Is Angular Momentum Truly Quantized in Quantum Mechanics?

[cepheid]

Hello,

I've come across some confusing areas in my Quantum Mechanics notes. This is an introductory course, taken over the summer semester, so I don't know my stuff too well yet.

My prof's notes begin with the Bohr model of the atom, stating that the entire model comes from classical physics with the additional condition that angular momentum is quantized:

$$L = n \hbar$$
$$n = 1, 2, ...$$

Directly underneath, he states that this condition can be replaced by

$$\oint \vec{p} \cdot d \vec{r} = nh = 2 \pi \hbar n$$

I'm wondering what this quantity in the path integral represents. If the two conditions are truly equivalent, then it should represent angular momentum. But angular momentum $$\inline \vec{r} \times \vec{p}$$, so how do you arrive at this integral? Also, if the two conditions are supposed to be equivalent, then why does the former have $$\inline n \hbar$$, and the latter $$\inline 2\pi n \hbar$$?

Now we skip a few pages, and we get to the point where he has defined the wave function for an electron as:

$$\psi (x,t) = Ae^{i \frac{px-Et}{\hbar} }$$

And underneath:

Note: $$\frac{\partial^2 \psi}{\partial x^2} = -\hbar^2 p^2 \psi (x,t)$$

I don't know about you, but when I differentiated, I actually got:

$$\frac{\partial^2 \psi}{\partial x^2} = -\frac{p^2}{\hbar^2} \psi (x,t)$$

It's entirely possible that my prof merely needs a new pair of glasses. But if I'm the one in error and am incapable of performing basic tasks such as differentiation, then I'd like to know sooner rather than later

 

[marlon]

I think your prof made a mistake in the derivation, your outcome is right...

The first question is solved in the attached word-doc, ok?

regards marlon

 

[R0man]

Hi there! It's great that you're taking an interest in Quantum Mechanics and looking for clarification on some confusing areas. I'll do my best to answer your questions and hopefully clear up any misunderstandings.

Regarding the first question about the equivalence of the two conditions for quantization of angular momentum, it's important to note that the path integral approach is a more general and powerful way of understanding quantum mechanics. The first condition, L = n \hbar, is a specific case of the path integral, where the path integral is taken over a circular path with radius r = n\hbar/p. This results in the expression \oint \vec{p} \cdot d\vec{r} = nh, which represents the quantization of angular momentum. So, in a sense, the path integral approach encompasses and generalizes the Bohr model.

For your second question, you are correct in your differentiation of the wave function. The expression given by your professor is incorrect, and it should be \frac{p^2}{\hbar^2} instead of -\hbar^2 p^2. It's always a good idea to double check the math and formulas presented by your professor, as they can make mistakes from time to time.

I hope this helps clarify things for you. Keep up the good work and don't be afraid to ask questions and seek clarification when needed. Good luck with the rest of your course!