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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

$\displaystyle \psi(x,t) = \psi_0 {\rm e}^{-{\rm i} (\omega t-k x)},$ (1094)

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 (1082) and (1080), 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

$\displaystyle E = K+ U,$ (1095)

where

$\displaystyle K = \frac{p^2}{2\,m}$ (1096)

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

$\displaystyle E\,\psi = (K+U)\,\psi$ (1097)

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

However, it follows from Equations (1080) and (1094) that

$\displaystyle \frac{\partial \psi}{\partial t} = -{\rm i} \omega \psi_0 {\rm e}^{-{\rm i} (\omega t-k x)} = -{\rm i} \frac{E}{\hbar} \psi,$ (1098)

which can be rearranged to give

$\displaystyle E \psi= {\rm i} \hbar \frac{\partial\psi}{\partial t}.$ (1099)

Likewise, from Equations (1082) and (1094),

$\displaystyle \frac{\partial^2\psi}{\partial x^2} = - k^2\,\psi_0 \,{\rm e}^{-{\rm i}\,(k\,x-\omega\,t)} = - \frac{p^2}{\hbar^2}\,\psi,$ (1100)

which can be rearranged to give

$\displaystyle K\,\psi=\frac{p^2}{2\,m}\,\psi = -\frac{\hbar^2}{2\,m}\,\frac{\partial^2\psi}{\partial x^2}.$ (1101)

Thus, combining Equations (1097), (1099), and (1101), we obtain

$\displaystyle {\rm i}\,\hbar\,\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2\,m}\,\frac{\partial^2\psi}{\partial x^2} + U(x)\,\psi.$ (1102)

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 (1094) is a solution of Schrödinger's equation, (1102), provided

$\displaystyle \omega = \frac{\hbar}{2\,m}\,k^2.$ (1103)

The previous 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

$\displaystyle v_p = \frac{\omega}{k} = \frac{\hbar\,k}{2\,m} = \frac{p}{2\,m},$ (1104)

where use has been made of Equation (1082). However, this phase velocity is only half the classical velocity, $ v=p/m$ , of a massive (non-relativistic) particle.


next up previous
Next: Probability Interpretation of Wavefunction Up: Wave Mechanics Previous: Representation of Waves via
Richard Fitzpatrick 2013-04-08