In quantum mechanics, given a particular Hamiltonian and an operator with corresponding eigenvalues and eigenvectors given by , the are said to be good quantum numbers if every eigenvector remains an eigenvector of with the same eigenvalue as time evolves.
In other words, the eigenvalues are good quantum numbers if the corresponding operator is a constant of motion. Good quantum numbers are often used to label initial and final states in experiments. For example, in particle colliders:
1. Particles are initially prepared in approximate momentum eigenstates; the particle momentum being a good quantum number for non-interacting particles.
2. The particles are made to collide. At this point, the momentum of each particle is undergoing change and thus the particles’ momenta are not a good quantum number for the interacting particles during the collision.
3. A significant time after the collision, particles are measured in momentum eigenstates. Momentum of each particle has stabilized and is again a good quantum number a long time after the collision.
Theorem: A necessary and sufficient condition for the to be good is that commutes with the Hamiltonian .
A proof of this theorem is given below. Note that the theorem holds even if the spectrum is continuous; the proof is slightly more difficult (but no more illuminating) in that case.
Proof for Discrete Spectrum
We will work in the Heisenberg picture. Let be a hermitian operator and let be a complete, orthonormal basis of eigenstates with eigenvalues . The unitary time-translation operator is and so that . The main observation behind the proof is
If commutes with the hamiltonian then the right side vanishes and we get that the are good quantum numbers. However, if we assume that the are good quantum numbers then the left hand side vanishes for all ; since the are complete, this implies that , which establishes the equivalence.
Ehrenfest Theorem and Good Quantum Numbers
The Ehrenfest Theorem gives the rate of change of the expectation value of operators. It reads as follows:
Commonly occurring operators don't depend explicitly on time. If such operators commute with the Hamiltonian, then their expectation value remains constant with time. Now, if the system is in one of the common eigenstates of the operator (and too), then the system remains in this eigenstate as time progresses. Any measurement of the quantity will give us the eigenvalue (or the good quantum number) associated with the eigenstates in which the particle is. This is actually a statement of conservation in quantum mechanics, and will be elaborated in more detail below.
Conservation in Quantum Mechanics
Case I: Stronger statement of conservation: When the system is in one of the common eigenstates of and
Let be an operator which commutes with the Hamiltonian . This implies that we can have common eigenstates of and . Assume that our system is in one of these common eigenstates. If we measure of , it will definitely yield an eigenvalue of (the good quantum number). Also, it is a well-known result that an eigenstate of the Hamiltonian is a stationary state, which means that even if the system is left to evolve for some time before the measurement is made, it will still yield the same eigenvalue. Therefore, If our system is in a common eigenstate, its eigenvalues of A (good quantum numbers) won't change with time.
Conclusion: If and the system is in a common eigenstate of and , the eigenvalues of (good quantum numbers) don't change with time.
Case II: Weaker statement of conservation: When the system is not in any of the common eigenstates of and
As assumed in case I, . But now the system is not in any of the common eigenstates of and . So the system must be in some linear combination of the basis formed by the common eigenstates of and . When a measurement of is made, it can yield any of the eigenvalues of . And then, if any number of subsequent measurements of are made, they are bound to yield the same result. In this case, a (weaker) statement of conservation holds: Using the Ehrenfest Theorem, doesn't explicitly depend on time:
This says that the expectation value of remains constant in time. When the measurement is made on identical systems again and again, it will generally yield different values, but the expectation value remains constant. This is a weaker conservation condition than the case when our system was a common eigenstate of and : The eigenvalues of are not ensured to remain constant, only its expectation value.