# QM tensor product

1. Jul 15, 2010

### johnsmi

Hi, I am reading this article for homework about a ring in a megnetic field. It starts off by giving a hamiltonian (an adiabatic part -never mind)
$$H_{0}= \frac{1}{2M} [ \Pi -A]^{2} -\mu B( \phi) \cdot \sigma$$

A- is a known operator

where $$\Pi=\frac{1}{2a} \frac{d}{d \phi} -\frac{eB_{z} \pi a}{2c}$$ is the generalized momentum operator

I know that the eigen states of $$\mu B( \phi) \cdot \sigma$$ (Spinors) are:
$$| \uparrow ( \phi) > =(i \alpha e^{-i \phi}$$ , $$-\beta)^{T}$$

$$| \downarrow ( \phi) > =(i\beta e^{-i \phi}$$ , $$\alpha)^{T}$$

Now, in this article I have he sais that the eigen states of H0 can be written as
$$| \uparrow ( \phi) > \otimes \psi ^{ \uparrow}_{n}$$ and

$$| \downarrow ( \phi) > \otimes \psi ^{ \downarrow}_{n}$$

When, $$\psi \^{ \uparrow}_{n}$$ for example is the eigenstate of a Hamiltonian:
$$H^{up}_{0}= \frac{1}{2M} [ \Pi -const]^{2} -\mu B$$

How did he get it (the last Hamiltonian)???
Someone told me to try and apply H0 on
$$| \uparrow ( \phi) > \otimes \psi ^{ \uparrow}_{n}$$
but I got something pretty awful
Can somone help me please???

Last edited: Jul 15, 2010