1.2: Recap- Position and Momentum States
Before proceeding, let us review the properties of quantum particles in free space. In a \(d\) -dimensional space, a coordinate vector \(\mathbf{r}\) is a real vector of \(d\) components. A quantum particle can be described by the position basis—a set of quantum states \(\{|\mathbf{r}\rangle\}\) , one for each possible \(\mathbf{r}\) . If we are studying a particle trapped in a finite region (e.g., a particle in a box), \(\mathbf{r}\) is restricted to that region; otherwise, \(\mathbf{r}\) is any real \(d\) -dimensional vector. In either case, the \(\mathbf{r}\) ’s are continuous, so the position eigenstates form an uncountably infinite set.
The position eigenstates are assumed to span the state space, so the identity operator can be resolved as
\[\hat{I} = \int d^dr \, |\mathbf{r}\rangle \,\langle\mathbf{r}|,\]
where the integral is taken over all allowed \(\mathbf{r}\) . It follows that
\[\langle \mathbf{r} | \mathbf{r}' \rangle = \delta^d(\mathbf{r}-\mathbf{r}').\]
The position eigenstates are thus said to be “delta-function normalized”, rather than being normalized to unity. In the above equation, \(\delta^d(\cdots)\) denotes the \(d\) -dimensional delta function; for example, in 2D,
\[\langle x,y \,|\, x',y' \rangle = \delta(x-x') \, \delta(y-y').\]
The position operator \(\hat{\mathbf{r}}\) is defined by taking \(|\mathbf{r}\rangle\) and \(\mathbf{r}\) as its eigenstates and eigenvalues:
\[\hat{\mathbf{r}} |\mathbf{r}\rangle \,=\, \mathbf{r}\, |\mathbf{r}\rangle.\]
Momentum eigenstates are constructed from position eigenstates via Fourier transforms. First, suppose the allowed region of space is a box of length \(L\) on each side, with periodic boundary conditions in every direction. Define the set of wave-vectors \(\mathbf{k}\) corresponding to plane waves satisfying the periodic boundary conditions at the box boundaries:
\[\nonumber\Big\{\mathbf{k} \; \Big| \;\; k_j = 2\pi m/L\;\;\mathrm{for} \;m\in\mathbb{Z}, \; j = 1, \dots,d\, \Big\}.\]
So long as \(L\) is finite, the \(\mathbf{k}\) vectors are discrete. Now define
\[|\mathbf{k}\rangle = \frac{1}{L^{d/2}} \, \int d^dr \; e^{i\mathbf{k}\cdot\mathbf{r}} |\mathbf{r}\rangle,\]
where the integral is taken over the box. These can be shown to satisfy
\[\langle\mathbf{k}|\mathbf{k}'\rangle = \delta_{\mathbf{k},\mathbf{k}'}, \quad \langle\mathbf{r}|\mathbf{k}'\rangle = \frac{1}{L^{d/2}} e^{i\mathbf{k}\cdot\mathbf{r}}, \quad I = \sum_{\mathbf{k}} |\mathbf{k}\rangle\,\langle\mathbf{k}|.\]
The momentum operator is defined so that its eigenstates are \(\{|\mathbf{k}\rangle\}\) , with \(\hbar\mathbf{k}\) as the corresponding eigenvalues:
\[\hat{\mathbf{p}} |\mathbf{k}\rangle \,=\, \hbar \mathbf{k}\, |\mathbf{k}\rangle.\]
Thus, for finite \(L\) , the momentum eigenstates are discrete and normalizable to unity. The momentum component in each direction is quantized to a multiple of \(\Delta p = 2\pi\hbar/L\) .
We then take the limit of an infinite box, \(L \rightarrow \infty\) . In this limit, \(\Delta p \rightarrow 0\) , so the momentum eigenvalues coalesce into a continuum. It is convenient to re-normalize the momentum eigenstates by taking
\[|\mathbf{k}\rangle \rightarrow \left(\frac{L}{2\pi}\right)^{d/2} |\mathbf{k}\rangle.\]
In the \(L\rightarrow\infty\) limit, the re-normalized momentum eigenstates satisfy
Definition: Re-normalized momentum eigenstates
\[\begin{align} |\mathbf{k}\rangle &= \frac{1}{(2\pi)^{d/2}} \, \int d^dr \; e^{i\mathbf{k}\cdot\mathbf{r}} |\mathbf{r}\rangle, \\ |\mathbf{r}\rangle &= \frac{1}{(2\pi)^{d/2}} \, \int d^dk \; e^{-i\mathbf{k}\cdot\mathbf{r}} |\mathbf{k}\rangle, \\ \langle\mathbf{k}|\mathbf{k}'\rangle = \delta^d(\mathbf{k}-\mathbf{k}'),& \quad \langle\mathbf{r}|\mathbf{k}\rangle = \frac{1}{(2\pi)^{d/2}} e^{i\mathbf{k}\cdot\mathbf{r}}, \quad I = \int d^dk \;|\mathbf{k}\rangle\,\langle\mathbf{k}|. \end{align}\]
The above integrals are taken over infinite space, and the position and momentum eigenstates are now on a similar footing: both are delta-function normalized. In deriving the above equations, it is helpful to use the formula
\[\int_{-\infty}^\infty dx\; \exp(ikx) \;=\; 2\pi\, \delta(k).\]
For an arbitrary quantum state \(|\psi\rangle\) , a wavefunction is defined as the projection onto the position basis: \(\psi(\mathbf{r}) = \langle \mathbf{r}|\psi\rangle\) . Using the momentum eigenstates, we can show that
\[\begin{align} \begin{aligned}\langle \mathbf{r}|\hat{\mathbf{p}}|\psi\rangle &= \int d^dk \; \langle\mathbf{r}|\mathbf{k}\rangle \; \hbar\mathbf{k} \; \langle\mathbf{k}|\psi\rangle \\ &= \int \frac{d^dk}{(2\pi)^{d/2}}\; \hbar\mathbf{k} \;e^{i\mathbf{k}\cdot\mathbf{r}} \langle\mathbf{k}|\psi\rangle \\ &= -i\hbar\nabla \int \frac{d^dk}{(2\pi)^{d/2}}\; \;e^{i\mathbf{k}\cdot\mathbf{r}} \langle\mathbf{k}|\psi\rangle \\ &= -i\hbar \nabla\psi(\mathbf{r}).\end{aligned}\end{align}\]
This result can also be used to prove Heisenberg’s commutation relation \([\hat{r}_i, \hat{p}_j] = i\hbar\delta_{ij}\) .