Mathematical Modelling

Solving the Schrodinger Equation for a particle on a nanowire surface

Psi is continuous in x and bounded in y.

In this problem, assume the circumference of the nanowire is circular and the sides vertical and linear. The the outer face can be represented as a 2D flat surface with length h and width c. The boundary conditions for this problem are that the wave function is continuous at x=\pm \frac{c}{2} and 0 at y=\pm \frac{h}{2}. In other words:

\Psi(\frac{c}{2},y) = \Psi(-\frac{c}{2},y)

Similarly:

\Psi(x,\frac{h}{2}) = \Psi(x, -\frac{h}{2}) = 0

The Schrodinger equation can then be solved using these boundary conditions by considering that the wave function has an energy component in x and in y, as well as assuming that the wave function can be expressed using separation of variables.

H\Psi = E\Psi

(-\frac{\hbar^2}{2m} (\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}) +V(x,y))\Psi (x,y) = E\Psi(x,y)

Let \Psi (x,y) = f(x)g(y); \Psi '(x,y) = g(y)f'(x)+f(x)g'(y); \Psi ''(x,y) = g(y)f''(x) +f(x)g''(y) + 2g'(y)f'(x):

-\frac{\hbar^2}{2m}(g(y) \frac{d^2f}{dx^2}+2\frac{df}{dx} \frac{dg}{dy} +f(x) \frac{d^2g}{dy^2}) + V(x)f(x)V(y)g(y) = Ef(x)g(y)