Half integer orbital momentum, following Griffiths

[Tomer]

Hello everyone, and thanks for reading.

I'm having a difficult time understanding something.
On yet another attempt to deepen my quantum mechanics understanding I referred to the widely recommended book of Griffiths.
I find the book indeed very good and pretty thorough in it's scope. There is one thing however I was unable to get from the book.
Griffiths develops the quantum number "l" in two different ways: In the first way - the "ugly" way, "l"'s origin is being a separation variable (more accurately, l(l+1) ) between the radial and angular expressions retrieved from the spherical Hamiltonian while looking for separable solutions. In this development however he concludes that l has to be a whole integer because of the cyclic nature $\phi$ imposes. ($e^{i\phi} = e^{i\phi + 2\pi})$.

Later, he develops the angular momentum operators which carry their "l"'s as their eigenvalues, and with help of ladder operators shows eventually the equivalence of this l with the last l.
However, in this way, he concludes that l isn't nessecarily whole, and that also half-integers are allowed.

What he doesn't bother to explain, is how come we didn't get these values from the first technique? How come we get a condition that forces that l's to be whole? And how do the half integers not contradict the cyclic nature of $\phi$?

Thanks a lot!

Tomer.

 

[dextercioby]

Well, essentially it boils down to what the small <l> stands for. In technical terms, it's the related to the spectral value of a linear operator which represents the square of the ORBITAL angular momentum operator, denoted by $\displaystyle{\hat{\vec{L}}}$ . Since we know that

$$\hat{\vec{L}} = \hat{\vec{r}} \times \hat{\vec{p}}$$

and we're asking for L's components and for L^2 to be self-adjoint, solving the spectral problem for L^2 (= solving 2 Sturm-Liouville ODE's derived from a linear PDE) we're forced to conclude that <l> must be a positive integer and <m> be linked to <l> in a specific way (<m> is the spectral value of $\displaystyle{\hat{L}}_z$).

When people talk in a general manner about angular momentum, not necessarily orbital one, they should use a different notation, namely J and <j>, respectively.

The general theory of J makes <j> to be either positive and integer, or positive and semi-integer. In a way, the set of all <l>'s in included in the set of all <j>'s.

 

[Tomer]

Alright, so orbital angular momentum has to be a whole integer, and the parallel, algebraic development shows the possible values of a somewhat "generalized" angular momentum which includes the spin?

 

[strangerep]

Alright, so orbital angular momentum has to be a whole integer, and the parallel, algebraic development shows the possible values of a somewhat "generalized" angular momentum which includes the spin?

The "algebraic" development (i.e., representing the generic rotation group generators as Hermitian operators on a Hilbert space and deriving their spectrum) should be regarded as more fundamental. From this comes the result that angular momentum is quantized to integer and half-integer values.

Adding the extra restriction on the form of the rotation generators, i.e., insisting that $L = r \times p$, has the consequence that only integral values of quantized angular momentum are allowed in that case.

Ballentine explains all this pretty well in his chapter on angular momentum.

 

[Tomer]

Thanks very much for the replies, I think I got it, more or less :-)