Where are charge and spin in Schrodinger's equation?

[SamRoss]

Schrödinger's equation is what is used to determine the probability of finding a particle somewhere, but where are charge and spin? How do you know if the probability wave you solved for is for a proton or electron or some other particle? I see m in the equation for plugging in mass, but nothing for charge, spin, or anything else. I'm guessing it has something to do with the potential V, but I'm not sure how.

 

[Jorriss]

The Schrödinger equation is for spin 0 particles. The spin is added on afterwards.

The charge come into play depending on the potential. If the potential between the particles is a coulomb potential, for example, there will be charge terms.

Do you know how to write a classical hamiltonian for a charged particle?

 

[DrDu]

http://en.wikipedia.org/wiki/Pauli_equation

 

[SamRoss]

Thank you for your reply. I don't know how to write a Hamiltonian operator. Do you recommend any websites or books?

 

[Jorriss]

Thank you for your reply. I don't know how to write a Hamiltonian operator. Do you recommend any websites or books?

Before I recommend anything, do you know what canonical quantization is explicitly?

The standard prescription for quantizing mechanics is to write the classical Hamiltonian and to change r and p to their corresponding operators. Is this a familiar process?

 

[SamRoss]

Before I recommend anything, do you know what canonical quantization is explicitly?

The standard prescription for quantizing mechanics is to write the classical Hamiltonian and to change r and p to their corresponding operators. Is this a familiar process?

I had not heard of canonical quantization before. After looking it up it does indeed seem to be a missing puzzle piece for me and I would like to learn more. Also, no, the process you described is not familiar to me. The closest I've come is David Griffiths' derivation of the position, velocity, and momentum operators in "Introduction to Quantum Mechanics".

 

[fargoth]

Well, a classical Hamiltonian is $H=E_{total}=\frac{P^2}{2m}+V$
In QM this turns to $\hat{H}=\frac{\hat{P}^2}{2m}+V$, where $\hat{P}=-i\hbar\frac{\partial}{\partial x}$ and $\hat{H}=i\hbar\frac{\partial}{\partial t}$.
So Schrödinger's equation is $\hat{H}\psi=\frac{\hat{P}^2}{2m}\psi+V\psi$
For an electrically charged particle it is: $\hat{P}=-i\hbar(\frac{\partial}{\partial x_{\mu}}-qA_{\mu})$, and remember to add the $\phi q$ to the V term.

For spin, you would have to move on to relativistic QM...
This is covered in Griffiths - Introduction to Elementary Particles, Zee - Quantum Field Theory in a Nutshell, and the classic Peskin, An Introduction to Quantum Field Theory.

be warned, before you tackle these books you need to know what a Lagrangian and Hamiltonian are, what variational math is (e.g. what is a functional), what are canonical variables, etc. (this is covered by analytical classical mechanics - best covered IMO in Goldstein, Classical Mechanics).
And you should know what a four vector is and how to handle tensors.

By the way, there is a *very* slow paced introductory course by Leonard Susskind on youtube.
He's very thorough. start here:
If that's too much for you, try his lectures on classical mechanics and special relativity first.

 

[Bill_K]

For spin, you would have to move on to relativistic QM...
This is covered in Griffiths - Introduction to Elementary Particles, Zee - Quantum Field Theory in a Nutshell, and the classic Peskin, An Introduction to Quantum Field Theory.

No, not really. The Schrödinger-Pauli Equation cited in #3 above is nonrelativistic.

 

[fargoth]

No, not really. The Schrödinger-Pauli Equation cited in #3 above is nonrelativistic.

right, forgot about pauli :)