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)(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

H_{0}= \frac{1}{2M} [ \Pi -A]^{2} -\mu B( \phi) \cdot \sigma

[/tex]

A- is a known operator

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

I know that the eigen states of [tex]\mu B( \phi) \cdot \sigma [/tex] (Spinors) are:

[tex]

| \uparrow ( \phi) > =(i \alpha e^{-i \phi}[/tex] , [tex] -\beta)^{T}

[/tex]

[tex]

| \downarrow ( \phi) > =(i\beta e^{-i \phi}[/tex] , [tex] \alpha)^{T}

[/tex]

Now, in this article I have he sais that the eigen states of H_{0}can be written as

[tex]

| \uparrow ( \phi) > \otimes \psi ^{ \uparrow}_{n}[/tex] and

[tex]

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

[/tex]

When, [tex] \psi \^{ \uparrow}_{n}[/tex] for example is the eigenstate of a Hamiltonian:

[tex]

H^{up}_{0}= \frac{1}{2M} [ \Pi -const]^{2} -\mu B

[/tex]

How did he get it (the last Hamiltonian)???

Someone told me to try and apply H_{0}on

[tex]

| \uparrow ( \phi) > \otimes \psi ^{ \uparrow}_{n} [/tex]

but I got something pretty awful

Can somone help me please???

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# QM tensor product

Can you offer guidance or do you also need help?

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