Derivation of time independent Schrodinger equation

The time-independent Schrödinger equation is obtained by combining the classical wave equation with de Broglie's relation $\lambda = h/p$. It describes the spatial part $\psi(x)$ of a particle's wave function in a potential $V(x)$.

Step 1: Classical wave

A wave travelling in the $+x$ direction is

$$y = A\sin 2\pi\left(ft - \frac{x}{\lambda}\right)$$

Differentiate with respect to $x$:

$$\frac{\partial y}{\partial x} = -\frac{2\pi}{\lambda}A\cos2\pi\left(ft - \frac{x}{\lambda}\right)$$ $$\frac{\partial^2y}{\partial x^2} = -\frac{4\pi^2}{\lambda^2}A\sin2\pi\left(ft - \frac{x}{\lambda}\right) = -\frac{4\pi^2}{\lambda^2}\,y$$

So

$$\frac{\partial^2y}{\partial x^2} + \frac{4\pi^2}{\lambda^2}y = 0 \qquad (1)$$

Step 2: Replace $y$ by the matter wave $\psi$

For a particle, the wave is described by the wave function $\psi$. Separating out the time part ($\psi(x,t) = \psi(x)\,e^{-j\omega t}$), the space part obeys the same equation:

$$\frac{d^2\psi}{dx^2} + \frac{4\pi^2}{\lambda^2}\psi = 0 \qquad (2)$$

Step 3: Use de Broglie's relation

$\lambda = \dfrac{h}{p}$, so $\dfrac{1}{\lambda^2} = \dfrac{p^2}{h^2}$.

The total energy is $E = \dfrac{p^2}{2m} + V$, so

$$p^2 = 2m(E - V) \quad\Rightarrow\quad \frac{1}{\lambda^2} = \frac{2m(E - V)}{h^2}$$

Step 4: Substitute in (2)

$$\frac{d^2\psi}{dx^2} + \frac{4\pi^2\cdot2m(E - V)}{h^2}\psi = 0$$ $$\frac{d^2\psi}{dx^2} + \frac{8\pi^2m}{h^2}(E - V)\psi = 0$$

Since $\hbar = h/2\pi$, $\dfrac{8\pi^2m}{h^2} = \dfrac{2m}{\hbar^2}$:

$$\frac{d^2\psi}{dx^2} + \frac{2m}{\hbar^2}(E - V)\psi = 0$$

or, in the usual form,

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

This is the time-independent Schrödinger equation in one dimension. In three dimensions $d^2/dx^2$ is replaced by $\nabla^2$:

$$\nabla^2\psi + \frac{2m}{\hbar^2}(E - V)\psi = 0$$

Here $\psi$ is the wave function, $m$ the mass, $E$ the total energy, $V$ the potential energy and $\hbar = h/2\pi$. It is used for problems where $V$ does not depend on time, such as the potential well, the barrier (tunneling) and the hydrogen atom.