Infinite Square Well
Overview
The Infinite square well is defined by a potential function {% V(x) %} as follows:
{% V(x) = 0 \; for \; 0 \leq x \leq a %}
{% V(x) = \infty \; otherwise %}
In the region where the potential is infinite, we take as an assumption that {% \psi(x) = 0 %}.
Inside the Well
The
time independent Schrodinger Equation
for the infinite square well is then
{% - \frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} = E \psi %}
which can be written as
{% \frac{d^2 \psi}{dx^2} = -k^2 \psi %}
where
{% k = \frac{\sqrt{2mE}}{\hbar} %}
The solution to this equation is
{% \psi(x) = A sin(kx) + B cos(kx) %}
If we assume that {% \psi %} is continuous then
{% \psi(0) = \psi(a) = 0 %}
which implies that {% B=0 %}, and so
{% \psi(x) = A sin (kx) %}
