[Click here for a PDF of an older version of post with nicer formatting]. Updates will be made in my old grad quantum notes.
In problem 1.17 of [2] we are to show that non-commuting operators that both commute with the Hamiltonian, have, in general, degenerate energy eigenvalues. That is
\begin{equation}\label{eqn:angularMomentumAndCentralForceCommutators:320} [A,H] = [B,H] = 0, \end{equation}
but
\begin{equation}\label{eqn:angularMomentumAndCentralForceCommutators:340} [A,B] \ne 0. \end{equation}
Matrix example of non-commuting commutators
I thought perhaps the problem at hand would be easier if I were to construct some example matrices representing operators that did not commute, but did commuted with a Hamiltonian. I came up with
\begin{equation}\label{eqn:angularMomentumAndCentralForceCommutators:360} \begin{aligned} A &= \begin{bmatrix} \sigma_z & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} \\ B &= \begin{bmatrix} \sigma_x & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} \\ H &= \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} \end{aligned} \end{equation}
This system has \antisymmetric{A}{H} = \antisymmetric{B}{H} = 0 , and
\begin{equation}\label{eqn:angularMomentumAndCentralForceCommutators:380} \antisymmetric{A}{B} = \begin{bmatrix} 0 & 2 & 0 \\ -2 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \end{equation}
There is one shared eigenvector between all of A, B, H
\begin{equation}\label{eqn:angularMomentumAndCentralForceCommutators:400} \ket{3} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}. \end{equation}
The other eigenvectors for A are
\begin{equation}\label{eqn:angularMomentumAndCentralForceCommutators:420}
\begin{aligned}
\ket{a_1} &=
\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix} \\
\ket{a_2} &=
\begin{bmatrix}
0 \\
1 \\
0
\end{bmatrix},
\end{aligned}
\end{equation}
and for B
\begin{equation}\label{eqn:angularMomentumAndCentralForceCommutators:440}
\begin{aligned}
\ket{b_1} &=
\inv{\sqrt{2}}
\begin{bmatrix}
1 \\
1 \\
0
\end{bmatrix} \\
\ket{b_2} &=
\inv{\sqrt{2}}
\begin{bmatrix}
1 \\
-1 \\
0
\end{bmatrix},
\end{aligned}
\end{equation}
This clearly has the degeneracy sought.
Looking to [1], it appears that it is possible to construct an even simpler example. Let
\begin{equation}\label{eqn:angularMomentumAndCentralForceCommutators:460} \begin{aligned} A &= \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \\ B &= \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \\ H &= \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}. \end{aligned} \end{equation}
Here \antisymmetric{A}{B} = -A , and \antisymmetric{A}{H} = \antisymmetric{B}{H} = 0 , but the Hamiltonian isn’t interesting at all physically.
A less boring example builds on this. Let
\begin{equation}\label{eqn:angularMomentumAndCentralForceCommutators:480} \begin{aligned} A &= \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \\ B &= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \\ H &= \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}. \end{aligned} \end{equation}
Here \antisymmetric{A}{B} \ne 0 , and \antisymmetric{A}{H} = \antisymmetric{B}{H} = 0 . I don’t see a way for any exception to be constructed.
The problem
The concrete examples above give some intuition for solving the more abstract problem. Suppose that we are working in a basis that simultaneously diagonalizes operator A and the Hamiltonian H . To make life easy consider the simplest case where this basis is also an eigenbasis for the second operator B for all but two of that operators eigenvectors. For such a system let’s write
\begin{equation}\label{eqn:angularMomentumAndCentralForceCommutators:160}
\begin{aligned}
H \ket{1} &= \epsilon_1 \ket{1} \\
H \ket{2} &= \epsilon_2 \ket{2} \\
A \ket{1} &= a_1 \ket{1} \\
A \ket{2} &= a_2 \ket{2},
\end{aligned}
\end{equation}
where \ket{1}, and \ket{2} are not eigenkets of B . Because B also commutes with H , we must have
\begin{equation}\label{eqn:angularMomentumAndCentralForceCommutators:180} H B \ket{1} = H \sum_n \ket{n}\bra{n} B \ket{1} = \sum_n \epsilon_n \ket{n} B_{n 1}, \end{equation}
and
\begin{equation}\label{eqn:angularMomentumAndCentralForceCommutators:200}
B H \ket{1}
= B \epsilon_1 \ket{1}
= \epsilon_1 \sum_n \ket{n}\bra{n} B \ket{1}
= \epsilon_1 \sum_n \ket{n} B_{n 1}.
\end{equation}
We can now compute the action of the commutators on \ket{1}, \ket{2} ,
\begin{equation}\label{eqn:angularMomentumAndCentralForceCommutators:220}
\antisymmetric{B}{H} \ket{1}
=
\sum_n \lr{ \epsilon_1 – \epsilon_n } \ket{n} B_{n 1}.
\end{equation}
Similarly
\begin{equation}\label{eqn:angularMomentumAndCentralForceCommutators:240}
\antisymmetric{B}{H} \ket{2}
=
\sum_n \lr{ \epsilon_2 – \epsilon_n } \ket{n} B_{n 2}.
\end{equation}
However, for those kets \ket{m} \in \setlr{ \ket{3}, \ket{4}, \cdots } that are eigenkets of B , with B \ket{m} = b_m \ket{m} , we have
\begin{equation}\label{eqn:angularMomentumAndCentralForceCommutators:280} \antisymmetric{B}{H} \ket{m} = B \epsilon_m \ket{m} – H b_m \ket{m} = b_m \epsilon_m \ket{m} – \epsilon_m b_m \ket{m} = 0, \end{equation}
The sums in \ref{eqn:angularMomentumAndCentralForceCommutators:220} and \ref{eqn:angularMomentumAndCentralForceCommutators:240} reduce to
\begin{equation}\label{eqn:angularMomentumAndCentralForceCommutators:500}
\antisymmetric{B}{H} \ket{1}
=
\sum_{n=1}^2 \lr{ \epsilon_1 – \epsilon_n } \ket{n} B_{n 1}
=
\lr{ \epsilon_1 – \epsilon_2 } \ket{2} B_{2 1},
\end{equation}
and
\begin{equation}\label{eqn:angularMomentumAndCentralForceCommutators:520}
\antisymmetric{B}{H} \ket{2}
=
\sum_{n=1}^2 \lr{ \epsilon_2 – \epsilon_n } \ket{n} B_{n 2}
=
\lr{ \epsilon_2 – \epsilon_1 } \ket{1} B_{1 2}.
\end{equation}
Since the commutator is zero, the matrix elements of the commutator must all be zero, in particular
\begin{equation}\label{eqn:angularMomentumAndCentralForceCommutators:260}
\begin{aligned}
\bra{1} \antisymmetric{B}{H} \ket{1} &= \lr{ \epsilon_1 – \epsilon_2 } B_{2 1} \braket{1}{2} = 0 \\
\bra{2} \antisymmetric{B}{H} \ket{1} &= \lr{ \epsilon_1 – \epsilon_2 } B_{2 1} \braket{1}{1} \\
\bra{1} \antisymmetric{B}{H} \ket{2} &= \lr{ \epsilon_2 – \epsilon_1 } B_{1 2} \braket{1}{2} = 0 \\
\bra{2} \antisymmetric{B}{H} \ket{2} &= \lr{ \epsilon_2 – \epsilon_1 } B_{1 2} \braket{2}{2}.
\end{aligned}
\end{equation}
We must either have
- B_{2 1} = B_{1 2} = 0 , or
- \epsilon_1 = \epsilon_2 .
If the first condition were true we would have
\begin{equation}\label{eqn:angularMomentumAndCentralForceCommutators:300} B \ket{1} = \ket{n}\bra{n} B \ket{1} = \ket{n} B_{n 1} = \ket{1} B_{1 1}, \end{equation}
and B \ket{2} = B_{2 2} \ket{2} . This contradicts the requirement that \ket{1}, \ket{2} not be eigenkets of B , leaving only the second option. That second option means there must be a degeneracy in the system.
References
[1] Ronald M. Aarts. Commuting Matrices, 2015. URL http://mathworld.wolfram.com/CommutingMatrices.html. [Online; accessed 22-Oct-2015].
[2] Jun John Sakurai and Jim J Napolitano. Modern quantum mechanics. Pearson Higher Ed, 2014.