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 (11.15)]

$$\psi(x,y,z,t) = \psi_0\,{\rm e}^{-{\rm i}\,(\omega\,t-{\bf k}\cdot{\bf r})},$$ (11.151)

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, $E$, according to [cf., Equation (11.3)]

$${\bf p} = \hbar\,{\bf k},$$ (11.152)

and [cf., Equation (11.1)]

$$E = \hbar\,\omega,$$ (11.153)

respectively. Generalizing the analysis of Section 11.5, the three-dimensional version of Schrödinger's equation is [cf., Equation (11.23)]

$${\rm i}\,\hbar\,\frac{\partial\psi}{\partial t} = - \frac{\hbar^{\,2}}{2\,m}\,\nabla^{\,2}\psi + U({\bf r})\,\psi,$$ (11.154)

where the differential operator

$$\nabla^{\,2} \equiv \frac{\partial^{\,2}}{\partial x^{\,2}} + \frac{\partial^{\,2} }{\partial y^{\,2}} + \frac{\partial^{\,2}}{\partial z^{\,2}}$$ (11.155)

is known as the Laplacian. The interpretation of a three-dimensional wavefunction is that the probability of simultaneously finding the particle between $x$ and $x+dx$, between $y$ and $y+dy$, and between $z$ and $z+dz$, at time $t$ is [cf., Equation (11.26)]

$$P(x,y,z,t) = \vert\psi(x,y,z,t)\vert^{\,2}\,dx\,dy\,dz.$$ (11.156)

Moreover, the normalization condition for the wavefunction becomes [cf., Equation (11.28)]

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

$${\mit\Delta} x\,{\mit\Delta} p_x\gtrsim \,\frac{\hbar}{2},$$ (11.158)

$${\mit\Delta} y\,{\mit\Delta} p_y\gtrsim \,\frac{\hbar}{2},$$ (11.159)

$${\mit\Delta} z\,{\mit\Delta} p_z\gtrsim \,\frac{\hbar}{2}.$$ (11.160)

Finally, a stationary state of energy $E$ is written [cf., Equation (11.60)]

$$\psi(x,y,z,t) = \psi(x,y,z)\,{\rm e}^{-{\rm i}\,(E/\hbar)\,t},$$ (11.161)

where the stationary wavefunction, $\psi(x,y,z)$, satisfies [cf., Equation (11.62)]

$$- \frac{\hbar^{\,2}}{2\,m}\,\nabla^{\,2}\psi + U({\bf r})\,\psi = E\,\psi.$$ (11.162)