[SOLVED] Perturbation of the simple harmonic oscillator


by nowits
Tags: harmonic, oscillator, perturbation, simple, solved
nowits
nowits is offline
#1
Dec3-07, 06:20 AM
P: 18
1. The problem statement, all variables and given/known data
An additional term V0e-ax2 is added to the potential of the simple harmonic oscillator (V and a are constants, V is small, a>0). Calculate the first-order correction of the ground state. How does the correction change when a gets bigger?

2. Relevant equations
[tex]E_0^1=<\psi_0^0|H'|\psi_0^0>[/tex]

3. The attempt at a solution
[tex]\alpha=\frac{m\omega}{\hbar}, E_0^1=\int ^{\infty}_{-\infty} (\frac{\alpha}{\pi})^{1/4}e^{-\alpha x^2/2}V_0e^{-ax^2}(\frac{\alpha}{\pi})^{1/4}e^{-\alpha x^2/2}dx=(\frac{\alpha}{\pi})^{1/2}V_0\int ^{\infty}_{-\infty} e^{(-\alpha -a)x^2}dx[/tex]
So I suppose this is not what is wanted.
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nrqed
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#2
Dec3-07, 06:22 AM
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Quote Quote by nowits View Post
1. The problem statement, all variables and given/known data
An additional term V0e-ax2 is added to the potential of the simple harmonic oscillator (V and a are constants, V is small, a>0). Calculate the first-order correction of the ground state. How does the correction change when a gets bigger?

2. Relevant equations
[tex]E_0^1=<\psi_0^0|H'|\psi_0^0>[/tex]

3. The attempt at a solution
[tex]\alpha=\frac{m\omega}{\hbar}, E_0^1=\int ^{\infty}_{-\infty} (\frac{\alpha}{\pi})^{1/4}e^{-\alpha x^2/2}V_0e^{-ax^2}(\frac{\alpha}{\pi})^{1/4}e^{-\alpha x^2/2}dx=(\frac{\alpha}{\pi})^{1/2}V_0\int ^{\infty}_{-\infty} e^{(-\alpha -a)x^2}dx[/tex]
So I suppose this is not what is wanted.
What makes you think this is not right?
Have you tried to compute the integral? It's a standard one.
nowits
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#3
Dec3-07, 07:10 AM
P: 18
Quote Quote by nrqed View Post
Have you tried to compute the integral? It's a standard one.
[tex]\int e^{-\xi x^2}=\frac{\sqrt{\pi}\ erf(\sqrt{\xi}x)}{2\sqrt{\xi}}\ \ \ ?[/tex]
I've never encountered an error function before in any homework problem, so I automatically assumed that I had done something wrong.

CompuChip
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#4
Dec3-07, 07:17 AM
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[SOLVED] Perturbation of the simple harmonic oscillator


That's right, the indefinite integral contains an erf.
But you have more information: you know the boundary conditions and (you should have) [itex]\xi > 0[/itex]. Using
[tex]\lim_{x \to \pm \infty} \operatorname{erf}(x) = \pm 1[/tex]
you can calculate it, and in fact it is just a Gaussian integral,
[tex]\int_{-\infty}^\infty e^{-\xi x^2} dx = \sqrt{\frac{\pi}{\xi}}[/tex]
Remember this -- it's ubiquitous in physics (at least, every sort of physics that has to do with any sort of statistics, amongst which QM, thermal physics, QFT, SFT).
nowits
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#5
Dec3-07, 07:40 AM
P: 18
Ok.

Thank you both!


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