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

$$\psi(x,t) = \psi_0\,{\rm e}^{-{\rm i}\,(\omega\,t-k\,x)},$$ (11.15)

where the complex amplitude, $\psi_0$, is arbitrary, while the wavenumber, $k$, and the angular frequency, $\omega$, are related to the particle momentum, $p$, and energy, $E$, via the fundamental relations (11.3) and (11.1), respectively. The previous one-dimensional wavefunction is the solution of a 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

$$E = K+ U,$$ (11.16)

where

$$K = \frac{p^{\,2}}{2\,m}$$ (11.17)

is the particle's kinetic energy (Fitzpatrick 2012). Hence,

$$E\,\psi = (K+U)\,\psi$$ (11.18)

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

However, it follows from Equations (11.1) and (11.15) that

$$\frac{\partial \psi}{\partial t} = -{\rm i}\,\omega\,\psi_0\,{\rm e}^{-{\rm i}\,(\omega\,t-k\,x)} = -{\rm i}\,\frac{E}{\hbar}\,\psi,$$ (11.19)

which can be rearranged to give

$$E\,\psi= {\rm i}\,\hbar\,\frac{\partial\psi}{\partial t}.$$ (11.20)

Likewise, from Equations (11.3) and (11.15),

$$\frac{\partial^{\,2}\psi}{\partial x^{\,2}} = - k^{\,2}\,\psi_0 \,{\rm e}^{-{\rm i}\,(k\,x-\omega\,t)} = - \frac{p^{\,2}}{\hbar^{\,2}}\,\psi,$$ (11.21)

which can be rearranged to give

$$K\,\psi=\frac{p^{\,2}}{2\,m}\,\psi = -\frac{\hbar^{\,2}}{2\,m}\,\frac{\partial^{\,2}\psi}{\partial x^{\,2}}.$$ (11.22)

Thus, combining Equations (11.18), (11.20), and (11.22), we obtain

$${\rm i}\,\hbar\,\frac{\partial\psi}{\partial t} = -\frac{\hbar^{\,2}}{2\,m}\,\frac{\partial^{\,2}\psi}{\partial x^{\,2}} + U(x)\,\psi.$$ (11.23)

This equation, which is known as Schrödinger's equation—because it was first formulated by Erwin Schrödinder in 1926—is the fundamental equation of wave mechanics (Dirac 1982).

For a massive particle moving in free space (i.e., $U=0$), the complex wavefunction (11.15) is a solution of Schrödinger's equation, (11.23), provided

$$\omega = \frac{\hbar}{2\,m}\,k^{\,2}.$$ (11.24)

The previous expression can be thought of as the dispersion relation (see Section 4.2) for matter waves in free space. The associated phase velocity (see Section 6.3) is

$$v_p = \frac{\omega}{k} = \frac{\hbar\,k}{2\,m} = \frac{p}{2\,m},$$ (11.25)

where use has been made of Equation (11.3). However, this phase velocity is only half the classical velocity, $v=p/m$, of a massive (non-relativistic) particle.