Matrix Mechanics, Hilbert Space & Operator Representations

1. Physical Impetus: The Double-Slit Experiment & Quantum Probability Amplitudes

The conceptual foundation of quantum mechanics rests upon the breakdown of classical probability theory, most dramatically demonstrated by electron diffraction through two closely spaced slits. In classical statistical mechanics, if an event can occur via two mutually exclusive classical alternatives (passing through slit 1 or slit 2), the resultant probability distribution on a detection screen is the direct sum of the individual probabilities:

In quantum mechanics, when no measurement is made to determine the specific path traversed by the electron, the detector records an interference pattern characteristic of wave phenomena. Quantum kinematics resolves this paradox by associating with every physical event a complex-valued probability amplitude $\psi(x) \in \mathbb{C}$, such that the total state is a coherent linear superposition of the two path alternatives:

The observable probability density $\mathcal{P}(x)$ is the modulus squared of the total amplitude:

The final term, $2\sqrt{\mathcal{P}_1\mathcal{P}_2}\cos(\phi_1 - \phi_2)$, is the quantum interference cross-term. If a non-destructive measurement is introduced to register which slit the particle traversed, the relative phase coherence is irrevocably destroyed by quantum backaction (decoherence), collapsing the interference pattern into the classical additive distribution $\mathcal{P}_1 + \mathcal{P}_2$.

2. The Dirac Bra and Ket Formalism

Paul Dirac synthesized the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg into an abstract vector space representation. A physical state of an isolated quantum system is represented by a ray in a complex vector space $\mathcal{H}$, termed a ket vector and denoted by $|\psi\rangle$.

Associated with the vector space $\mathcal{H}$ is its dual space $\mathcal{H}^*$, consisting of all continuous linear functionals mapping $\mathcal{H} \to \mathbb{C}$. Dirac denoted elements of the dual space as bra vectors, $\langle \phi| \in \mathcal{H}^*$. By the Riesz representation theorem, for every ket $|\psi\rangle \in \mathcal{H}$, there exists a unique conjugate dual bra $\langle \psi| \in \mathcal{H}^*$ under an anti-linear bijective map termed the Hermitian adjoint:

The action of a functional $\langle \phi|$ on a state ket $|\psi\rangle$ forms the Dirac bracket, representing the inner product:

The inner product satisfies three defining axioms:

1. Skew-symmetry (Hermitian conjugate property): $\langle \phi | \psi \rangle = \langle \psi | \phi \rangle^*$
2. Linearity in the ket argument: $\langle \phi | (c_1 |\psi_1\rangle + c_2 |\psi_2\rangle) = c_1 \langle \phi | \psi_1\rangle + c_2 \langle \phi | \psi_2\rangle$
3. Positive-definiteness: $\langle \psi | \psi \rangle \ge 0$, with equality holding if and only if $|\psi\rangle = 0$.

1. Definition of Complex Hilbert Space

An abstract Hilbert space $\mathcal{H}$ is a complete complex inner product space. Completeness ensures that every Cauchy sequence of state vectors $\{|\psi_n\rangle\}_{n=1}^\infty$ converges to an element within $\mathcal{H}$ with respect to the induced norm:

2. The Cauchy-Schwarz and Triangle Inequalities

For any two state kets $|\psi\rangle, |\phi\rangle \in \mathcal{H}$, consider the arbitrary real or complex parameter $\lambda$. Since the norm of the vector $|\chi\rangle = |\psi\rangle + \lambda |\phi\rangle$ is strictly non-negative:

Choosing the specific variation $\lambda = -\frac{\langle \phi | \psi \rangle}{\langle \phi | \phi \rangle}$ (assuming $\langle \phi | \phi \rangle \ne 0$) yields:

This is the foundational Cauchy-Schwarz inequality:

Equality holds if and only if the kets are linearly dependent: $|\psi\rangle = c|\phi\rangle$. Using this result, the Minkowski (triangle) inequality directly follows:

3. Orthonormal Bases and Completeness (Closure) Relations

Let $\{|u_n\rangle\}_{n=1}^N$ (where $N$ may be finite or countably infinite) be an orthonormal set spanning $\mathcal{H}$, satisfying the orthonormality condition:

Any arbitrary state $|\psi\rangle \in \mathcal{H}$ can be expanded uniquely as:

Substituting $c_n$ into the expansion reveals the identity operator $\hat{I}$:

This equation is the completeness relation (or resolution of the identity). The norm of $|\psi\rangle$ expressed through basis coordinates gives Parseval's identity:

1. Linear Operators and Dyadic Outer Products

A linear operator $\hat{A}: \mathcal{H} \to \mathcal{H}$ maps kets to kets while preserving linear combinations:

The outer product between a ket $|\phi\rangle$ and a bra $\langle \chi|$ forms a rank-1 linear operator (a dyad):

2. Matrix Representation in a Discrete Basis

By inserting the completeness relation on both sides of an operator $\hat{A}$, we express $\hat{A}$ in terms of its matrix elements:

Defining the matrix element $A_{nm} \equiv \langle u_n | \hat{A} | u_m \rangle$, the operator acts as a matrix acting upon coordinate column vectors:

3. Hermitian Adjoint and Observable Operators

The Hermitian adjoint $\hat{A}^\dagger$ of an operator $\hat{A}$ is defined by the condition:

In matrix terms, $(A^\dagger)_{nm} = (A_{mn})^*$, representing the conjugate transpose. An operator represents a physical observable if and only if it is self-adjoint (Hermitian):

Hermitian operators possess two vital physical properties:

1. Real Eigenvalues: If $\hat{A}|a_n\rangle = a_n |a_n\rangle$, then $a_n = a_n^* \in \mathbb{R}$.
2. Orthogonality of Eigenvectors: If $\hat{A}|a_n\rangle = a_n|a_n\rangle$ and $\hat{A}|a_m\rangle = a_m|a_m\rangle$ with $a_n \ne a_m$, then $\langle a_n | a_m \rangle = 0$.

4. Projection Operators and the Spectral Decomposition Theorem

A projection operator $\hat{P}_n$ onto the one-dimensional subspace spanned by the normalized ket $|u_n\rangle$ is defined as:

Projection operators satisfy the algebraic properties of idempotency and Hermiticity:

According to the Spectral Decomposition Theorem, any Hermitian observable $\hat{A}$ with a non-degenerate discrete spectrum can be represented as a weighted sum of its projection operators:

For any analytic function $f(\hat{A})$, the spectral representation evaluates directly to:

1. Change of Basis and Unitary Operators

Consider two distinct complete orthonormal bases $\{|u_n\rangle\}$ and $\{|v_k\rangle\}$ spanning the same Hilbert space $\mathcal{H}$. The transformation connecting the two bases is mediated by a linear operator $\hat{U}$:

Evaluating the adjoint operator:

Thus, $\hat{U}^\dagger = \hat{U}^{-1}$, meaning $\hat{U}$ is a unitary operator. Unitary transformations preserve inner products, vector norms, and the algebraic spectra of operators:

2. Continuous Spectra: Position and Momentum Representations

When an observable exhibits a continuous spectrum, such as position $\hat{x}$ or momentum $\hat{p}$, the Kronecker delta is replaced by the Dirac delta distribution:

The Schrödinger wave function $\psi(x)$ is the coordinate-basis projection of the state ket $|\psi\rangle$:

The transformation kernel connecting coordinate and momentum space is the overlap bracket $\langle x | p \rangle$. Using the fundamental commutator $[\hat{x}, \hat{p}] = i\hbar\hat{I}$ and the differential representation $\langle x | \hat{p} | \psi \rangle = -i\hbar \frac{\partial}{\partial x}\psi(x)$:

Consequently, transforming from coordinate to momentum wavefunctions is mathematically identical to a continuous Fourier transformation:

1. The Spatial Inversion (Parity) Operator

The parity operator $\hat{\Pi}$ acts on spatial coordinate eigenkets by inverting all Cartesian axes through the origin:

Acting twice upon any arbitrary state returns the original spatial coordinates:

Since $\hat{\Pi}$ is also unitary ($\hat{\Pi}^\dagger \hat{\Pi} = \hat{I}$), it is simultaneously Hermitian:

The eigenvalues $\pi_k$ of the parity operator must satisfy:

States with eigenvalue $+1$ are termed even parity states ($\psi(-x) = +\psi(x)$), while states with eigenvalue $-1$ are termed odd parity states ($\psi(-x) = -\psi(x)$).

2. Transformation of Dynamical Observables Under Parity

Under coordinate inversion, the position and momentum operators change sign:

Conversely, the orbital angular momentum operator $\hat{\vec{L}} = \hat{\vec{r}} \times \hat{\vec{p}}$ is an axial vector (pseudovector) and commutes with parity:

3. Parity Conservation and Laporte's Selection Rule

If the Hamiltonian of a system is invariant under spatial inversion, $V(-\vec{r}) = V(\vec{r})$, then:

Consequently, stationary states can be chosen as simultaneous eigenstates of $\hat{H}$ and $\hat{\Pi}$. Furthermore, parity is a constant of motion:

For electric dipole transitions mediated by the vector dipole operator $\hat{\vec{d}} = q\hat{\vec{r}}$, the transition matrix element between states $|\psi_i\rangle$ and $|\psi_f\rangle$ with definite parities $\pi_i$ and $\pi_f$ satisfies:

Thus, the matrix element is identically zero unless $\pi_f \pi_i = -1$. This establishes Laporte's selection rule: electric dipole transitions can only occur between states of opposite parity ($\Delta l = \pm 1$).

1. Limitations of Pure State Vectors and Statistical Ensembles

A single state ket $|\psi\rangle$ describes a pure state, in which complete maximal knowledge about the quantum preparation is available. However, in realistic experimental scenarios (such as an unpolarized thermal beam of particles, or a subsystem entangled with an unobserved environment), the system is described by a statistical ensemble: it has classical probability $p_k \ge 0$ of being in state $|\psi_k\rangle$, where $\sum_k p_k = 1$. Note that the kets $\{|\psi_k\rangle\}$ need not be mutually orthogonal.

2. The Density Operator

To describe such a statistical mixture, John von Neumann introduced the density operator (or density matrix) $\hat{\rho}$:

The ensemble average expectation value of any physical observable $\hat{A}$ is given by:

3. Mathematical Properties of the Density Operator

1. Hermiticity: $\hat{\rho}^\dagger = \sum_k p_k (|\psi_k\rangle \langle \psi_k|)^\dagger = \hat{\rho}$
2. Unit Trace (Conservation of Total Probability): $\text{Tr}(\hat{\rho}) = \sum_k p_k \langle \psi_k | \psi_k \rangle = \sum_k p_k = 1$
3. Positive Semi-Definiteness: For any ket $|\phi\rangle$, $\langle \phi | \hat{\rho} | \phi \rangle = \sum_k p_k |\langle \phi | \psi_k \rangle|^2 \ge 0$
4. Purity Criterion: $$\text{Tr}(\hat{\rho}^2) \le 1$$ Equality $\text{Tr}(\hat{\rho}^2) = 1 \iff \hat{\rho}^2 = \hat{\rho}$ holds if and only if the state is pure ($\hat{\rho} = |\psi\rangle \langle \psi|$). For a strictly mixed state in an $N$-dimensional Hilbert space, $\frac{1}{N} \le \text{Tr}(\hat{\rho}^2) < 1$.

4. von Neumann Entropy and Thermal Density Operators

The quantum mechanical generalization of Gibbs-Shannon entropy is the von Neumann entropy:

where $\{\lambda_j\}$ are the eigenvalues of $\hat{\rho}$. For any pure state, $S(\hat{\rho}) = 0$. For a maximally mixed state in $N$ dimensions ($\hat{\rho} = \frac{1}{N}\hat{I}$), $S = k_B \ln N$.

For a system in canonical thermal equilibrium at temperature $T = (k_B \beta)^{-1}$, the thermal density operator is:

1. Factorization of the Harmonic Oscillator Hamiltonian

Consider the one-dimensional quantum harmonic oscillator Hamiltonian:

We define the dimensionless non-Hermitian creation (raising) operator $\hat{a}^\dagger$ and annihilation (lowering) operator $\hat{a}$:

Evaluating the commutator of $\hat{a}$ and $\hat{a}^\dagger$ using $[\hat{x}, \hat{p}] = i\hbar$:

Multiplying $\hat{a}^\dagger \hat{a}$:

Defining the Hermitian number operator $\hat{N} \equiv \hat{a}^\dagger \hat{a}$, the Hamiltonian takes the canonical diagonal form:

2. The Fock State Eigenvalue Spectrum

The commutators of $\hat{N}$ with $\hat{a}$ and $\hat{a}^\dagger$ are:

Let $|n\rangle$ denote an eigenstate of $\hat{N}$ with eigenvalue $n$: $\hat{N}|n\rangle = n|n\rangle$. Then:

Thus, $\hat{a}$ decreases $n$ by 1, and $\hat{a}^\dagger$ increases $n$ by 1. Since $\langle n | \hat{N} | n \rangle = \|\hat{a}|n\rangle\|^2 \ge 0$, the spectrum must be bounded from below. There must exist a unique ground state $|0\rangle$ such that:

Repeated application of $\hat{a}^\dagger$ generates the complete discrete spectrum:

3. Exact Matrix Representation of Operators

In the Fock basis $\{|0\rangle, |1\rangle, |2\rangle, \dots\}$, the operators have the infinite-dimensional matrix representations:

Inverting the definitions of $\hat{a}$ and $\hat{a}^\dagger$ expresses the coordinate and momentum operators purely in terms of ladder operators:

This yields the off-diagonal Heisenberg matrix mechanics: