Perturbation of the simple harmonic oscillator

In summary, the additional term V0e-ax2 perturbs the ground state of the simple harmonic oscillator, and the correction changes when a gets bigger.
  • #1
nowits
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[SOLVED] Perturbation of the simple harmonic oscillator

Homework Statement


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?

Homework Equations


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

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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  • #2
nowits said:

Homework Statement


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?

Homework Equations


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

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.
 
  • #3
nrqed said:
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.
 
  • #4
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).
 
Last edited:
  • #5
Ok.

Thank you both!
 

1. What is a simple harmonic oscillator?

A simple harmonic oscillator is a system that exhibits a repeating pattern of motion, typically back and forth around a central equilibrium point, with a restoring force proportional to its displacement from the equilibrium point. Examples include a mass on a spring and a pendulum.

2. How is the simple harmonic oscillator perturbed?

The simple harmonic oscillator can be perturbed by applying an external force to the system, changing the system's initial conditions, or altering the parameters of the system such as the mass or spring constant. This perturbation causes the system to deviate from its regular harmonic motion.

3. What is the equation of motion for a perturbed simple harmonic oscillator?

The equation of motion for a perturbed simple harmonic oscillator is given by m(d^2x/dt^2) + kx = F(t), where m is the mass, x is the displacement from equilibrium, k is the spring constant, and F(t) is the external force as a function of time.

4. How does the amplitude of the oscillator change with perturbation?

The amplitude of the oscillator changes with perturbation depending on the frequency and magnitude of the external force. In some cases, the amplitude may increase, causing the oscillator to oscillate with larger swings. In other cases, the amplitude may decrease, causing the oscillator to dampen and eventually come to rest.

5. How is the phase of the oscillator affected by perturbation?

The phase of the oscillator, or the timing of its oscillations, is affected by perturbation in a similar manner to the amplitude. The frequency and magnitude of the external force can cause the oscillator to shift its phase, leading to changes in the timing of its oscillations.

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