Shelf in a box, treating the shelf as a weak perturbation

Futurestar33

Problem Statement
Shelf in a box, treating the shelf as a weak perturbation. The pertubation is not given what should I use?
Relevant Equations
H= Ho+ H'
E_n = <Y_n|H'|Y_n>
In this problem I am supposed to treat the shelf as a weak perturbation. However it doesn't give us what the perturbed state H' is. At the step V(x) = Vo, but that is all that is given and isn't needed to determine H'.

This isn't in a weak magnetic field so I wouldn't you use H'=qEx and then treat X as an operator.

The other option I would use is H'=lamdaX, (but that is usually given in a problem as well)

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DrClaude

Mentor
At the step V(x) = Vo, but that is all that is given and isn't needed to determine H'.
What do you mean? This is exactly what is needed to know $H'$.

Start with the Hamiltonian in the absence of the shelf, then figure out how it is different if there is a shelf.

Futurestar33

Im confused at this then.
When there is no shelf the H= ∇^2Ψ=-K^2Ψ where K= √(2mE)/hbar
General expression being Ψ= Csin(kx)+Dcos(kx) , k=npi/a

When there is a shelf H = ∇^2Ψ = -η^2Ψ where η=√(2mE-Vo)/hbar
General expression being Ψ= Ae^(κx)+Be^(-κx).

I know Im not supposed to plug those in for H'

are you saying I just use En = <Yn|H'| Yn> ==> <Yn| Vo |Yn> ?

Futurestar33

Ok I got it, I was just thinking that usually it is given or some form of an operator.
but in this case it does just end up up being the above.

DrClaude

Mentor
are you saying I just use En = <Yn|H'| Yn> ==> <Yn| Vo |Yn> ?
Yes. Remember that the Hamiltonian is kinetic energy + potential energy. This is not evident in the case of the particle in a box, because in the domain where the wave function is non-zero, the potential is zero.

$H'$ is defined by parts, but calculating $\langle \psi_n |H' | \psi_n \rangle$ is easy since its an integration over a constant potential in a finite region of space.

"Shelf in a box, treating the shelf as a weak perturbation"

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