Three-Dimensional Wave Mechanics

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 (4.10)]

$\displaystyle \psi(x,y,z,t) = \psi_0\,{\rm e}^{{\rm i}\,({\bf k}\cdot{\bf r}-\omega\,t)},$

(4.181)

where $\psi_0$ is a complex constant, and ${\bf r} = (x,\,y,\,z)$ . Here, the wavevector, ${\bf k}$ , and the angular frequency, $\omega$ , are related to the particle momentum, ${\bf p}$ , and energy, , according to [cf., Equation (4.9)]

$\displaystyle {\bf p } = \hbar\,{\bf k},$

(4.182)

and [cf., Equation (4.8)]

$\displaystyle E =\hbar\,\omega,$

(4.183)

respectively. Generalizing the analysis of Section 4.2.2, the three-dimensional version of Schrödinger's equation is [cf., Equation (4.22)]

$\displaystyle {\rm i}\,\hbar\,\frac{\partial\psi}{\partial t} = - \frac{\hbar^{2}}{2\,m}\,\nabla^{2}\psi + U({\bf r})\,\psi,$

(4.184)

where the differential operator

$\displaystyle \nabla^{2} \equiv \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2} }{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}$

(4.185)

is known as the Laplacian. (See Section A.21.) 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 (4.25)]

$\displaystyle P(x,y,z,t) = \vert\psi(x,y,z,t)\vert^{2}\,dx\,dy\,dz.$

(4.186)

Moreover, the normalization condition for the wavefunction becomes [cf., Equation (4.27)]

$\displaystyle \int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty\vert\psi(x,y,z,t)\vert^{2}\,dx\,dy\,dz =1.$

(4.187)

It can be demonstrated that Schrödinger's equation, (4.184), preserves the normalization condition, (4.187), of a localized wavefunction. Heisenberg's uncertainty principle generalizes to [cf., Equation (4.65)]

$\displaystyle {\mit\Delta} x\,{\mit\Delta} p_x\gtrsim$	$\displaystyle \,\frac{\hbar}{2},$	(4.188)
$\displaystyle {\mit\Delta} y\,{\mit\Delta} p_y\gtrsim$	$\displaystyle \,\frac{\hbar}{2},$	(4.189)
$\displaystyle {\mit\Delta} z\,{\mit\Delta} p_z\gtrsim$	$\displaystyle \,\frac{\hbar}{2}.$	(4.190)

Finally, a stationary state of energy is written [cf., Equation (4.69)]

$\displaystyle \psi(x,y,z,t) = \psi(x,y,z)\,{\rm e}^{-{\rm i}\,(E/\hbar)\,t},$

(4.191)

where the stationary wavefunction, $\psi(x,y,z)$ , satisfies [cf., Equation (4.71)]

$\displaystyle - \frac{\hbar^{2}}{2\,m}\,\nabla^{2}\psi + U({\bf r})\,\psi = E\,\psi.$

(4.192)