3.5: Time-variation of expectation values - Degeneracy and constants of motion
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The time variation of the expectation value of an operator \(\hat{A}\) which commutes with the Hamiltonian is:
\[\frac{d}{dt} \langle \Phi |\hat{A}| \Phi \rangle = \int d^3 r \frac{d \Phi^*}{dt} \hat{A} \Phi + \Phi^*\hat{A} \frac{d \Phi}{dt} \nonumber\]
but since \(i \hbar \frac{d \Phi}{dt} = \hat{H} \Phi\) and \(−i \hbar \frac{d \Phi^*}{dt} = \hat{H}^* \Phi^* \)
\[−i \hbar \frac{d}{dt} \langle \Phi |\hat{A}| \Phi \rangle = \int (\hat{H}^* \Phi^*\hat{A} \Phi − \Phi^* \hat{A}\hat{H} \Phi )d^3 r = h \Phi |[\hat{H}, \hat{A}]| \Phi \rangle \nonumber\]
Where we also use the fact that \(\hat{H}\) is Hermitian. Thus if \(\hat{H}\) commutes with \(\hat{A} ([\hat{H}, \hat{A}] = 0)\), the expectation value of A is independent of time. It is a conserved quantity.
As we have seen above, if we have degenerate eigenstates of the Hamiltonian, \(\hat{H}\), then there must be some other operator \(\hat{A}\) which commutes with the Hamiltonian for which there are energydegenerate eigenstates with different eigenvalues A. These eigenvalues, A, are then constants of the motion. Moreover, if \(\Phi\) is an eigenfunction of \(\hat{H}\), then \(\hat{A} \Phi\) is also an eigenfunction of \(\hat{H}\).
\[\hat{H} (\hat{A} \Phi ) = \hat{A}\hat{H} \Phi = \hat{A}(E \Phi ) = E(\hat{A} \Phi ) \nonumber\]
There is a three way link between symmetry, degeneracy and conserved quantities.
