QM: position and spin dependent potential

[Heirot]

Homework Statement

A spin 1/2 particle of mass m is described by the Hamiltonian: H = p^2/(2m) + 1/2 mw^2 x^2 + g * x * sigma_x, where sigma_x is one of the Pauli matrices.

Homework Equations

The Attempt at a Solution

I have no idea where to start. It's obvious that the harmonic oscillator part of H should be multiplied by unit matrix. Does this mean that I need to diagonalize the whole matrix H? That would give me two eigenvalues containing operators p and x. When I solve the two equations, I should get the energies for spin up and spin down? I'm not certain is it ok to mix the spin and position variables in such a way.

 

[fzero]

Think of the wavefunction as a two-component vector

[tex] \begin{pmatrix} \psi_+(x) \\ \psi_-(x) \end{pmatrix}. [/tex]

 

[Heirot]

Yes, that's exactly what I did and it leads to the following equations:
[tex] (\frac{p^2}{2m}+\frac{m\omega^2}{2}x^2-E_+)\Psi_+ + gx\Psi_-=0 \qquad
(\frac{p^2}{2m}+\frac{m\omega^2}{2}x^2-E_-)\Psi_- + gx\Psi_+=0[/tex]

Now, for this system to be consistent, it's necessary for the determinant to vanish. But here the coefficients are not numbers but operators! I'm not sure whether I'm allowed to calculate the determinant of an operator?

 

[fzero]

Yes, that's exactly what I did and it leads to the following equations:
[tex] (\frac{p^2}{2m}+\frac{m\omega^2}{2}x^2-E_+)\Psi_+ + gx\Psi_-=0 \qquad
(\frac{p^2}{2m}+\frac{m\omega^2}{2}x^2-E_-)\Psi_- + gx\Psi_+=0[/tex]

Now, for this system to be consistent, it's necessary for the determinant to vanish. But here the coefficients are not numbers but operators! I'm not sure whether I'm allowed to calculate the determinant of an operator?

You still want to solve

[tex]H \psi = E \psi,[/tex]

so don't introduce [tex]E_\pm[/tex] at this point. I was a bit hasty to introduce [tex]\psi_\pm[/tex]. Things will be cleaner if you use [tex](\Psi_1, \Psi_2)[/tex], since we're not working in the basis that diagonalizes [tex]\sigma_x[/tex]. All together, we'll have

[tex] (\frac{p^2}{2m}+\frac{m\omega^2}{2}x^2-E)\Psi_1 + gx\Psi_2=0 \qquad
(\frac{p^2}{2m}+\frac{m\omega^2}{2}x^2-E)\Psi_2 + gx\Psi_1=0.[/tex]

Don't worry about any determinants yet, just consider the sum and difference of these equations and find the spectrum. The determinant of those equations is zero whenever [tex]E=E_\pm[/tex], the energy eigenvalues, so it's consistent.

 

[paranormal]

I can provide some guidance on how to approach this problem. First, it is important to understand the components of the Hamiltonian and what they represent. The first term, p^2/(2m), represents the kinetic energy of the particle. The second term, 1/2 mw^2 x^2, represents the potential energy of a harmonic oscillator. The third term, g * x * sigma_x, is a position and spin dependent potential term.

To solve this problem, you will need to use the Schrodinger equation, which is the fundamental equation of quantum mechanics. This equation describes how the wave function of a system evolves over time. In this case, the wave function will depend on both position and spin variables.

To find the energy eigenvalues, you will need to diagonalize the Hamiltonian matrix. This means finding the eigenvalues and eigenvectors of the matrix. The eigenvalues will give you the energy levels of the system, and the eigenvectors will give you the corresponding wave functions.

It is important to note that in this case, the eigenvalues will depend on both position and spin. This means that the energies for spin up and spin down will be different. This is due to the spin-dependent potential term in the Hamiltonian.

In summary, to solve this problem, you will need to use the Schrodinger equation and diagonalize the Hamiltonian matrix to find the energy eigenvalues and corresponding wave functions. This will give you the energies for spin up and spin down. Good luck with your homework!