Perturbation Theory - expressing the perturbation

[Zero1010]

Homework Statement

Specify the perturbation for the Coulomb model of hydrogen.

Relevant Equations

$V(x) = \begin{cases} -\frac{e^{2}}{4\pi\epsilon_{0}}\frac{b}{r^{2}} \text{if } 0<r \leq 0 \\ -\frac{e^{2}}{4\pi\epsilon_{0}}\frac{1}{r} \text{if } r > 0 \end{cases}$

Hi,

I just need someone to check if I am on the right track here

Below is a mutual Coulomb potential energy between the electron and proton in a hydrogen atom which is the perturbed system:

$V(x) = \begin{cases} - \frac{e^{2}}{4\pi\epsilon_{0}}\frac{b}{r^{2}} \text{for } 0<r \leq 0 \\ - \frac{e^{2}}{4\pi\epsilon_{0}}\frac{1}{r} \text{for } r > 0 \end{cases}$

Where e = mag of electron charge, B = a constant length, $4\pi\epsilon_{0}$ = permittivity of free space

Using this I need to specify the perturbation of the system.

Since V(x) is the perturbed system is the total perturbation the sum of both these functions across the length?

In which case I get:

$\delta\hat H = - \frac{e^{2}}{4\pi\epsilon_{0}}\frac{b}{r^{2}} + (- \frac{e^{2}}{4\pi\epsilon_{0}}\frac{1}{r})$

$\delta\hat H = - \frac{e^{2}}{4\pi\epsilon_{0}}(\frac{b}{r^{2}} + \frac{1}{r})$

I just want to check if this is the correct way to go about this?

Thanks in advance.

 

[vanhees71]

I don't understand the potential you are looking at. Usually for the hydrogen atom you just use the simple Coulomb potential. What does the first line mean? $0<r \leq 0$ is self-contradictory, because $r>0$ already implies that $r \neq 0$.

 

[Zero1010]

Thanks for the reply.

Sorry that's my typo it should be:

$0 \lt r \leq b$

and the second one should be

$r \gt b$

Also in the question it says to suppose there is a deviation from Coulombs law at very small distances (even though there is no evidence for it)

 

[nrqed]

Homework Statement:: Specify the perturbation for the Coulomb model of hydrogen.
Relevant Equations:: $V(x) = \begin{cases} -\frac{e^{2}}{4\pi\epsilon_{0}}\frac{b}{r^{2}} \text{if } 0<r \leq 0 \\ -\frac{e^{2}}{4\pi\epsilon_{0}}\frac{1}{r} \text{if } r > 0 \end{cases}$

Hi,

I just need someone to check if I am on the right track here

Below is a mutual Coulomb potential energy between the electron and proton in a hydrogen atom which is the perturbed system:

$V(x) = \begin{cases} - \frac{e^{2}}{4\pi\epsilon_{0}}\frac{b}{r^{2}} \text{for } 0<r \leq 0 \\ - \frac{e^{2}}{4\pi\epsilon_{0}}\frac{1}{r} \text{for } r > 0 \end{cases}$

Where e = mag of electron charge, B = a constant length, $4\pi\epsilon_{0}$ = permittivity of free space

Using this I need to specify the perturbation of the system.

Since V(x) is the perturbed system is the total perturbation the sum of both these functions across the length?

In which case I get:

$\delta\hat H = - \frac{e^{2}}{4\pi\epsilon_{0}}\frac{b}{r^{2}} + (- \frac{e^{2}}{4\pi\epsilon_{0}}\frac{1}{r})$

$\delta\hat H = - \frac{e^{2}}{4\pi\epsilon_{0}}(\frac{b}{r^{2}} + \frac{1}{r})$

I just want to check if this is the correct way to go about this?

Thanks in advance.

You should be subtracting the Coulomb potential in $\delta\hat H$.