Required math: calculus
Required physics: 3-d Schrödinger equation
References: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education Section 4.3.1 & Problem 4.18.
The angular momentum operator in quantum mechanics has three components that are not mutually observable. In the calculation of the eigenvalues of and , we made use of the raising and lowering operators , defined as follows:
We showed that the effect of these operators on an eigenfunction of and is to generate a new eigenfunction with the properties
That is, the new eigenfunction has the same eigenvalue as the original eigenfunction with respect to the operator and an eigenvalue raised or lowered by with respect to .
It is useful to derive a formula which gives the exact operation of the raising and lowering operators on the eigenfunctions. That is, we seek a formula of form
for some constant . It turns out we can find the value of this constant without knowing anything more about the eigenfunctions than that they are normalized.
To do this, we need to show that is the hermitian conjugate of . To be strict about this, we need first to show that and are hermitian. Since they are observables, they have to be, but we can show it explicitly anyway.
To prove that the components of angular momentum are Hermitian, we can use the same technique as that used for proving that linear momentum is Hermitian. The difference here is that matrix elements are integrated over 3 dimensions rather than one. We can see how the proof goes for the case of .
We can consider the first term in this integral, since all other terms in this and the other integrals for the other components work in the same way. Using integration by parts over we get
The first term is zero for the usual reason that any function that is physically meaningful must be normalizable and so must go to zero at infinity. The second term uses the fact that is a constant when taking the derivative with respect to . If we plug this result and the similar one for the second term in the integral back into the original equation, we get
Once we know that and are Hermitian, then it follows that .
Now to get the constant , we have
The second line uses the equation derived earlier:
and the third line uses the eigenvalues of and .
Assuming all the are normalized,
Applying to or to results in being zero, as required.