Particle in a potential- variation method

[Rorshach]

Homework Statement

Okay, I have no idea about the method they want me to solve it with. What in this case is the indicator that a function is appropriate?
A particle mass m affects a potential of the form $V(x)=V_0 \frac{|x|}{a}$ where $V_0$ and $a$ are positive constants.

a) Draw a sketch of the ground state wave function and indicate the characteristics of this function.

b) Use the variation method to determine an approximate value of the ground state energy. Use functions selected from the list below. Some of these functions are unsuitable while others are less good. Give reasons for all if they are worse or better! Then do the calculation with the function that you find most suitable.

$Nsin(\alpha x)exp(-\alpha |x|)$
$Nexp(-\alpha x^2)$
$\frac{N}{(x^2+\alpha^2)}$
$Nexp(-\alpha x^2)$
$\frac{N}{\sqrt{|x|+\alpha}}$

Homework Equations

The Attempt at a Solution

 

[AdkinsJr]

You've never learned about the variational method? Just look it up.

In quantum I don't think there was any way to choose an "appropriate" trial function, but with a given trial function you just normalize it and then minimize the expectation value of the Hamiltonian with respect to the variational parameter (alpha).

 

[TSny]

What in this case is the indicator that a function is appropriate?

The wavefunction should be normalizable, for example. Should the ground state have any "nodes" (places where ψ = 0 other than at infinity)?

Your 2nd and 4th wavefunctions look identical to me.

For a rough sketch of the ground state, note that your potential is very roughly like a harmonic oscillator potential. So, roughly, you can expect a graph of the ground state wavefunction to have the general appearance of the ground state of the harmonic oscillator.

 

[Rorshach]

The ground state should be flat, without any nodes, right? You are right, the second function was supposed to be $Nxexp(-\alpha x^2)$ I decided that fourth function is the one I should use, so I normalized it and the constant is equal to $N=\sqrt[4]{\frac{2}{\pi}\alpha}$. The next thing according to the book is the equation that looks more or less like this: $\int_{-\infty}^{+\infty} exp(-2\alpha x^2)(-\frac{2\hbar^2 \alpha^2 x^2}{m}+\frac{\hbar^2 \alpha}{2m}+V_0 \frac{|x|}{\alpha}) \,dx$
http://www.wolframalpha.com/input/?...|x|/a))),[x,-inf,+inf]&a=*C.V-_*RomanNumeral-
But I have a problem with solving this integral. How should it look like?

 

[andrien]

you have to setup a variation integral for the energy and then minimize it with respect to alpha to get alpha.Then you will have wave function and energy both.

 

[TSny]

The ground state should be flat, without any nodes, right? You are right, the second function was supposed to be $Nxexp(-\alpha x^2)$ I decided that fourth function is the one I should use, ...

I haven't worked it out, but why not also investigate the third function and compare with result for fourth function?