Schrödinger equation



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Equality and Equivalence



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 mm, impulse pp moving in the space 3\mathbb{R}^3 with a given potential function VV, the energy of it is the sum of kinetic and potential energy:

E=p 22m+V E = \frac{p^2}{2 m} + V

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

Eit E \to i \hbar \frac{\partial}{\partial t}
pi p \to -i \hbar \nabla

with the Planck constant hh and

=h2π \hbar = \frac{h}{2 \pi}

the reduced Planck constant.

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

itψ(t,x)= 22m 2ψ(t,x)+V(t,x)ψ(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 VV and ψ\psi.

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

iψ t=Hψ i \hbar \psi_t = H \psi




Decomposition into phase and amplitude

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

itΨ= 22m 2 2xΨ+VΨ, 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 Ψ\Psi – away from its zero locus – by a complex phase function

S:× S \;\colon\; \mathbb{R}\times \mathbb{R} \longrightarrow \mathbb{R}

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

ρ:× \sqrt{\rho} \;\colon\; \mathbb{R}\times \mathbb{R} \longrightarrow \mathbb{R}

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

Ψexp(iS/)ρ. \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:

tS=12m(xS) 2V+ 22m1ρ 2 2xρ \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}


tρ=x(1m(xS)ρ). \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 SS and the canonical momentum

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

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

    Q∶− 22m1ρ 2 2xρ 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 ρ\rho.)

  2. The second equation has the form of the continuity equation? of the flow expressed by 1mp\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.)



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

Revised on December 20, 2016 16:18:29 by Urs Schreiber (