -\ket{\uparrow} \\
\end{aligned}
}
\end{equation}

Note that acting the time reversal operator twice has the effects

\begin{equation}\label{eqn:qmLecture13:660}
\Theta^2 \ket{\uparrow} = \Theta \ket{\downarrow} = – \ket{\uparrow}
\end{equation}
\begin{equation}\label{eqn:qmLecture13:680}
\Theta^2 \ket{\downarrow} = \Theta (-\ket{\uparrow}) = – \ket{\uparrow}.
\end{equation}

We end up with the same state we started with, but with the opposite sign. This means that as an operator

\begin{equation}\label{eqn:qmLecture13:700}
\boxed{
\Theta^2 = -1.
}
\end{equation}

This is try for half integer particles (Fermions) \( S = 1/2, 3/2, 5/2, \cdots \), but for Bosons with integer spin \( S \).

\begin{equation}\label{eqn:qmLecture13:720}
\boxed{
\Theta^2 = 1.
}
\end{equation}

Kramer’s degeneracy for Spin 1/2 (Fermions)

Suppose we imagine there is state for which the action of the time reversal operator products the same state, just different in phase

\begin{equation}\label{eqn:qmLecture13:800}
\begin{aligned}
\Theta \ket{\Psi_n}
&= \ket{\tilde{\Psi}_n} \\
&= e^{i \delta} \ket{\tilde{\Psi}_n},
\end{aligned}
\end{equation}

then
\begin{equation}\label{eqn:qmLecture13:840}
\begin{aligned}
\Theta^2 \ket{\Psi_n}
&= \Theta e^{i \delta} \ket{\tilde{\Psi}_n} \\
&= e^{i \delta} e^{i \delta} \ket{\tilde{\Psi}_n},
\end{aligned}
\end{equation}

but

\begin{equation}\label{eqn:qmLecture13:860}
\begin{aligned}
\Theta e^{i \delta} \ket{\tilde{\Psi}_n}
&=
e^{-i \delta} \Theta \ket{\tilde{\Psi}_n} \\
&=
e^{-i \delta} e^{i \delta} \ket{\tilde{\Psi}_n} \\
&=
\ket{\tilde{\Psi}_n}
\ne
– \ket{\tilde{\Psi}_n}.
\end{aligned}
\end{equation}

This is a contradiction, so we must have at least a two-fold degeneracy. This is called Kramer’s degeneracy. In the homework we will show that this is not the case for integer spin particles.

Angular momentum

In classical mechanics the (orbital) angular momentum is

\begin{equation}\label{eqn:qmLecture13:880}
\BL = \Br \cross \Bp.
\end{equation}

Here “orbital” is to distinguish from spin angular momentum.

In quantum mechanics, the mapping to operators, in component form, is

\begin{equation}\label{eqn:qmLecture13:900}
\hat{L}_i = \epsilon_{ijk} \hat{r}_j \hat{p}_k.
\end{equation}

These operators do not commute
\begin{equation}\label{eqn:qmLecture13:920}
\antisymmetric{\hat{L}_i}{\hat{L}_j}
=
i \Hbar \epsilon_{ijk} \hat{L}_k.
\end{equation}

which means that we can’t simultaneously determine \( \hat{L}_i \) for all \( i \).

Aside: In quantum mechanics, we define an operator \( \hat{\BV} \) to be a vector operator if

\begin{equation}\label{eqn:qmLecture13:940}
\antisymmetric{\hat{L}_i}{\hatV_j}
=
i \Hbar \epsilon_{ijk} \hatV_k.
\end{equation}

The commutator of the squared angular momentum operator with any \( \hat{L}_i \), say \( \hat{L}_x \) is zero

\begin{equation}\label{eqn:qmLecture13:960}
\begin{aligned}
\antisymmetric{
\hat{L}_x^2 +
\hat{L}_y^2 +
\hat{L}_z^2
}
{\hat{L}_x}
&=
\hat{L}_y \hat{L}_y \hat{L}_x
– \hat{L}_x \hat{L}_y \hat{L}_y
+
\hat{L}_z \hat{L}_z \hat{L}_x
– \hat{L}_x \hat{L}_z \hat{L}_z \\
&=
\hat{L}_y \lr{ \antisymmetric{\hat{L}_y}{\hat{L}_x} + {\hat{L}_x \hat{L}_y} }
-\lr{ \antisymmetric{\hat{L}_x}{\hat{L}_y} + {\hat{L}_y \hat{L}_x} } \hat{L}_y \\
&\quad +\hat{L}_z \lr{ \antisymmetric{\hat{L}_z}{\hat{L}_x} + {\hat{L}_x \hat{L}_z} }
-\lr{ \antisymmetric{\hat{L}_x}{\hat{L}_z} + {\hat{L}_z \hat{L}_x} } \hat{L}_z \\
&=
\hat{L}_y \antisymmetric{\hat{L}_y}{\hat{L}_x}
-\antisymmetric{\hat{L}_x}{\hat{L}_y} \hat{L}_y
+\hat{L}_z \antisymmetric{\hat{L}_z}{\hat{L}_x}
-\antisymmetric{\hat{L}_x}{\hat{L}_z} \hat{L}_z \\
&=
i \Hbar \lr{
-\hat{L}_y \hat{L}_z
– \hat{L}_z \hat{L}_y
+\hat{L}_z \hat{L}_y
+ \hat{L}_y \hat{L}_z
} \\
&=
0.
\end{aligned}
\end{equation}

Suppose we have a state \( \ket{\Psi} \) with a well defined \( \hat{L}_z \) eigenvalue and well defined \( \hat{\BL^2} \) eigenvalue, written as

\begin{equation}\label{eqn:qmLecture13:1000}
\ket{\Psi} = \ket{a, b},
\end{equation}

where the label \( a \) is used for the eigenvalue of \( \hat{\BL}^2 \) and \( b \) labels the eigenvalue of \( \hat{L}_z \). Then

\begin{equation}\label{eqn:qmLecture13:1020}
\begin{aligned}
\hat{\BL}^2 \ket{a , b} &= \Hbar^2 a \ket{a ,b} \\
\hat{L}_z \ket{a , b} &= \Hbar b \ket{a ,b}.
\end{aligned}
\end{equation}

Things aren’t so nice when we act with other angular momentum operators, producing a scrambled mess

\begin{equation}\label{eqn:qmLecture13:1040}
\begin{aligned}
\hat{L}_x \ket{a , b} &= \sum_{a’, b’} \mathcal{A}^x_{a, b, a’, b’} \ket{a’, b’} \\
\hat{L}_y \ket{a , b} &= \sum_{a’, b’} \mathcal{A}^y_{a, b, a’, b’} \ket{a’, b’} \\
\end{aligned}
\end{equation}

With this representation, we have

\begin{equation}\label{eqn:qmLecture13:1060}
\hat{L}_x \hat{\BL}^2 \ket{a, b}
=
\hat{L}_x \Hbar^2 a
\sum_{a’, b’} \mathcal{A}^x_{a, b, a’, b’} \ket{a’, b’}.
\end{equation}

\begin{equation}\label{eqn:qmLecture13:1080}
\hat{\BL}^2 \hat{L}_x \ket{a, b}
=
\Hbar^2
\sum_{a’, b’} a’ \mathcal{A}^x_{a, b, a’, b’} \ket{a’, b’}.
\end{equation}

Since \( \hat{\BL}^2, \hat{L}_x \) commute, we must have

\begin{equation}\label{eqn:qmLecture13:1100}
\mathcal{A}^x_{a, b, a’, b’} = \delta_{a, a’} \mathcal{A}^x_{a’; b, b’},
\end{equation}

and similarly
\begin{equation}\label{eqn:qmLecture13:1120}
\mathcal{A}^y_{a, b, a’, b’} = \delta_{a, a’} \mathcal{A}^y_{a’; b, b’}.
\end{equation}

Simplifying things we can write the action of \( \hat{L}_x, \hat{L}_y \) on the state as

\begin{equation}\label{eqn:qmLecture13:1140}
\begin{aligned}
\hat{L}_x \ket{a , b} &= \sum_{ b’} \mathcal{A}^x_{a; b, b’} \ket{a, b’} \\
\hat{L}_y \ket{a , b} &= \sum_{ b’} \mathcal{A}^y_{a; b, b’} \ket{a, b’} \\
\end{aligned}
\end{equation}

Let’s define
\begin{equation}\label{eqn:qmLecture13:1160}
\begin{aligned}
\hat{L}_{+} &\equiv \hat{L}_x + i \hat{L}_y \\
\hat{L}_{-} &\equiv \hat{L}_x – i \hat{L}_y \\
\end{aligned}
\end{equation}

Because these are sums of \( \hat{L}_x, \hat{L}_y \) they must also commute with \( \hat{\BL}^2 \)

\begin{equation}\label{eqn:qmLecture13:1180}
\antisymmetric{\hat{\BL}^2}{\hat{L}_{\pm}} = 0.
\end{equation}

The commutators with \( \hat{L}_z \) are non-zero

\begin{equation}\label{eqn:qmLecture13:1740}
\begin{aligned}
\antisymmetric{\hat{L}_z}{\hat{L}_{\pm}}
&=
\hat{L}_z \lr{ \hat{L}_x \pm i \hat{L}_y }
– \lr{ \hat{L}_x \pm i \hat{L}_y } \hat{L}_z \\
&=
\antisymmetric{\hat{L}_z}{\hat{L}_x}
\pm i
\antisymmetric{\hat{L}_z}{\hat{L}_y} \\
&=
i \Hbar \lr{
\hat{L}_y \mp i \hat{L}_x
} \\
&=
\Hbar \lr{ i \hat{L}_y \pm \hat{L}_x } \\
&=
\pm \Hbar \lr{ \hat{L}_x \pm i \hat{L}_y } \\
&=
\pm \Hbar \hat{L}_{\pm}.
\end{aligned}
\end{equation}

Explicitly, that is

\begin{equation}\label{eqn:qmLecture13:1220}
\begin{aligned}
\hat{L}_z \hat{L}_{+} – \hat{L}_{+} \hat{L}_z &= \Hbar \hat{L}_{+} \\
\hat{L}_z \hat{L}_{-} – \hat{L}_{-} \hat{L}_z &= -\Hbar \hat{L}_{-}
\end{aligned}
\end{equation}

Now we are set to compute actions of these (assumed) raising and lowering operators on the eigenstate of \( \hat{L}_z, \hat{\BL}^2 \)

\begin{equation}\label{eqn:qmLecture13:1240}
\begin{aligned}
\hat{L}_z \hat{L}_{\pm} \ket{a, b}
&=
\Hbar \hat{L}_{\pm} \ket{a,b} \pm \hat{L}_{\pm} \hat{L}_z \ket{a,b} \\
&=
\Hbar \hat{L}_{\pm} \ket{a,b} \pm \Hbar b \hat{L}_{\pm} \ket{a,b} \\
&=
\Hbar \lr{ b \pm 1 } \hat{L}_{\pm} \ket{a, b} .
\end{aligned}
\end{equation}

There must be a proportionality of the form

\begin{equation}\label{eqn:qmLecture13:1260}
\ket{\hat{L}_{\pm}} \propto \ket{a, b \pm 1},
\end{equation}

The products of the raising and lowering operators are

\begin{equation}\label{eqn:qmLecture13:1280}
\begin{aligned}
\hat{L}_{-} \hat{L}_{+}
&=
\lr{ \hat{L}_x – i \hat{L}_y }
\lr{ \hat{L}_x + i \hat{L}_y } \\
&=
\hat{L}_x^2 + \hat{L}_y^2 + i \hat{L}_x \hat{L}_y – i \hat{L}_y \hat{L}_x \\
&=
\lr{ \hat{\BL}^2 – \hat{L}_z^2 } + i \antisymmetric{\hat{L}_x}{\hat{L}_y} \\
&=
\hat{\BL}^2 – \hat{L}_z^2 – \Hbar \hat{L}_z,
\end{aligned}
\end{equation}

and
\begin{equation}\label{eqn:qmLecture13:1300}
\begin{aligned}
\hat{L}_{+} \hat{L}_{-}
&=
\lr{ \hat{L}_x + i \hat{L}_y }
\lr{ \hat{L}_x – i \hat{L}_y } \\
&=
\hat{L}_x^2 + \hat{L}_y^2 – i \hat{L}_x \hat{L}_y + i \hat{L}_y \hat{L}_x \\
&=
\lr{ \hat{\BL}^2 – \hat{L}_z^2 } – i \antisymmetric{\hat{L}_x}{\hat{L}_y} \\
&=
\hat{\BL}^2 – \hat{L}_z^2 + \Hbar \hat{L}_z,
\end{aligned}
\end{equation}

So we must have

\begin{equation}\label{eqn:qmLecture13:1320}
\begin{aligned}
0
&\le \bra{a, b} \hat{L}_{-} \hat{L}_{+} \ket{a, b} \\
&=
\bra{a, b}
\lr{ \hat{\BL}^2 – \hat{L}_z^2 – \Hbar \hat{L}_z }
\ket{a, b} \\
&=
\Hbar^2 a – \Hbar^2 b^2 – \Hbar^2 b,
\end{aligned}
\end{equation}

and

\begin{equation}\label{eqn:qmLecture13:1340}
\begin{aligned}
0
&\le \bra{a, b} \hat{L}_{+} \hat{L}_{-} \ket{a, b} \\
&=
\bra{a, b}
\lr{ \hat{\BL}^2 – \hat{L}_z^2 + \Hbar \hat{L}_z }
\ket{a, b} \\
&=
\Hbar^2 a – \Hbar^2 b^2 + \Hbar^2 b.
\end{aligned}
\end{equation}

This puts constraints on \( a, b \), roughly of the form

  1. \begin{equation}\label{eqn:qmLecture13:1360}
    a – b( b + 1) \ge 0
    \end{equation}

    With \( b_{\textrm{max}} > 0 \), \( b_{\textrm{max}} \approx \sqrt{a} \).

  2. \begin{equation}\label{eqn:qmLecture13:1380}
    a – b( b – 1) \ge 0
    \end{equation}

    With \( b_{\textrm{min}} < 0 \), \( b_{\textrm{max}} \approx -\sqrt{a} \).

Question: Angular momentum commutators

Using \( \hat{L}_i = \epsilon_{ijk} \hat{r}_j \hat{p}_k \), show that

\begin{equation}\label{eqn:qmLecture13:1620}
\antisymmetric{\hat{L}_i}{\hat{L}_j} = i \Hbar \epsilon_{ijk} \hat{L}_k
\end{equation}

Answer

Let’s start without using abstract index expressions, computing the commutator for \( \hat{L}_1, \hat{L}_2 \), which should show the basic steps required

\begin{equation}\label{eqn:qmLecture13:1640}
\begin{aligned}
\antisymmetric{\hat{L}_1}{\hat{L}_2}
&=
\antisymmetric{\hat{r}_2 \hat{p}_3 – \hat{r}_3 \hat{p}_2}{\hat{r}_3 \hat{p}_1
– \hat{r}_1 \hat{p}_3} \\
&=
\antisymmetric{\hat{r}_2 \hat{p}_3}{\hat{r}_3 \hat{p}_1}
-\antisymmetric{\hat{r}_2 \hat{p}_3}{\hat{r}_1 \hat{p}_3}
-\antisymmetric{\hat{r}_3 \hat{p}_2}{\hat{r}_3 \hat{p}_1}
+\antisymmetric{\hat{r}_3 \hat{p}_2}{\hat{r}_1 \hat{p}_3}.
\end{aligned}
\end{equation}

The first of these commutators is

\begin{equation}\label{eqn:qmLecture13:1660}
\begin{aligned}
\antisymmetric{\hat{r}_2 \hat{p}_3}{\hat{r}_3 \hat{p}_1}
&=
{\hat{r}_2 \hat{p}_3}{\hat{r}_3 \hat{p}_1}

{\hat{r}_3 \hat{p}_1}
{\hat{r}_2 \hat{p}_3} \\
&=
\hat{r}_2 \hat{p}_1 \antisymmetric{\hat{p}_3}{\hat{r}_3} \\
&=
-i \Hbar \hat{r}_2 \hat{p}_1.
\end{aligned}
\end{equation}

We see that any factors in the commutator don’t have like indexes (i.e. \( \hat{r}_k, \hat{p}_k \)) on both position and momentum terms, can be pulled out of the commutator. This leaves

\begin{equation}\label{eqn:qmLecture13:1680}
\begin{aligned}
\antisymmetric{\hat{L}_1}{\hat{L}_2}
&=
\hat{r}_2 \hat{p}_1 \antisymmetric{\hat{p}_3}{\hat{r}_3}
-{\antisymmetric{\hat{r}_2 \hat{p}_3}{\hat{r}_1 \hat{p}_3}}
-{\antisymmetric{\hat{r}_3 \hat{p}_2}{\hat{r}_3 \hat{p}_1}}
+\hat{r}_1 \hat{p}_2 \antisymmetric{\hat{r}_3}{\hat{p}_3} \\
&=
i \Hbar \lr{ \hat{r}_1 \hat{p}_2 – \hat{r}_2 \hat{p}_1 } \\
&=
i \Hbar \hat{L}_3.
\end{aligned}
\end{equation}

With cyclic permutation this is really enough to consider \ref{eqn:qmLecture13:1620} proven. However, can we do this in the general case with the abstract index expression? The quantity to simplify looks forbidding

\begin{equation}\label{eqn:qmLecture13:1700}
\antisymmetric{\hat{L}_i}{\hat{L}_j}
=
\epsilon_{i a b }
\epsilon_{j s t }
\antisymmetric{ \hat{r}_a \hat{p}_b }{ \hat{r}_s \hat{p}_t }
\end{equation}

Because there are no repeated indexes, this doesn’t submit to any of the normal reduction identities. Note however, since we only care about the \( i \ne j \) case, that one of the indexes \( a, b \) must be \( j \) for this quantity to be non-zero. Therefore (for \( i \ne j \))

\begin{equation}\label{eqn:qmLecture13:1720}
\begin{aligned}
\antisymmetric{\hat{L}_i}{\hat{L}_j}
&=
\epsilon_{i j b }
\epsilon_{j s t }
\antisymmetric{ \hat{r}_j \hat{p}_b }{ \hat{r}_s \hat{p}_t }
+
\epsilon_{i a j }
\epsilon_{j s t }
\antisymmetric{ \hat{r}_a \hat{p}_j }{ \hat{r}_s \hat{p}_t } \\
&=
\epsilon_{i j b }
\epsilon_{j s t }
\lr{
\antisymmetric{ \hat{r}_j \hat{p}_b }{ \hat{r}_s \hat{p}_t }

\antisymmetric{ \hat{r}_b \hat{p}_j }{ \hat{r}_s \hat{p}_t }
} \\
&=
-\delta^{s t}_{[i b]}
\antisymmetric{ \hat{r}_j \hat{p}_b – \hat{r}_b \hat{p}_j }{ \hat{r}_s
\hat{p}_t } \\
&=
\antisymmetric{ \hat{r}_j \hat{p}_b – \hat{r}_b \hat{p}_j }{ \hat{r}_b
\hat{p}_i – \hat{r}_i \hat{p}_b } \\
&=
\antisymmetric{ \hat{r}_j \hat{p}_b }{ \hat{r}_b \hat{p}_i }
– {\antisymmetric{ \hat{r}_j \hat{p}_b }{ \hat{r}_i \hat{p}_b }}
– {\antisymmetric{ \hat{r}_b \hat{p}_j }{ \hat{r}_b \hat{p}_i }}
+ \antisymmetric{ \hat{r}_b \hat{p}_j }{ \hat{r}_i \hat{p}_b } \\
&=
\hat{r}_j \hat{p}_i \antisymmetric{ \hat{p}_b }{ \hat{r}_b }
+ \hat{r}_i \hat{p}_j \antisymmetric{ \hat{r}_b }{ \hat{p}_b } \\
&=
i \Hbar \lr{ \hat{r}_i \hat{p}_j – \hat{r}_j \hat{p}_i } \\
&=
i \Hbar \epsilon_{i j k} \hat{r}_i \hat{p}_j .
\end{aligned}
\end{equation}

References

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

PHY1520H Graduate Quantum Mechanics. Lecture 12: Symmetry (cont.). Taught by Prof. Arun Paramekanti

November 5, 2015 phy1520 No comments , , , , , , ,

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

Disclaimer

Peeter’s lecture notes from class. These may be incoherent and rough.

These are notes for the UofT course PHY1520, Graduate Quantum Mechanics, taught by Prof. Paramekanti, covering chap. 4 content from [1].

Parity (review)

\begin{equation}\label{eqn:qmLecture12:20}
\hat{\Pi} \hat{x} \hat{\Pi} = – \hat{x}
\end{equation}
\begin{equation}\label{eqn:qmLecture12:40}
\hat{\Pi} \hat{p} \hat{\Pi} = – \hat{p}
\end{equation}

These are polar vectors, in contrast to an axial vector such as \( \BL = \Br \cross \Bp \).

\begin{equation}\label{eqn:qmLecture12:60}
\hat{\Pi}^2 = 1
\end{equation}

\begin{equation}\label{eqn:qmLecture12:80}
\Psi(x) \rightarrow \Psi(-x)
\end{equation}

If \( \antisymmetric{\hat{\Pi}}{\hat{H}} = 0 \) then all the eigenstates are either

  • even: \( \hat{\Pi} \) eigenvalue is \( + 1 \).
  • odd: \( \hat{\Pi} \) eigenvalue is \( – 1 \).

We are done with discrete symmetry operators for now.

Translations

Define a (continuous) translation operator

\begin{equation}\label{eqn:qmLecture12:100}
\hat{T}_\epsilon \ket{x} = \ket{x + \epsilon}
\end{equation}

The action of this operator is sketched in fig. 1.

lecture12Fig1

fig. 1. Translation operation.

 

This is a unitary operator

\begin{equation}\label{eqn:qmLecture12:120}
\hat{T}_{-\epsilon} = \hat{T}_{\epsilon}^\dagger = \hat{T}_{\epsilon}^{-1}
\end{equation}

In a position basis, the action of this operator is

\begin{equation}\label{eqn:qmLecture12:140}
\bra{x} \hat{T}_{\epsilon} \ket{\psi} = \braket{x-\epsilon}{\psi} = \psi(x – \epsilon)
\end{equation}

\begin{equation}\label{eqn:qmLecture12:160}
\Psi(x – \epsilon) \approx \Psi(x) – \epsilon \PD{x}{\Psi}
\end{equation}

\begin{equation}\label{eqn:qmLecture12:180}
\bra{x} \hat{T}_{\epsilon} \ket{\Psi}
= \braket{x}{\Psi} – \frac{\epsilon}{\Hbar} \bra{ x} i \hat{p} \ket{\Psi}
\end{equation}

\begin{equation}\label{eqn:qmLecture12:200}
\hat{T}_{\epsilon} \approx \lr{ 1 – i \frac{\epsilon}{\Hbar} \hat{p} }
\end{equation}

A non-infinitesimal translation can be composed of many small translations, as sketched in fig. 2.

fig. 2. Composition of small translations

fig. 2. Composition of small translations

For \( \epsilon \rightarrow 0, N \rightarrow \infty, N \epsilon = a \), the total translation operator is

\begin{equation}\label{eqn:qmLecture12:220}
\begin{aligned}
\hat{T}_{a}
&= \hat{T}_{\epsilon}^N \\
&= \lim_{\epsilon \rightarrow 0, N \rightarrow \infty, N \epsilon = a }
\lr{ 1 – \frac{\epsilon}{\Hbar} \hat{p} }^N \\
&= e^{-i a \hat{p}/\Hbar}
\end{aligned}
\end{equation}

The momentum \( \hat{p} \) is called a “Generator” generator of translations. If a Hamiltonian \( H \) is translationally invariant, then

\begin{equation}\label{eqn:qmLecture12:240}
\antisymmetric{\hat{T}_{a}}{H} = 0, \qquad \forall a.
\end{equation}

This means that momentum will be a good quantum number

\begin{equation}\label{eqn:qmLecture12:260}
\antisymmetric{\hat{p}}{H} = 0.
\end{equation}

Rotations

Rotations form a non-Abelian group, since the order of rotations \( \hatR_1 \hatR_2 \ne \hatR_2 \hatR_1 \).

Given a rotation acting on a ket

\begin{equation}\label{eqn:qmLecture12:280}
\hatR \ket{\Br} = \ket{R \Br},
\end{equation}

observe that the action of the rotation operator on a wave function is inverted

\begin{equation}\label{eqn:qmLecture12:300}
\bra{\Br} \hatR \ket{\Psi}
=
\bra{R^{-1} \Br} \ket{\Psi}
= \Psi(R^{-1} \Br).
\end{equation}

Example: Z axis normal rotation

Consider an infinitesimal rotation about the z-axis as sketched in fig. 3(a),(b)

lecture12Fig3

fig 3(a). Rotation about z-axis.

fig 3(b). Rotation about z-axis.

fig 3(b). Rotation about z-axis.

\begin{equation}\label{eqn:qmLecture12:320}
\begin{aligned}
x’ &= x – \epsilon y \\
y’ &= y + \epsilon y \\
z’ &= z
\end{aligned}
\end{equation}

The rotated wave function is

\begin{equation}\label{eqn:qmLecture12:340}
\tilde{\Psi}(x,y,z)
= \Psi( x + \epsilon y, y – \epsilon x, z )
=
\Psi( x, y, z )
+
\epsilon y \underbrace{\PD{x}{\Psi}}_{i \hat{p}_x/\Hbar}

\epsilon x \underbrace{\PD{y}{\Psi}}_{i \hat{p}_y/\Hbar}.
\end{equation}

The state must then transform as

\begin{equation}\label{eqn:qmLecture12:360}
\ket{\tilde{\Psi}}
=
\lr{
1
+ i \frac{\epsilon}{\Hbar} \hat{y} \hat{p}_x
– i \frac{\epsilon}{\Hbar} \hat{x} \hat{p}_y
}
\ket{\Psi}.
\end{equation}

Observe that the combination \( \hat{x} \hat{p}_y – \hat{y} \hat{p}_x \) is the \( \hat{L}_z \) component of angular momentum \( \hat{\BL} = \hat{\Br} \cross \hat{\Bp} \), so the infinitesimal rotation can be written

\begin{equation}\label{eqn:qmLecture12:380}
\boxed{
\hatR_z(\epsilon) \ket{\Psi}
=
\lr{ 1 – i \frac{\epsilon}{\Hbar} \hat{L}_z } \ket{\Psi}.
}
\end{equation}

For a finite rotation \( \epsilon \rightarrow 0, N \rightarrow \infty, \phi = \epsilon N \), the total rotation is

\begin{equation}\label{eqn:qmLecture12:420}
\hatR_z(\phi)
=
\lr{ 1 – \frac{i \epsilon}{\Hbar} \hat{L}_z }^N,
\end{equation}

or
\begin{equation}\label{eqn:qmLecture12:440}
\boxed{
\hatR_z(\phi)
=
e^{-i \frac{\phi}{\Hbar} \hat{L}_z}.
}
\end{equation}

Note that \( \antisymmetric{\hat{L}_x}{\hat{L}_y} \ne 0 \).

By construction using Euler angles or any other method, a general rotation will include contributions from components of all the angular momentum operator, and will have the structure

\begin{equation}\label{eqn:qmLecture12:480}
\boxed{
\hatR_\ncap(\phi)
=
e^{-i \frac{\phi}{\Hbar} \lr{ \hat{\BL} \cdot \ncap }}.
}
\end{equation}

Rotationally invariant \( \hat{H} \).

Given a rotationally invariant Hamiltonian

\begin{equation}\label{eqn:qmLecture12:520}
\antisymmetric{\hat{R}_\ncap(\phi)}{\hat{H}} = 0 \qquad \forall \ncap, \phi,
\end{equation}

then every

\begin{equation}\label{eqn:qmLecture12:540}
\antisymmetric{\BL \cdot \ncap}{\hat{H}} = 0,
\end{equation}

or
\begin{equation}\label{eqn:qmLecture12:560}
\antisymmetric{L_i}{\hat{H}} = 0,
\end{equation}

Non-Abelian implies degeneracies in the spectrum.

Time-reversal

Imagine that we have something moving along a curve at time \( t = 0 \), and ending up at the final position at time \( t = t_f \).

fig. 4. Time reversal trajectory.

fig. 4. Time reversal trajectory.

Imagine that we flip the direction of motion (i.e. flipping the velocity) and run time backwards so the final-time state becomes the initial state.

If the time reversal operator is designated \( \hat{\Theta} \), with operation

\begin{equation}\label{eqn:qmLecture12:580}
\hat{\Theta} \ket{\Psi} = \ket{\tilde{\Psi}},
\end{equation}

so that

\begin{equation}\label{eqn:qmLecture12:600}
\hat{\Theta}^{-1} e^{-i \hat{H} t/\Hbar} \hat{\Theta} \ket{\Psi(t)} = \ket{\Psi(0)},
\end{equation}

or

\begin{equation}\label{eqn:qmLecture12:620}
\hat{\Theta}^{-1} e^{-i \hat{H} t/\Hbar} \hat{\Theta} \ket{\Psi(0)} = \ket{\Psi(-t)}.
\end{equation}

References

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