cepheid

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**1. Homework Statement**

Find the best bound state on E

_{gs}for the one-dimensional harmonic oscillator using a trial wave function of the form

[tex] \psi(x) = \frac{A}{x^2 + b^2} [/tex]

where A is determined by normalization and b is an adjustable parameter.

**2. Homework Equations**

The variational principle

[tex] \langle \psi |H| \psi \rangle \geq E_{gs} [/tex]

**3. The Attempt at a Solution**

This problem has some really nasty integrals. E.g., just to normalize the wavefunction, you need to calculate:

[tex] \int_{-\infty}^{\infty} \frac{dx}{(x^2 + b^2)^2} [/tex]

I was able to find most of them using residue theory. It was a last resort...the only method I could think of! I had to dig through my old complex analysis notes to remind myself of the technique using a semicircular contour of radius r and then letting r go to infinity. The integral of f(z) over the arc is zero, leaving the integral on the real axis, which is the part I want to evaluate. Anyway, I'm actually pretty sure of my answers, which are as follows

The best value of b is:

[tex] b^2 = \frac{\hbar}{m\omega \sqrt{2}} [/tex]

[tex] \langle \psi |H| \psi \rangle = \frac{\hbar \omega}{\sqrt{2}} \geq E_{gs} [/tex]

My question is, in a subsequent problem, Griffiths asks us to generalize this to a trial wavefunction of the form

[tex] \psi(x) = \frac{A}{(x^2 + b^2)^n} [/tex]

Huh? Did I miss something?! I had enough trouble with the integrals the first time around!! I don't think contour integration is going to help me. Am I missing some more obvious method for evaluating these integrals?

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