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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?

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

1 Answer 1

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Just like your previous question about this subject, you've neglected the normalization factor.

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.

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

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