How is Schrödinger's time-independent wave equation derived for a particle in one dimension?

In one line

Starting from the classical wave equation and substituting de Broglie's \(\lambda = h/p\) with \(p^2 = 2m(E - V)\), we get \(\dfrac{d^2\psi}{dx^2} + \dfrac{8\pi^2m}{h^2}(E - V)\psi = 0\).

Explained simply

We already know how ordinary waves behave. The trick is to take the standard wave equation and replace the wavelength with de Broglie's wavelength, which depends on the particle's energy. The time part separates out, and we are left with an equation describing the wave's shape in space for a fixed energy.

How it works

1. Classical wave equation: \(\dfrac{\partial^2\Psi}{\partial x^2} = \dfrac{1}{u^2}\dfrac{\partial^2\Psi}{\partial t^2}\), where \(u\) is the wave velocity.
2. Assume a solution \(\Psi(x,t) = \psi(x)e^{-i\omega t}\). Then \(\dfrac{\partial^2\Psi}{\partial t^2} = -\omega^2\Psi\).
3. Substituting: \(\dfrac{d^2\psi}{dx^2} + \dfrac{\omega^2}{u^2}\psi = 0\).
4. Since \(\omega = 2\pi\nu\) and \(u = \nu\lambda\), \(\omega/u = 2\pi/\lambda\): \(\dfrac{d^2\psi}{dx^2} + \dfrac{4\pi^2}{\lambda^2}\psi = 0\).
5. de Broglie: \(\lambda = h/p\), so \(\dfrac{4\pi^2}{\lambda^2} = \dfrac{4\pi^2p^2}{h^2}\).
6. Energy: \(E = \dfrac{p^2}{2m} + V\), so \(p^2 = 2m(E - V)\).
7. Therefore \(\dfrac{d^2\psi}{dx^2} + \dfrac{8\pi^2m}{h^2}(E - V)\psi = 0\).
8. In 3D: \(\nabla^2\psi + \dfrac{2m}{\hbar^2}(E - V)\psi = 0\). For a free particle, \(V = 0\).

Key formula

\(\dfrac{d^2\psi}{dx^2} + \dfrac{8\pi^2m}{h^2}(E - V)\psi = 0\), equivalently \(-\dfrac{\hbar^2}{2m}\dfrac{d^2\psi}{dx^2} + V\psi = E\psi\). \(\psi(x)\): space part of wave function, \(m\): mass (kg), \(E\): total energy (J), \(V\): potential energy (J), \(h\): Planck's constant, \(\hbar = h/2\pi\).

Quick example

For a free particle (\(V = 0\)), the equation becomes \(d^2\psi/dx^2 + k^2\psi = 0\) with \(k = \sqrt{2mE}/\hbar\). Its solution \(\psi = Ae^{ikx}\) is a wave of wavelength \(2\pi/k = h/\sqrt{2mE}\), exactly the de Broglie wavelength, showing the equation is consistent.

Common mistakes

• Writing \(4\pi^2 m/h^2\) instead of \(8\pi^2 m/h^2\) (missing the factor 2 from \(p^2 = 2m(E-V)\)).
• Using \((E + V)\) instead of \((E - V)\).
• Keeping \(t\) in the final time-independent equation.

How MU asks it

One of the most expected 5-mark derivations: "Derive Schrödinger's time-independent wave equation". Write every step from the classical wave equation, state the substitution of \(\lambda = h/p\) and \(p^2 = 2m(E-V)\), and box the final 1D and 3D forms.