- #1
Zero1010
- 40
- 2
- 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.
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.
Last edited: