Quantum Mechanics and Momentum
Part 10: Ladder Operators
Last week, we introduced the quantum angular momentum operator L, and found some properties of its magnitude and components. Most notably, that only one component may be fixed at a time, and that component, usually the z component, and the magnitude squared may be fixed simultaneously. So now, let us look at the common eigenfunctions of L2 and Lz. We want eigenfunction Y so that and .
Now, let us define two “ladder operators” L+ and L– by
Let us examine the operation of L+ on our Lz eigenvalue equation:
Now, from the definition of the commutator,
and from our definition of L+,
So , and
which tells us that is also an eigenstate of Lz, with “raised” eigenvalue . Thus, we call L+ the “raising operator.” Repeated application of this shows that for n applications of the raising operator,
Similarly, we can show that gives
and so it is called the “lowering operator.”
Now, let us examine how L2 behaves on these raised an lowered eigenstates. The commutators of L2 with our raising and lowering operators are
and the ladder of eigenvalues …b-2ℏ, b-ℏ, b, b+ℏ b+2ℏ,… generated by the raising and lowering operators are all eigenstates of L2 with eigenvalue a.
We have eigenfunctions , and , with eigenvalues .
Subtracting this from , and recalling that , we find:
Note that the middle term above corresponds to a non-negative physical quantity; thus we expect that cannot be negative: . This, in turn, means that our ladder must be bounded both above and below; there exists and such that and .
So then we have . But
Analogously, using , and , one finds
Taking the difference of these two equations, and thus cancelling a, one finds:
Combine this with the knowledge that all eigenvalues on the ladder are separated by units of ℏ, we see , for some integer n, and so
giving us our eigenvalue ladder .
And as , we find