- #1

ShayanJ

Gold Member

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## Homework Statement

I'm struggling with the problem below:

If [itex] E_n^{(0)} \ \ (n\in \mathbb N)[/itex] are the energy eigenvalues of a system with Hamiltonian [itex] H_0=\frac{p^2}{2m}+V(x) [/itex], what are the exact energy eigenvalues of the system if the Hamiltonian is changed to [itex] H=H_0+\frac{\lambda}{m} p \ [/itex]?

## Homework Equations

[/B]

## The Attempt at a Solution

[itex] H=\frac{p^2}{2m}+V(x)+\frac{\lambda}{m} p+\frac{\lambda^2}{2m}-\frac{\lambda^2}{2m}=\frac{(p+\lambda)^2}{2m}+V(x)-\frac{\lambda^2}{2m}[/itex]

So it seems that I should write:

[itex]

H \psi_n=E_n \psi_n \Rightarrow [\frac{(p+\lambda)^2}{2m}+V(x)]\psi_n=(E_n+\frac{\lambda^2}{2m})\psi_n

[/itex]

But from the statement of the problem, we know that:

[itex]

H_0 \phi_n=E_n^{(0)} \phi_n \Rightarrow [\frac{p^2}{2m}+V(x)]\phi_n=E_n^{(0)}\phi_n

[/itex]

My problem is, I can't find a connection between the two equations. One idea I have is that because the first equation has its momentum shifted by a constant, we may have [itex] \psi_n=e^{i\varphi} \phi_n [/itex] probably with [itex] \varphi=\frac{\lambda}{\hbar} [/itex]. But it seems to me that this can be true only for plane waves and so we should decompose the wavefunctions to plane waves and sum them again later. But this again has the problem that because of the presence of the potential, plane waves are not solutions to the TISE in general.

Any ideas?

Thanks