Aproximate ground state wave functions

[dingo_d]

Homework Statement

So to test the variational method of simple harmonic oscillator I am using some functions that can be a good approximations, such as:

Gaussian: $$\psi(x)=Ae^{-bx^2}$$
Polynomial: $$\psi(x)=1-bx^2+\frac{b^2 x^4}{2}$$ (I just expanded the Gaussian into Taylor, I can use just quadratic term, it doesn't matter)
Rational function: $$\psi(x)=\frac{A}{x^2+b^2}$$

So can you suggest some more functions that can describe the ground state, and perhaps the first excited state for SHO?

Thanks...

 

[Tangent87]

Why can't you use the exact functions?

 

[dingo_d]

Because I'm using variational method and the point is that I use a trial wave function, find eigenvalue and then minimize it by some small parameter to see what is the deviation from the exact solution...

 

[elkement]

Because I'm using variational method and the point is that I use a trial wave function, find eigenvalue and then minimize it by some small parameter to see what is the deviation from the exact solution...

So I would work backwards actually: Write down the exact functions (I guess you know them from your lecture? Otherwise they are easy to find) and build the to-be-varied functions by adding some more orders (polynomial / exponential, products of exponential and polynomial functions). Replace coefficients by parameters to be varied.

 

[dingo_d]

I mean are there any more types of functions that would approximate ground state and first excited state of SHO? Or do I just, as you said, try to plot them in Mathematica and see what looks kinda ok?

 

[elkement]

My idea was to generate a test function that already contains the correct functions if some of the coefficients become zero. (I hope that I understood the problem correctly and this is a permitted approach.)

The exact functions are built from products of polynomial expressions and exponential functions (Hermite polynomials).

If you start from something like this, you should get A = 0 for the first excited state and B = 0 for the ground state.

$$\psi(x)=Ae^{-bx^2} + Bxe^{-bx^2}$$

If you need more parameters to be varied I would add e.g.

$$Cx^2e^{-bx^2}$$

The point I wanted to make is: I would use a test function that contains 'mixed' terms, not only Gaussian functions or polynomials.

 

[dingo_d]

I see, cool approach, but I guess I just need straightforward functions that I can just use in variational method.

I think that those 3 will do fine for the ground state and that I'll just have to try to see what other I can use for excited state.

Thanks for the help :)