In quantum mechanics, the eigenvalue
is said to be a good quantum number if the observable
In other words, the quantum number is good if the corresponding observable commutes with the Hamiltonian.
If the system starts from the eigenstate with an eigenvalue
, it remains on that state as the system evolves in time, and the measurement of
[1] Good quantum numbers are often used to label initial and final states in experiments.
For example, in particle colliders:[citation needed] Let
[2] Assume that our system is in one of these common eigenstates.
Also, it is a well-known result that an eigenstate of the Hamiltonian is a stationary state,[3] 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.
[4] Therefore, If our system is in a common eigenstate, its eigenvalues of
(good quantum numbers) won't change with time.
States which can be labelled by good quantum numbers are eigenstates of the Hamiltonian.
[5] They are so called because the system remains in the same state as time elapses, in every observable way.
The evolution of the state ket is governed by the Schrödinger equation: It gives the time evolution of the state of the system as: The time evolution only involves a steady change of a complex phase factor, which can't be observed.
In the case of the hydrogen atom (with the assumption that there is no spin-orbit coupling), the observables that commute with Hamiltonian are the orbital angular momentum, spin angular momentum, the sum of the spin angular momentum and orbital angular momentum, and the
Thus, the good quantum numbers in this case, (which are the eigenvalues of these observables) are
, since it always is constant for an electron and carries no significance as far the labeling of states is concerned.
However, all the good quantum numbers in the above case of the hydrogen atom (with negligible spin-orbit coupling), namely
can't be used simultaneously to specify a state.
Here is when CSCO (Complete set of commuting observables) comes into play.
don't form a commuting set.
So, are in this case, they form a set of good quantum numbers.
too form a set of good quantum numbers.
To take the spin-orbit interaction is taken into account, we have to add an extra term in Hamiltonian[7] where the prefactor
determines the strength of the spin-orbit coupling.
, which is the total angular momentum operator.
are no longer good quantum numbers, but
are (in addition to the principal quantum number
And since, good quantum numbers are used to label the eigenstates, the relevant formulae of interest are expressed in terms of them.
[dubious – discuss] For example, the expectation value of the spin-orbit interaction energy is given by[8] where The above expressions contain the good quantum numbers characterizing the eigenstate.