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Here is the partial differential equation for which the solutions Y are are the wave functions for the rigid-rotator (Equation 5.5 in McQuarrie and Simon's Physical Chemistry).

$$\sin\theta\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial Y}{\partial \theta}\right) + \frac{\partial^2 Y}{\partial \phi^2} + \left(\beta \sin^2\theta\right)Y = 0$$

In McQuarrie and Simon's Physical Chemistry, it is said that

When we solve Equation 5.5, it turns out naturally that $\beta$ ... must obey the condition: $$\beta = J(J+1)$$

My question is why must $\boldsymbol{\beta}$ take the form $\boldsymbol{J(J+1)}$ (where $\boldsymbol{J}$ is a natural number))? Since $\beta$ is essentially a constant, this implies that using any value of $\beta$ (e.g., 1.5) would lead to a partial differential equation with no solutions. (1) is this assumption correct and (2) if so, what is the mathematical proof that $\beta$ in any other form would lead to no solutions?

orthocresol
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Eli Jones
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    $\beta$ must take this form because any other value of $\beta$ would lead to a partial differential equation with no *nice* solutions. As for the mathematical proof, it can be neither told nor heard without doing quite a bit of math. – Ivan Neretin Jan 25 '23 at 21:00
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    I guess, in the linked question I only explicitly show that the term must be of the form J(J+1), and I don't actually prove that J must be a non-negative integer (assuming that we're talking only about orbital angular momentum). As Ivan's comment notes, the proof of that is pretty ugly (you must explicitly solve the differential equations), and you are best served by looking it up in a textbook of your choice. I think the solutions are called associated Legendre polynomials, so that will give you a search term. Btw, if you look it up, skip the chemistry books—go straight to physics or maths. – orthocresol Jan 25 '23 at 21:11

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