Up to now, we have only discussed wave mechanics for a particle moving in one dimension. However, the
generalization to a particle moving in three dimensions is fairly straightforward.
A massive particle moving in three dimensions
has a complex wavefunction of the form [cf., Equation (11.15)]
![$\displaystyle \psi(x,y,z,t) = \psi_0\,{\rm e}^{-{\rm i}\,(\omega\,t-{\bf k}\cdot{\bf r})},$](img4134.png) |
(11.151) |
where
is a complex constant, and
. Here, the wavevector,
, and
the angular frequency,
, are related to the particle momentum,
, and energy,
, according
to [cf., Equation (11.3)]
![$\displaystyle {\bf p} = \hbar\,{\bf k},$](img4136.png) |
(11.152) |
and [cf., Equation (11.1)]
![$\displaystyle E = \hbar\,\omega,$](img4137.png) |
(11.153) |
respectively. Generalizing the
analysis of Section 11.5, the three-dimensional version of Schrödinger's
equation is [cf., Equation (11.23)]
![$\displaystyle {\rm i}\,\hbar\,\frac{\partial\psi}{\partial t} = - \frac{\hbar^{\,2}}{2\,m}\,\nabla^{\,2}\psi + U({\bf r})\,\psi,$](img4138.png) |
(11.154) |
where the differential operator
![$\displaystyle \nabla^{\,2} \equiv \frac{\partial^{\,2}}{\partial x^{\,2}} + \frac{\partial^{\,2} }{\partial y^{\,2}} + \frac{\partial^{\,2}}{\partial z^{\,2}}$](img4139.png) |
(11.155) |
is known as the Laplacian. The interpretation of a three-dimensional wavefunction is that the
probability of simultaneously finding the particle between
and
, between
and
, and
between
and
, at time
is [cf., Equation (11.26)]
![$\displaystyle P(x,y,z,t) = \vert\psi(x,y,z,t)\vert^{\,2}\,dx\,dy\,dz.$](img4140.png) |
(11.156) |
Moreover, the normalization condition for the wavefunction becomes [cf., Equation (11.28)]
![$\displaystyle \int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty\vert\psi(x,y,z,t)\vert^{\,2}\,dx\,dy\,dz =1.$](img4141.png) |
(11.157) |
It can be demonstrated that Schrödinger's equation, (11.154), preserves the normalization
condition, (11.157), of a localized wavefunction (Gasiorowicz 1996).
Heisenberg's uncertainty principle generalizes to [cf., Equation (11.56)]
![$\displaystyle {\mit\Delta} x\,{\mit\Delta} p_x\gtrsim$](img4142.png) |
![$\displaystyle \,\frac{\hbar}{2},$](img4143.png) |
(11.158) |
![$\displaystyle {\mit\Delta} y\,{\mit\Delta} p_y\gtrsim$](img4144.png) |
![$\displaystyle \,\frac{\hbar}{2},$](img4143.png) |
(11.159) |
![$\displaystyle {\mit\Delta} z\,{\mit\Delta} p_z\gtrsim$](img4145.png) |
![$\displaystyle \,\frac{\hbar}{2}.$](img4146.png) |
(11.160) |
Finally, a stationary state of energy
is written [cf., Equation (11.60)]
![$\displaystyle \psi(x,y,z,t) = \psi(x,y,z)\,{\rm e}^{-{\rm i}\,(E/\hbar)\,t},$](img4147.png) |
(11.161) |
where the stationary wavefunction,
, satisfies [cf., Equation (11.62)]
![$\displaystyle - \frac{\hbar^{\,2}}{2\,m}\,\nabla^{\,2}\psi + U({\bf r})\,\psi = E\,\psi.$](img4149.png) |
(11.162) |