- #1
rwooduk
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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 realize 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.