Schrödinger's Equation
The basic premise of wave mechanics is that a massive particle of energy
and linear momentum
, moving in the
-direction (say),
can be represented by a one-dimensional complex wavefunction of the form
![$\displaystyle \psi(x,t) = \psi_0\,{\rm e}^{-{\rm i}\,(\omega\,t-k\,x)},$](img3822.png) |
(11.15) |
where the complex amplitude,
, is arbitrary, while the wavenumber,
, and the angular frequency,
,
are related to the particle momentum,
, and energy,
, 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
, moving in a one-dimensional potential
, satisfies the energy conservation
equation
![$\displaystyle E = K+ U,$](img3828.png) |
(11.16) |
where
![$\displaystyle K = \frac{p^{\,2}}{2\,m}$](img3829.png) |
(11.17) |
is the particle's kinetic energy (Fitzpatrick 2012). Hence,
![$\displaystyle E\,\psi = (K+U)\,\psi$](img3830.png) |
(11.18) |
is a valid, but not obviously useful, wave equation.
However, it follows from Equations (11.1) and (11.15) that
![$\displaystyle \frac{\partial \psi}{\partial t} = -{\rm i}\,\omega\,\psi_0\,{\rm e}^{-{\rm i}\,(\omega\,t-k\,x)} = -{\rm i}\,\frac{E}{\hbar}\,\psi,$](img3831.png) |
(11.19) |
which can be rearranged to give
![$\displaystyle E\,\psi= {\rm i}\,\hbar\,\frac{\partial\psi}{\partial t}.$](img3832.png) |
(11.20) |
Likewise, from Equations (11.3) and (11.15),
![$\displaystyle \frac{\partial^{\,2}\psi}{\partial x^{\,2}} = - k^{\,2}\,\psi_0 \,{\rm e}^{-{\rm i}\,(k\,x-\omega\,t)} = - \frac{p^{\,2}}{\hbar^{\,2}}\,\psi,$](img3833.png) |
(11.21) |
which can be rearranged to give
![$\displaystyle K\,\psi=\frac{p^{\,2}}{2\,m}\,\psi = -\frac{\hbar^{\,2}}{2\,m}\,\frac{\partial^{\,2}\psi}{\partial x^{\,2}}.$](img3834.png) |
(11.22) |
Thus, combining Equations (11.18), (11.20), and (11.22), we obtain
![$\displaystyle {\rm i}\,\hbar\,\frac{\partial\psi}{\partial t} = -\frac{\hbar^{\,2}}{2\,m}\,\frac{\partial^{\,2}\psi}{\partial x^{\,2}} + U(x)\,\psi.$](img3835.png) |
(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.,
), the complex wavefunction (11.15) is a
solution of Schrödinger's equation, (11.23), provided
![$\displaystyle \omega = \frac{\hbar}{2\,m}\,k^{\,2}.$](img3837.png) |
(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
![$\displaystyle v_p = \frac{\omega}{k} = \frac{\hbar\,k}{2\,m} = \frac{p}{2\,m},$](img3838.png) |
(11.25) |
where use has been made of Equation (11.3). However, this phase velocity is only half the classical velocity,
,
of a massive (non-relativistic) particle.