Schrodinger's Equation

Schrodinger's equation is a second-order linear differential equation that describes, in quantum-mechanical terms, the behavior of a particle in a specified potential. In one dimension, the equation takes the form:

$$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\psi = E\psi\,,$$

where $V(x)$ is the potential, $E$ is the energy, and $\vert\psi(x)\vert^2 dx$ is the probability of finding the particle in $(x, x+dx)\,.$ In the classical system, we would have $E = \frac{1}{2}mv^2 + V(x)$.

The equation is a boundary-value problem because the value of $\psi$ (or its derivatives) must be specified at two points. Instead of simply setting the value of $\psi$, typical applications might instead specify the relation between $\psi$ and $\psi^\prime \equiv d\psi/dx$. Similarly, the boundary conditions may be imposed in some limit, for example $\vert x\vert\rightarrow\infty$, instead of being imposed at particular points.

The form of the equations and the boundary conditions are such that only certain values of the energy $E$ lead to a nontrivial solution. The values of $E$ for which a solution exists are called eigenvalues, and the associated solutions $\psi$ are eigenfunctions. In many circumstances, the eigenvalues form a discrete set (that is, the energy of the system is quantized). An eigenvalue problem can be recast into a form accessible to the numerical methods described earlier, as follows. The general problem can be written in the form

$$\frac{dy}{dx} = f(y, x; \lambda)\,,$$

where $y = (y_1,...,y_N)$ is a vector of dimension $N$ and $\lambda$ is a parameter. If we introduce the new variables $z_i = y_i~~(y = 1,...,N),~z_{N+1} = \lambda$, with

$$\frac{dz_{N+1}}{dx} = 0\,,$$

then the $N+1$-dimensional system is in standard form:

$$\frac{dz}{dx} = f(z, x)\,.$$

With this information, we can now proceed to solve eigenvalue problems using the boundary-value problem tools we already have in hand.