# Is there any missing constraint to deduce that angular momentum is quantized in pure math?

In Susskind's Advanced Quantum Mechanics Lecture 2, he shows the quantization of eigenvalues of $$L_z$$ using the fact that if $$m$$ is an eigenvalue with eigenvector $$\psi$$, then $$L_\pm\psi$$ is also an eigenvector (up to a normalization factor) with eigenvalue $$m\pm 1$$, until being killed. Thus, the eigenvalues of $$L_z$$ must be $$l,l-1,\cdots,-l+1,-l$$ where $$l$$ is a (half-)integer.

In the lecture I believe he used following two assumption:

• The number of eigenvalues is finite, since those states are what physicists care.
• The system is rotational symmetrical.

However, I failed to deduce the spectrum of $$L_z$$ in pure mathematics based on those assumptions.

The first assumption indicates the spectrum has a maximum $$l$$ and minimum $$m$$, which can be any real number.

The second assumption indicates that $$m=-l$$ and the spectrum is reflective symmetrical.

Up to now, we can conclude mathematically that the spectrum contains $$\{l,l-1,\cdots,-l+1,-l\}$$ as a subset. But we still do not eliminate following cases:

$$\{1, 0.7, 0.3, -0.3, -0.7, -1\}$$

I guess $$L_+$$ killing both $$\psi_{0.7}$$ and $$\psi_1$$ may violate some hidden assumption. Could anyone help?

• Are you familiar with the identity $\left[L_{z},L_{\pm}\right]=\pm\hbar L_{\pm}$? Dec 23, 2018 at 17:25
• @eranreches yes, but does it help? Dec 23, 2018 at 17:31

The reason that $$\{-1, 0, 1\}$$ form a set of valid values is because, if you keep trying to raise beyond $$1$$ or lower beyond $$-1$$, the normalization factor will be zero. That is, there is no state beyond $$\pm 1$$.

If you had $$0.3$$, it's easy to check this will never happen: either you will be able to raise upward forever, or lower downward forever, or both. (Try it yourself, by plugging in various values for $$\ell$$.) This is covered in detail in any book on quantum mechanics, such as Griffiths.

In general, I wouldn't recommend using Susskind's lectures as a serious introduction. They're okay for getting the gist, but there's not enough detail given to actually check the mathematical steps properly. If you want to learn the subject, try a book or normal lecture course.

• Got it, thanks! and I planned to try a book after finishing his introductory lectures. BTW, any recommendations? Dec 23, 2018 at 17:30
• @HanXu Griffiths is the most common, Shankar is at a similar level but more mathematically careful. Dec 23, 2018 at 17:51