nLab
Schrödinger equation

Idea

The Schrödinger equation (named after Erwin Schrödinger) is the evolution equation of quantum mechanics in the Schrödinger picture. Its simplest version results from replacing the classical expressions in the nonrelativistic, mechanical equation for the energy of a pointparticle, by operators on a Hilbert space:

We start with a point particle with mass $m$, impulse $p$ moving in the space ${ℝ}^3$ with a given potential function $V$, the energy of it is the sum of kinetic and potential energy:

$$E=\frac{p^2}{2m}+V$$E = \frac{p^2}{2 m} + V

Quantizing this equation means replacing the coordinate $x\in {ℝ}^3$ with the Hilbert space $L^2({ℝ}^3)$ and

$$E\to i\hslash \frac{\partial }{\partial t}$$E \to i \hbar \frac{\partial}{\partial t}

$$p\to -i\hslash \nabla$$p \to -i \hbar \nabla

with the Planck constant $h$ and

$$\hslash =\frac{h}{2\pi }$$\hbar = \frac{h}{2 \pi}

the reduced Planck constant.

This results in the Schrödinger equation for a single particle in a potential:

$$i\hslash \frac{\partial }{\partial t}\psi (t,x)=-\frac{{\hslash }^2}{2m}{\nabla }^2\psi (t,x)+V(t,x)\psi (t,x)$$i \hbar \frac{\partial}{\partial t} \psi(t, x) = - \frac{\hbar^2}{2 m} \nabla^2 \psi(t, x) + V(t, x) \psi(t, x)

The last term is the multiplication of the functions $V$ and $\psi$.

The right hand side is called the Hamilton operator $H$, the Schrödinger equation is therefore mostly stated in this form:

$$i\hslash {\psi }_t=H\psi$$i \hbar \psi_t = H \psi

Abstract

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Definition

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Properties

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Examples

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References

Any introductory textbook about quantum mechanics will explain the Schrödinger equation (from the viewpoint of physicists mostly).

• Wikipedia: Schrödinger equation