• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Harmonic oscillator in electric field

  • Thread starter skrat
  • Start date
744
8
1. Homework Statement
Potential energy of electron in harmonic potential can be described as ##V(x)=\frac{m\omega _0^2x^2}{2}-eEx##, where E is electric field that has no gradient.
What are the energies of eigenstates of an electron in potential ##V(x)##? Also calculate ##<ex>##.

HINT: Use ##(x-a)^2=x^2-2ax+a^2##


2. Homework Equations



3. The Attempt at a Solution

I am sorry, to say, but I have no idea how start here.

I know that if there were no electric field, the energies would be ##E_n=\hbar \omega (n+1/2)##. Since there is Electric field, I assume I have to solve ##\hat{H}\psi =E\psi ## for ##\hat{H}=\hat{T}+\hat{V}##... but, how on earth can i do that?
 

TSny

Homework Helper
Gold Member
12,046
2,619
The hint is to "complete the square" in the expression ##V(x)=\frac{m\omega _0^2x^2}{2}-eEx##.
That is, can you write it as ##V(x) = b(x-a)^2 - c## for certain constants ##a##, ##b## and ##c##?
 
744
8
Ok, I got that but I can't see how this makes my life any easier..

##V(x)=\frac{m\omega _0^2x^2}{2}-eEx=(\sqrt{\frac{m}{2}}\omega _0x-\frac{eE}{\sqrt{2m}\omega _0})^2-\frac{e^2E^2}{2m\omega _0^2}##
 

TSny

Homework Helper
Gold Member
12,046
2,619
Ok, I got that but I can't see how this makes my life any easier..

##V(x)=\frac{m\omega _0^2x^2}{2}-eEx=(\sqrt{\frac{m}{2}}\omega _0x-\frac{eE}{\sqrt{2m}\omega _0})^2-\frac{e^2E^2}{2m\omega _0^2}##
Try to factor out something inside the parentheses so that the x has a coefficient of 1 inside the parentheses. Then define a new variable in terms of x and solve the problem in the new variable.
 
744
8
Hmm,

##V(x)=(\sqrt{\frac{m}{2}}\omega _0x-\frac{eE}{\sqrt{2m}\omega _0})^2-\frac{e^2E^2}{2m\omega _0^2}##

##V(x)=(\sqrt{\frac{m}{2}}\omega _0x-\frac{eE\sqrt{m}\omega_0}{\sqrt{2}m\omega _0^2})^2-\frac{e^2E^2}{2m\omega _0^2}## and finally

##V(x)=(\frac{m}{2})^{1/4}\omega_0^{1/2}(x-\frac{eE}{m\omega _0^2})^2-\frac{e^2E^2}{2m\omega _0^2}##

now let's say ##u=x-\frac{eE}{m\omega _0^2}## than ##V(u)=(\frac{m}{2}\omega_0^2)^{1/4}u^2-\frac{e^2E^2}{2m\omega _0^2}##

Now ##\hat{H}\psi = E_n \psi ##

##\hat{V}\psi=((\frac{m}{2}\omega_0^2)^{1/4}\hat{u}^2-\frac{e^2E^2}{2m\omega _0^2})\psi =E_n \psi##

Are the energies of eigenstates than ##W_n=(\hbar \omega(n+1/2)+\frac{e^2E^2}{2m\omega _0^2})(\frac{m}{2}\omega_0^2)^{-1/4}##?
 

Want to reply to this thread?

"Harmonic oscillator in electric field" You must log in or register to reply here.

Related Threads for: Harmonic oscillator in electric field

Replies
6
Views
2K
Replies
1
Views
7K
Replies
2
Views
12K
Replies
5
Views
4K
Replies
1
Views
1K
Replies
8
Views
6K
Replies
1
Views
924

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top