Quantum Mechanics — Lecture notes for PHYS223 XI Perturbation theory XIII Three-dimensional examples and applications

XII Quantum mechanics in three dimensions

XII.1 Coordinates and wavefunction

Three-dimensional space ${\bf r}=x{\bf i}+y{\bf j}+z{\bf k}$ is spanned by three basis vectors ${\bf i}$ , ${\bf j}$ , ${\bf k}$ with coordinates $x$ , $y$ and $z$ .

The state of a system is described by a wavefunction $\psi({\bf r})=\psi(x,y,z)$ .

XII.2 Position operators

The coordinates are associated with three position operators $\hat{x}$ , $\hat{y}$ , $\hat{z}$ which act as

\hat{x}\psi({\bf r})=x\psi({\bf r}),\quad\hat{y}\psi({\bf r})=y\psi({\bf r}),% \quad\hat{z}\psi({\bf r})=z\psi({\bf r}).

(184)

These coordinates commute since $(\hat{x}\hat{y}-\hat{y}\hat{x})\psi(x,y,z)=xy\psi(x,y,z)-yx\psi(x,y,z)=0$ etc. Hence $[\hat{x},\hat{y}]=0$ , $[\hat{x},\hat{z}]=0$ , $[\hat{y},\hat{z}]=0$ . Therefore, $x$ , $y$ and $z$ are simultaneous observables (they can be measured simultaneously without affecting each other). Indeed, Heisenberg’s uncertainty relation gives, e.g., $\Delta x\Delta y\geq 0$ , so that it is possible to determine both $x$ and $y$ with no uncertainty, $\Delta x=\Delta y=0$ .

XII.3 Momentum operators

Momentum ${\bf p}=p_{x}{\bf i}+p_{y}{\bf j}+p_{z}{\bf k}$ is associated with momentum operators

\hat{p}_{x}=-i\hbar\frac{\partial}{\partial x},\quad\hat{p}_{y}=-i\hbar\frac{% \partial}{\partial y},\quad\hat{p}_{z}=-i\hbar\frac{\partial}{\partial z},

(185)

which act as

\hat{p}_{x}\psi({\bf r})=-i\hbar\frac{\partial\psi({\bf r})}{\partial x},

(186)

\hat{p}_{y}\psi({\bf r})=-i\hbar\frac{\partial\psi({\bf r})}{\partial y},

(187)

\hat{p}_{z}\psi({\bf r})=-i\hbar\frac{\partial\psi({\bf r})}{\partial z}.

(188)

The momentum operators commute with each other because the order of differentiation does not matter for any function $\psi({\bf r})$ :

\frac{\partial^{2}\psi({\bf r})}{\partial x\partial y}=\frac{\partial^{2}\psi(% {\bf r})}{\partial y\partial x}.

(189)

Hence $[\hat{p}_{x},\hat{p}_{y}]=0$ , $[\hat{p}_{x},\hat{p}_{z}]=0$ , $[\hat{p}_{y},\hat{p}_{z}]=0$ .

XII.4 Commutators between position and momentum

From one dimension we already know $[\hat{x},\hat{p}_{x}]=i\hbar$ . This also translates to the commutators $[\hat{y},\hat{p}_{y}]=i\hbar$ , $[\hat{z},\hat{p}_{z}]=i\hbar$ .

However, the following commutators vanish: $[\hat{x},\hat{p}_{y}]=0$ , $[\hat{x},\hat{p}_{z}]=0$ , $[\hat{y},\hat{p}_{x}]=0$ , $[\hat{y},\hat{p}_{z}]=0$ , $[\hat{z},\hat{p}_{x}]=0$ , $[\hat{z},\hat{p}_{y}]=0$ .

XII.5 Momentum eigenstates

The normalised momentum eigenfunctions in three dimensions are given by

\displaystyle\psi_{\bf p}({\bf r})

\displaystyle=

\displaystyle(2\pi\hbar)^{-3/2}\exp(i{\bf p}\cdot{\bf r}/\hbar)

(190)

where ${\bf p}=p_{x}{\bf i}+p_{y}{\bf j}+p_{z}{\bf k}$ .

They can also be written as

\displaystyle\psi_{\bf p}({\bf r})

\displaystyle=

\displaystyle\psi_{p_{x}}(x)\psi_{p_{y}}(y)\psi_{p_{z}}(z)

(191)

where $\psi_{p}(x)=(2\pi\hbar)^{-1/2}\exp(ipx/\hbar)$ .

Indeed we find

\hat{p}_{x}\psi_{\bf p}({\bf r})=p_{x}\psi_{\bf p}({\bf r}),

(192)

\hat{p}_{y}\psi_{\bf p}({\bf r})=p_{y}\psi_{\bf p}({\bf r}),

(193)

\hat{p}_{z}\psi_{\bf p}({\bf r})=p_{z}\psi_{\bf p}({\bf r}).

(194)

XII.6 Dirac notation

In Dirac notation, we denote states as $|\psi\rangle$ . In order to establish the connection to the wave function $\psi({\bf r})$ in three dimensions, we employ the position basis $|{\bf r}\rangle$ with $\hat{x}|{\bf r}\rangle=x|{\bf r}\rangle$ etc, and write

|\psi\rangle=\iiint d{\bf r}\,\psi({\bf r})|{\bf r}\rangle.

(195)

Alternatively, we may use the momentum basis $|{\bf p}\rangle$ with $\hat{p}_{x}|{\bf p}\rangle=p_{x}|{\bf p}\rangle$ etc, and write

|\psi\rangle=\iiint d{\bf p}\,\tilde{\psi}({\bf p})|{\bf p}\rangle.

(196)

As $\langle{\bf r}|{\bf p}\rangle=(2\pi\hbar)^{-3/2}\exp(i{\bf p}\cdot{\bf r}/\hbar)$ , the expansion coefficients $\psi({\bf r})=\langle{\bf r}|\psi\rangle$ and $\tilde{\psi}({\bf p})=\langle{\bf p}|\psi\rangle$ in both basis sets are related by a three-dimensional Fourier transformation,

\psi({\bf r})=\iiint d{\bf p}\,\tilde{\psi}({\bf p})\langle{\bf r}|{\bf p}\rangle.

(197)

XII.7 Schrödinger equation in three dimensions

In three dimensions the Hamiltonian for a point particle of mass $m$ is given by

\hat{H}=\frac{\hat{p}_{x}^{2}+\hat{p}_{y}^{2}+\hat{p}_{z}^{2}}{2m}+V(\hat{\bf r% })=-\frac{\hbar^{2}}{2m}\Delta+V(\hat{\bf r})

(198)

where $\Delta=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}% +\frac{\partial^{2}}{\partial z^{2}}$ is the Laplace operator. In position representation, the stationary Schrödinger equation $E|\psi\rangle=\hat{H}|\psi\rangle$ is given by

E\psi({\bf r})=-\frac{\hbar^{2}}{2m}\Delta\psi({\bf r})+V({\bf r})\psi({\bf r}).

(199)