# Where can I find detailed description of cononical quantization?

Leamas
For example, I want to know how to quantize a free particle in the spherical coordinates. Given a classical Hamiltonian $$H(r, \theta, \phi, p_r, p_{\theta}, p_{\phi})$$, the standard procedure tells us to let $$r$$, $$\theta$$, $$\phi$$ be operators and they form a complete set. And The corresponding generalized momentums satisfy the well-known commutation relation. But starting from here, how can we obtain the specific matrix form in the coordinate representation? I looked this up in Sakurai's modern quantum mechanics, it seems to me that following the same reasoning there, we can get $$p_r=-i\hbar\frac{\partial}{\partial r}$$, but the expression resulting from the usual quantization procedure is $$p_r=-i\hbar(\frac{\partial}{\partial r}+\frac{1}{r})$$.

Another problem is that how to know the right form of some "combined" variables, like angular momentum, from the classical definition of those variables. The essential issue is the order of basic variables which the combined variables are made out of.

I know this is a very classical problem. But I can't find detailed presentation of the stuff. Does anyone know any textbook containing this thing? Thanks.

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## Answers and Replies

Homework Helper
Gold Member
what is a "usual" quantization procedure?

Leamas
The usual procedure is like this: first quantize it in $$x$$, $$y$$, $$z$$ coordinate system, one obtains: $$H=-\frac{\hbar^2}{2m}(\frac{\partial^2}{\partial x}+\frac{\partial^2}{\partial y}+\frac{\partial^2}{\partial z})$$. Then change it to the spherical coordinate system, using simple calculus, one gets: $$H=-\frac{\hbar^2}{2m}\frac{1}{r}\frac{\partial^2}{\partial r^2}r+\frac{\bold{l}^2}{2mr^2}$$, with $$\bold{l}$$ the angular momentum, then one identifys the part relating to $$r$$ is $$-\frac{\hbar^2}{2m}\frac{1}{r}\frac{\partial^2}{\partial r^2}r$$, which is $$\frac{p_r^2}{2m}$$ where $$p_r=-i\hbar(\frac{\partial}{\partial r}+\frac{1}{r})$$.

Homework Helper
Gold Member
Why is that the usual quantization process?

Just take the laplace-operator in spherical coordinates.

Where is $$p_r=-i\bar{h}\frac{\partial}{\partial r}$$
given in Sakurai?