Hilbert Spaces, Fourier Analysis, and Representations

A Hilbert space is a complete inner product space. Completeness means that every Cauchy sequence of kets converges to a ket in the space. Finite-dimensional inner product spaces are automatically complete, but infinite-dimensional spaces such as spaces of functions require this extra condition.

The additional structure of Hilbert spaces allows us to retain most of the intuition from finite-dimensional linear algebra while dealing with continuous variables. We still have inner products, norms, orthonormal bases, linear operators, adjoints, and spectral theory, though some technical issues regarding domains arise.

Examples of Hilbert spaces include $\mathbb{C}^n$ with the standard inner product and the infinite-dimensional space $L^2(\mathbb{R})$ of square-integrable wave functions. The latter is the natural home for the states of a single particle moving in one dimension.

In infinite dimensions, the spectral theorem becomes more general. Instead of a sum over eigenvectors, it often involves integrals over projection-valued measures. Discrete spectra correspond to sums, continuous spectra to integrals, and many physical systems have both.

The space $L^2(\mathbb{R})$ consists of square-integrable wave functions. In the position representation, a state $|\psi\rangle$ is described by the wave function $\psi(x) = \langle x|\psi\rangle$, which satisfies $\int_{-\infty}^{+\infty} |\psi(x)|^2 \,dx < \infty$. The inner product is $\langle\phi|\psi\rangle = \int_{-\infty}^{+\infty} \phi^*(x)\psi(x) \,dx$. Functions that differ only on a set of measure zero are identified.

This space is complete under the metric induced by the inner product, making it a Hilbert space. It is also separable, meaning it has a countable orthonormal basis. The Hermite functions, harmonic-oscillator eigenfunctions, form one such basis.

States in quantum mechanics are elements of $L^2(\mathbb{R})$. The condition $\int |\psi|^2 = 1$ is the normalization condition. Expectation values of functions of position are computed by integrals against $|\psi(x)|^2$.

Not every state in $L^2(\mathbb{R})$ is a smooth function. The space contains distributions such as the Dirac delta only in a generalized sense. Rigorous treatment of these objects requires the theory of distributions or rigged Hilbert spaces.

In Dirac notation, the position basis consists of generalized eigenstates $|x\rangle$ of the position operator $\hat{x}$, with $\hat{x}|x\rangle = x|x\rangle$. These states are not normalizable in the usual sense, but they satisfy the completeness relation $\int |x\rangle\langle x| \,dx = I$ and the orthogonality relation $\langle x|x^\prime\rangle = \delta(x - x^\prime)$.

A general state $|\psi\rangle$ can be expanded as $|\psi\rangle = \int \psi(x) |x\rangle \,dx$, where the coefficient $\psi(x) = \langle x|\psi\rangle$ is the position-space wave function. The probability density for finding the particle near x is $|\psi(x)|^2$.

Similarly, the momentum basis $|p\rangle$ satisfies $\hat{p}|p\rangle = p|p\rangle$, $\int |p\rangle\langle p| \,dp = I$, and $\langle p|p^\prime\rangle = \delta(p - p^\prime)$. The momentum-space wave function is $\tilde{\psi}(p) = \langle p|\psi\rangle$.

The position and momentum bases are related by a Fourier transform. This relationship is the rigorous statement of wave-particle duality: a state localized in position is delocalized in momentum, and vice versa.

The overlap between position and momentum eigenstates is $\langle x|p\rangle = \frac{1}{\sqrt{2\pi\hbar}} e^{ipx/\hbar}$. This complex exponential is the kernel of the Fourier transform. Expanding $|p\rangle$ in the $|x\rangle$ basis gives $|p\rangle = \int \langle x|p\rangle |x\rangle \,dx$.

Therefore the momentum-space wave function is the Fourier transform of the position-space wave function: $\tilde{\psi}(p) = \langle p|\psi\rangle = \int \langle p|x\rangle\langle x|\psi\rangle \,dx = \frac{1}{\sqrt{2\pi\hbar}} \int e^{-ipx/\hbar} \psi(x) \,dx$. The inverse transform recovers $\psi(x)$.

The Fourier transform is a unitary map from $L^2(\mathbb{R})$ to $L^2(\mathbb{R})$. It preserves inner products and norms, as expressed by Plancherel's theorem. This unitarity is why the position and momentum representations are physically equivalent.

Changing between position and momentum representations is exactly analogous to changing between two orthonormal bases in finite dimensions. The role of the unitary change-of-basis matrix is played by the integral kernel $\langle x|p\rangle$.

Continuous eigenstates such as $|x\rangle$ and $|p\rangle$ cannot be normalized to 1 because they are infinitely localized. Instead, they are normalized to a Dirac delta function. The relation $\langle x|x^\prime\rangle = \delta(x - x^\prime)$ expresses this generalized orthonormality.

The Dirac delta $\delta(x - x^\prime)$ is not a function in the ordinary sense but a distribution. It is defined by its action on test functions: $\int f(x) \delta(x - x^\prime) \,dx = f(x^\prime)$. It can be viewed as the limit of a sequence of sharply peaked, normalized functions.

Completeness relations with delta normalization integrate to the identity. For example, $\int |x\rangle\langle x| \,dx = I$ means that for any $|\psi\rangle$, $\int \langle\phi|x\rangle\langle x|\psi\rangle \,dx = \langle\phi|\psi\rangle$. This is a continuous version of the discrete completeness relation $\sum_i |e_i\rangle\langle e_i| = I$.

Delta-function normalization is the price of working with idealized eigenstates of operators with continuous spectra. Physical states are always normalizable wave packets, which can be expanded as integrals over delta-normalized basis states.

An operator has a different form in different representations. The position operator $\hat{x}$ acts as multiplication by x in position space: $\langle x|\hat{x}|\psi\rangle = x \psi(x)$. In momentum space it becomes a derivative: $\langle p|\hat{x}|\psi\rangle = i\hbar \frac{d}{dp} \tilde{\psi}(p)$.

Conversely, the momentum operator $\hat{p}$ acts as $-i\hbar \frac{d}{dx}$ in position space and as multiplication by p in momentum space. These differential forms follow directly from the Fourier transform and the canonical commutation relation $[\hat{x}, \hat{p}] = i\hbar I$.

The expectation value of any observable can be computed in either representation. For example, $\langle x\rangle = \int x |\psi(x)|^2 \,dx = \int \tilde{\psi}^*(p) \left(i\hbar \frac{d}{dp}\right) \tilde{\psi}(p) \,dp$. Both integrals give the same number because the representations are unitarily equivalent.

Switching representations is particularly useful when solving the Schrödinger equation. A free particle is simplest in momentum space, where the Hamiltonian is diagonal, while a particle in a position-dependent potential is usually simpler in position space.

Consider the normalized Gaussian state $|\psi\rangle$ whose position-space wave function is $\psi(x) = \langle x|\psi\rangle = (2a/\pi)^{1/4} e^{-a x^2}$ with $a > 0$. Find its momentum-space representation $\tilde{\psi}(p)=\langle p|\psi\rangle$ and verify that it is also Gaussian.

Step 1 -- Fourier transform. The momentum-space wave function is $\tilde{\psi}(p) = \langle p|\psi\rangle = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{+\infty} e^{-ipx/\hbar} \left(\frac{2a}{\pi}\right)^{1/4} e^{-a x^2} \,dx$. This is a Gaussian Fourier integral.

Step 2 -- Evaluate the integral. Using $\int_{-\infty}^{+\infty} e^{-a x^2 - i b x} \,dx = \sqrt{\frac{\pi}{a}} e^{-b^2/(4a)}$ with $b = p/\hbar$, we obtain $\tilde{\psi}(p) = \left(\frac{1}{2\pi a \hbar^2}\right)^{1/4} e^{-p^2/(4a\hbar^2)}$.

Step 3 -- Interpret. The momentum-space wave function is a Gaussian centered at $p = 0$ with width parameter proportional to $\sqrt{a}$. A narrower position-space Gaussian (large a) corresponds to a broader momentum-space Gaussian. This is the qualitative content of the Heisenberg uncertainty principle, which we can verify gives $\sigma_x \sigma_p = \hbar/2$ for this state.