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 point particle, by operators on a Hilbert space:

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

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

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

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

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

with the Planck constant $h$ and

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

the reduced Planck constant.

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

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

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:

$$i \hbar \psi_t = H \psi$$

Abstract

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Definition

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Properties

Decomposition into phase and amplitude

Consider for simplicity, the mechanical system of a particle of mass $m$ propagating on the real line $\mathbb{R}$ and subject to a potential $V \in C^\infty(\mathbb{R})$, so that the Schrödinger equation is the differential equation on complex-valued functions $\Psi \colon \mathbb{R}\times \mathbb{R} \to \mathbb{C}$ given by

$$i \hbar \frac{\partial}{\partial t} \Psi = \frac{\hbar^2}{2m} \frac{\partial^2}{\partial^2 x} \Psi + V \Psi \,,$$

where $\hbar$ denotes Planck's constant.

By the nature of complex numbers and by the discussion at phase and phase space in physics, it is natural to parameterize $\Psi$ – away from its zero locus – by a complex phase function

$$S \;\colon\; \mathbb{R}\times \mathbb{R} \longrightarrow \mathbb{R}$$

and an absolute value function $\sqrt{\rho}$

$$\sqrt{\rho} \;\colon\; \mathbb{R}\times \mathbb{R} \longrightarrow \mathbb{R}$$

which is positive, $\sqrt{\rho} \gt 0$, as

$$\Psi \coloneqq \exp\left(\frac{i}{\hbar} S / \hbar\right) \sqrt{\rho} \,.$$

Entering this Ansatz into the above Schrödinger equation, that complex equation becomes equivalent to the following two real equations:

$$\frac{\partial}{\partial t} S = - \frac{1}{2m} \left(\frac{\partial}{\partial x}S\right)^2 - V + \frac{\hbar^2}{2m} \frac{1}{\sqrt{\rho}}\frac{\partial^2}{\partial^2 x} \sqrt{\rho}$$

and

$$\frac{\partial}{\partial t} \rho = - \frac{\partial}{\partial x} \left( \frac{1}{m} \left(\frac{\partial}{\partial x}S\right) \rho \right) \,.$$

Now in this form one may notice a similarity of the form of these two equations with other equations from classical mechanics and statistical mechanics:

1. The first equation is similar to the Hamilton-Jacobi equation that expresses the classical action functional $S$ and the canonical momentum

   $$p \coloneqq \frac{\partial}{\partial x} S$$

   except that in addition to the ordinary potential energy $V$ there is an additional term

   $$Q \coloneq \frac{\hbar^2}{2m} \frac{1}{\sqrt{\rho}}\frac{\partial^2}{\partial^2 x} \sqrt{\rho}$$

   which is unlike what may appar in an ordinary Hamilton-Jacobi equation. The perspective of Bohmian mechanics is to regard this as a correction of quantum physics to classical Hamilton-Jacobi theory, it is then called the quantum potential. Notice that unlike ordinary potentials, this “quantum potential” is a function of the density that is subject to the potential. (Notice that this works only away from the zero locus of $\rho$.)

2. The second equation has the form of the continuity equation? of the flow expressed by $\frac{1}{m}p$.

(In the context of Bohmian mechanics one regard this equivalent rewriting of the Schrödinger equation as providing a hidden variable theory formulation of quantum mechanics.)

Examples

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Related concepts

• Tomonaga-Schwinger equation

• Schrödinger picture

• heat equation

• wave equation, Klein-Gordon equation

• Dirac equation

References

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

• Wikipedia: Schrödinger equation