Schrödinger's Equation

The basic premise of wave mechanics is that a massive particle of energy $E$ and linear momentum $p$, moving in the $x$-direction (say), can be represented by a one-dimensional complex wavefunction of the form

$$\psi(x,t) = \psi_0\,{\rm e}^{\,{\rm i}\,(k\,x-\omega\,t)},$$ (784)

where the complex amplitude, $\psi_0$, is arbitrary, whilst the wavenumber, $k$, and the angular frequency, $\omega$, are related to the particle momentum, $p$, and energy, $E$, via the fundamental relations (773) and (771), respectively. Now, the above one-dimensional wavefunction is, presumably, the solution of some one-dimensional wave equation that determines how the wavefunction evolves in time. As described below, we can guess the form of this wave equation by drawing an analogy with classical physics.

A classical particle of mass $m$, moving in a one-dimensional potential $U(x)$, satisfies the energy conservation equation

$$E = K+ U,$$ (785)

where

$$K = \frac{p^2}{2\,m}$$ (786)

is the particle's kinetic energy. Hence,

$$E\,\psi = (K+U)\,\psi$$ (787)

is a valid, but not obviously useful, wave equation.

However, it follows from Equations (785) and (771) that

$$\frac{\partial \psi}{\partial t} = -{\rm i}\,\omega\,\psi_0\... ...,{\rm i}\,(k\,x-\omega\,t)} =- {\rm i}\,\frac{E}{\hbar}\,\psi,$$ (788)

which can be rearranged to give

$$E\,\psi= {\rm i}\,\hbar\,\frac{\partial\psi}{\partial t}.$$ (789)

Likewise, from (785) and (773),

$$\frac{\partial^2\psi}{\partial x^2} = - k^2\,\psi_0 \,{\rm e}^{\,{\rm i}\,(k\,x-\omega\,t)} = - \frac{p^2}{\hbar^2}\,\psi,$$ (790)

which can be rearranged to give

$$\frac{p^2}{2\,m}\,\psi = -\frac{\hbar^2}{2\,m}\,\frac{\partial^2\psi}{\partial x^2}.$$ (791)

Thus, combining Equations (788), (790), and (792), we obtain

$${\rm i}\,\hbar\,\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2\,m}\,\frac{\partial^2\psi}{\partial x^2} + U(x)\,\psi.$$ (792)

This equation, which is known as Schrödinger's equation--since it was first formulated by Erwin Schrödinder in 1926--is the fundamental equation of wave mechanics.

Now, for a massive particle moving in free space (i.e., $U=0$), the complex wavefunction (785) is a solution of Schrödinger's equation, (793), provided that

$$\omega = \frac{\hbar}{2\,m}\,k^2.$$ (793)

The above expression can be thought of as the dispersion relation (see Section 5.1) for matter waves in free space. The associated phase velocity (see Section 7.2) is

$$v_p = \frac{\omega}{k} = \frac{\hbar\,k}{2\,m} = \frac{p}{2\,m},$$ (794)

where use has been made of (773). Note that this phase velocity is only half the classical velocity, $v=p/m$, of a massive (non-relativistic) particle.