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## Homework Statement

I'm going to list two questions as they offer the same problem with more choices, hopefully it will help realise the method (?) used

**(A)**

An electron, confined in the two dimensional region 0<x<L and 0<y<L with infinite potential walls, is subject to the potential:

[tex]V(x,y)=V_{1}x + V_{2}y^{2}[/tex]

Which of the following trial wave functions is better suited for approximating the wave function of the electron:

[tex]\psi_{T} (x,y) = cos(ax+by)[/tex]

[tex]\psi_{T} (x,y) = exp -(ax^{2} + by^{4})[/tex]

[tex]\psi_{T} (x,y) = xy(a-x)(b-y)[/tex]

**(B)**

Consider the infinite potential well V(x) = ∞ at x = ±L/2 and for -L/2 < x < L/2 it is:

[tex]V(x) = \frac{1}{2} k(x-x_{0})^2[/tex]

Which trial wavefunction should we consider:

[tex]\psi_{T} (x) = x(x-L)exp -\gamma(x-x^{0})^2[/tex]

[tex]\psi_{T} (x) = (x- \frac{L}{2} )(x+ \frac{L}{2} )exp -\gamma(x-x_{0})^2[/tex]

[tex]\psi_{T} (x) = \alpha(x- \frac{L}{2} )(x+ \frac{L}{2} ) + \beta exp -\gamma (x-x_{0})^2[/tex]

[tex]\psi_{T} (x) = (x- \frac{L}{2} )^{\alpha}(x+ \frac{L}{2} )^{\beta} + \beta exp -\gamma (x-x_{0})^2[/tex]

**More examples from other questions, where the trial wavefunction was already "chosen"**

[tex]if.... V(r) = A exp (- \frac{r}{a} ) ----> \psi_{T} = c exp (- \frac{\alpha r}{2a} )[/tex]

[tex]if.... V(r)= \frac{V_{0} x}{L} ----> \psi_{T} = x(x-L) [/tex]

[tex]if.... V(r)=0 ----> \psi_{T} = sin(ax+b) [/tex]

## Homework Equations

Intuition? Boundary conditions? Should it be of the same form as the potential?

## The Attempt at a Solution

Sometimes it seems the trial wavefunction is picked at random out of a hat, I really don't see

__how__it is chosen. Is there a general method? I asked the lecturer and he said to look at the boundary conditions for the problem, but how would that help me decide?

I really think this question could help a lot of people as I've searched on the internet and most of the time it's simply defined.

**Any**ideas would be welcome.