I am currently self-studying quantum mechanics, and I'm working problems on angular momentum. The problem I'm currently working on asks one to consider a particle subjected to a spherically symmetric potential with wave function given by

\begin{equation*} \psi(\mathbf{x}) = (x+y+3z)f(r). \end{equation*}

It first asks whether or not $\psi$ is an eigenfunction of $\mathbf{L}^{2}$ and if so, for which value of $\ell$. By rewriting

\begin{equation*} \psi(\mathbf{x}) = \left[c_{-1}Y_{1}^{-1}(\theta,\phi) + c_{1}Y_{1}^{1}(\theta,\phi) + c_{0}Y_{1}^{0}(\theta,\phi)\right]rf(r) \end{equation*}


\begin{equation*} c_{-1} = \sqrt{\frac{2\pi}{3}}(1+i),\hspace{1pc}c_{1} = \sqrt{\frac{2\pi}{3}}(i-1),\hspace{1pc}\mbox{ and }\hspace{1pc}c_{0} = 2\sqrt{3\pi}, \end{equation*}

I have determined that $\psi$ is indeed an eigenfunction of $\mathbf{L}^{2}$ corresponding to $\ell = 1$, because for $\mathbf{L}^{2}$, the eigenvalue of $Y_{\ell}^{m}(\theta,\phi)$ is independent of $m$ and so any linear combination of $Y_{\ell}^{m}$ (with $\ell$ fixed) is an eigenfunction of $\mathbf{L}^{2}$.

The problem then asks what the probabilities are that the particle is found in its various $m_{\ell}$ states (i.e., $m_{\ell} = -1,0,1$). The solutions I have for the problem say tersely that the probability of $\psi$ being found in the state $|1,m\rangle$ is given by

\begin{equation*} P(m) = \frac{\left|c_{m}\right|^{2}}{\sum_{m=-1}^{1}{\left|c_{m}\right|^{2}}}. \end{equation*}

I'm not sure how to arrive at this conclusion, though. My thought process is as follows. If the particle is found in the $m_{\ell} = 1$ state, its wavefunction is, upon normalization,

\begin{equation*} \psi^{1}(\theta,\phi) = \tilde{c}_{1}Y_{1}^{1}(\theta,\phi) rf(r). \end{equation*}

Therefore, the probability of the particle being found in this state is

\begin{equation*} \int{\left[\psi^{1}(\theta,\phi)\right]^{\ast}\psi(\theta,\phi)\mathrm{d}^{3}x} = \int_{0}^{\infty}{\int_{0}^{\pi}{\int_{0}^{2\pi}{\tilde{c}_{1}^{\ast}\left[Y_{1}^{1}(\theta,\phi)\right]^{\ast}rf(r)\left[c_{-1}Y_{1}^{-1}(\theta,\phi) + c_{1}Y_{1}^{1}(\theta,\phi) + c_{0}Y_{1}^{0}(\theta,\phi)\right]rf(r)r^{2}\sin{(\theta)}\mathrm{d}\phi}\mathrm{d}\theta}\mathrm{d}r} \end{equation*}

and then to use the orthonormality of the spherical harmonics.

Is this a correct way to think about this problem?

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    $\begingroup$ I don't really understand the close votes. Sure, this is from a homework problem, but essentially the OP asks how to compute probabilities concerning observables which are defined only on a subsystem (here the angular part) - which is completely conceptual! $\endgroup$ Commented May 21, 2023 at 15:43
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    $\begingroup$ @OP: If this is really what you want to ask, you could edit the question to make this more explicit... $\endgroup$ Commented May 22, 2023 at 7:01
  • $\begingroup$ @TobiasFünke this is totally an assignment question. Indeed it is taken from a textbook: I know because I assigned it (although I can’t immediately recall the exact book). If you disagree it’s an assignment question, it qualifies as a check-my-work question (“what mistake am I making”?) $\endgroup$ Commented May 22, 2023 at 12:56
  • $\begingroup$ @ZeroTheHero I disagree here in the sense that the wording of the question makes it sound as a check-my-work question, but I interpret the question differently. The "mistake" OP is doing (the question they are asking) in computing the probabilities is of a conceptual kind (at least how I read the question), so I think this is fine $\endgroup$ Commented May 22, 2023 at 12:59
  • $\begingroup$ @OP Given the above comment, you should really consider to rephrase the question accordingly. I also have removed your last edit, because questions (and answers) should stand for their own and are not suitable for a discussion of this kind. That being said, I think you can safely cut-off the first half of the question by just stating the wave function and the problem ("given $\psi$, I want to compute the probabilities...") and then state the problem you encounter to fulfill this task. No need to mention the other parts of the task, for example. $\endgroup$ Commented May 22, 2023 at 13:02

1 Answer 1


From a more general point of view, we have $L^2(\mathbb R^3)\cong L^2(\Omega)\otimes L^2(\mathbb R_+)$ with $\psi =\varphi\otimes g$, which means $\psi(\mathbf x):= \varphi(\phi,\theta)\, g(r)$. Note that the $\psi$ in the question is indeed a function of this form.

The inner product between any two such (tensor) product functions is simply $$\langle \psi_1,\psi_2\rangle_{L^2(\mathbb R^3)}= \langle \varphi_1,\varphi_2\rangle_{L^2(\Omega)} \langle g_1,g_2\rangle_{L^2(\mathbb R_+)} \quad ,$$

with $$\langle \varphi_1,\varphi_2\rangle_{L^2(\Omega)}:=\int \int\mathrm d\phi\, \mathrm d\theta \sin(\theta)\, \varphi_1(\phi,\theta)\, \varphi_2(\phi,\theta) $$ and $$\langle g_1,g_2\rangle_{L^2(\mathbb R_+)}:=\int\limits_{0}^{\infty}\mathrm dr\, r^2 g_1(r)\, g_2(r) \quad .$$

Now you have to make use of the following general fact (axiom in some sense): Given a bipartite Hilbert space $H=H_1\otimes H_2$, the probability to measure in a normalized state $\psi \in H$ some value $o$ for some observable $O$ on $H_1$ is given by $\langle \psi, P_o \otimes \mathbb I_2 \, \psi\rangle_H$, where $\mathbb I_2$ is the identity on $H_2$ and $P_o$ the projector appearing in the spectral decomposition of $O$ corresponding to the eigenvalue $o$.

This should be explained in any textbook on e.g. quantum information or good book on QM - and after a little bit of thought it also seems rather intuitive.

Thus: The probability to measure the value $m$ (given $\ell$) is, in your case: $$\langle \psi, |\ell, m\rangle\langle \ell, m| \otimes \mathbb I_{L^2(\mathbb R_+)} \,\psi\rangle_{L^2(\mathbb R^3)} = \langle \varphi |\ell,m\rangle\langle \ell,m| \,\varphi\rangle_{L^2(\Omega)}\, \langle g,g\rangle_{L^2(\mathbb R_+)} \quad . $$

The orthonormality of the spherical harmonics now yields that $ \langle \varphi |\ell,m\rangle\langle \ell,m| \,\varphi\rangle_{L^2(\Omega)} = |c_m|^2$.

It seems the task did not normalize $\psi$ before, so you have to divide $\psi$ by $||\psi||=||\varphi|| ||g||$ first, which then should yield the desired result.

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    $\begingroup$ This answers my question and then some--it makes explicit what my current textbook seems nervous to mention for fear of scaring of physicists with too much math. Is it worth editing my question to try to re-open it? I guess the way to avoid getting closed for "Homework-like questions" is just to avoid any mention that your problem originated from a textbook. Also, do you have any recommendations for a quantum mechanics text that treats the mathematics involved more rigorously? $\endgroup$
    – kandb
    Commented May 22, 2023 at 7:22
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    $\begingroup$ Hi @kandb, hmhm whether or not it is worth is up to you. Glad my answer helped. Yes, you're right, many books are (sometimes) sloppy. Regarding the book: It really depends what you are looking for: From a conceptional point of view, I really (really) like Isham's book on Quantum Theory! He explains many things you cannot find in other books, but therefore skips the usual model systems, such as QHO, particle in a box etc. Also, he only refers to rigor at some points, but is is not rigorous himself, but still everything is very clear and formally written down. $\endgroup$ Commented May 22, 2023 at 7:25
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    $\begingroup$ Other than that, I also enjoy Jochen Rau's book: it is really nice because he has a different way to explain QM, more from an operational point of view, and similarly to Isham he formally develops everything very nice, but not as rigorous as a mathematician would - and still there are the common models missing. A completely rigorous book could be one of Valter Moretti's book, but also the books from Hall (2013) or Browers (although I only skimmed through it). I also like the book by Heinosaari and Ziman... $\endgroup$ Commented May 22, 2023 at 7:29
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    $\begingroup$ Long story short: For rigorous books: Moretti or Hall, for example. For conceptional books (but also at a formal mathematical level very clear): Isham. Let me know if you still have questions. Edit: Ballentine is very nice, too! I guess it is a perfect mix between the "standard courses" regarding the examples etc., but at the same time is clear at a mathematical and conceptual level! The same goes for Galindo and Pascual! Hope this helps. $\endgroup$ Commented May 22, 2023 at 7:30
  • $\begingroup$ @kandb if you think people close homework question just because they are tagged as such you are sorely mistaken. I VtC and your question was reopened: good for you but it's still an assignment question (at least as phrased) and should be closed as such. $\endgroup$ Commented May 22, 2023 at 13:17

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