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Matrix Mechanics

Field: Quantum Theory

A formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan.

Sequence of Expressions

Let H\mathcal{H} be a separable Hilbert space representing the state space of a quantum system. The state is described by the wavefunction Ψ(x,t)∈H\Psi(\mathbf{x}, t) \in \mathcal{H}. The time evolution of Ψ\Psi is governed by the time-dependent Schrödinger equation, which mandates that the Hamiltonian operator H^:H→H\hat{H}: \mathcal{H} \to \mathcal{H} must be Hermitian, such that:\n∂∂tΨ(x,t)=−iH^ℏΨ(x,t)\frac{\partial}{\partial t} \Psi(\mathbf{x}, t) = -i \frac{\hat{H}}{\hbar} \Psi(\mathbf{x}, t)\nFurthermore, the probability density function P(x,t)P(\mathbf{x}, t) is defined by the Born rule, requiring the normalization condition:\n∫R3∣Ψ(x,t)∣2d3x=1\int_{\mathbb{R}^3} |\Psi(\mathbf{x}, t)|^2 d^3\mathbf{x} = 1
Let H\mathcal{H} be a complex Hilbert space, equipped with inner product ⟨⋅,⋅⟩\langle \cdot, \cdot \rangle. A linear operator A^:H→H\hat{A}: \mathcal{H} \to \mathcal{H} is defined as a mapping satisfying: 1. Linearity: A^(c1ψ1+c2ψ2)=c1A^ψ1+c2A^ψ2\hat{A}(c_1 \psi_1 + c_2 \psi_2) = c_1 \hat{A}\psi_1 + c_2 \hat{A}\psi_2 for all ψ1,ψ2∈H\psi_1, \psi_2 \in \mathcal{H} and c1,c2∈Cc_1, c_2 \in \mathbb{C}. 2. Boundedness: There exists a finite constant M=sup⁡ψ∈H,ψ≠0∣⟨A^ψ,A^ψ⟩∣⟨ψ,ψ⟩<∞M = \sup_{\psi \in \mathcal{H}, \psi \neq 0} \frac{|\langle \hat{A}\psi, \hat{A}\psi \rangle|}{\langle \psi, \psi \rangle} < \infty. Such an operator is often represented by a matrix A\mathbf{A} in a chosen basis, such that A^↔A∈L(H)\hat{A} \leftrightarrow \mathbf{A} \in \mathcal{L}(\mathcal{H}). If A^\hat{A} is self-adjoint (Hermitian), it satisfies A^=A^†\hat{A} = \hat{A}^{\dagger}, ensuring that its eigenvalues are real and that it represents a physical observable.
Let H\mathcal{H} be a separable Hilbert space, and let A^\hat{A} and B^\hat{B} be two self-adjoint operators acting on H\mathcal{H}. Define the expectation values ΔA=⟨ψ∣(A^−⟨A^⟩I^)2∣ψ⟩\Delta A = \sqrt{\langle \psi | (\hat{A} - \langle \hat{A} \rangle \hat{I})^2 | \psi \rangle} and ΔB=⟨ψ∣(B^−⟨B^⟩I^)2∣ψ⟩\Delta B = \sqrt{\langle \psi | (\hat{B} - \langle \hat{B} \rangle \hat{I})^2 | \psi \rangle} for a state ∣ψ⟩∈H|\psi\rangle \in \mathcal{H}. If A^=X^\hat{A} = \hat{X} (position operator) and B^=P^\hat{B} = \hat{P} (momentum operator), then the generalized uncertainty principle dictates the lower bound on the product of the standard deviations: \ΔXΔP≥12∣⟨[X^,P^]⟩∣\Delta X \Delta P \geq \frac{1}{2} | \langle [\hat{X}, \hat{P}] \rangle |. Given the canonical commutation relation [X^,P^]=iℏI^[\hat{X}, \hat{P}] = i\hbar \hat{I}, the minimum uncertainty product is established as: \ΔXΔP≥ℏ2\Delta X \Delta P \geq \frac{\hbar}{2}.
Let H\mathcal{H} be a separable Hilbert space over C\mathbb{C}. Define the state vector ∣Ψ⟩∈H|\Psi\rangle \in \mathcal{H} and its corresponding dual element ⟨Ψ∣\langle\Psi| as the linear functional on H\mathcal{H} such that ⟨Ψ∣Φ⟩=⟨Φ∣Ψ⟩∗\langle\Psi|\Phi\rangle = \langle\Phi|\Psi\rangle^*. The inner product ⟨Ψ1∣Ψ2⟩\langle\Psi_1|\Psi_2\rangle is defined by the Hermitian inner product ⟨Ψ1,Ψ2⟩H\langle\Psi_1, \Psi_2\rangle_{\mathcal{H}}. The bra-ket notation is formalized by the identity operator I=∑k=1N∣k⟩⟨k∣\mathbf{I} = \sum_{k=1}^{N} |k\rangle\langle k| (or I=∫∣k⟩⟨k∣dμ(k)\mathbf{I} = \int |k\rangle\langle k| d\mu(k) in continuous bases), which ensures the completeness relation: ⟨Ψ∣Φ⟩=⟨Ψ,Φ⟩H\langle\Psi|\Phi\rangle = \langle\Psi, \Phi\rangle_{\mathcal{H}}. Furthermore, the expectation value of a self-adjoint operator A^\hat{A} for state ∣Ψ⟩|\Psi\rangle is given by ⟨A^⟩=⟨Ψ∣A^∣Ψ⟩=⟨Ψ,A^Ψ⟩H\langle\hat{A}\rangle = \langle\Psi|\hat{A}|\Psi\rangle = \langle\Psi, \hat{A}\Psi\rangle_{\mathcal{H}}. This structure defines the action of operators A^:H→H\hat{A}: \mathcal{H} \to \mathcal{H} via matrix multiplication in a chosen basis.
Let A∈Cm×k\mathbf{A} \in \mathbb{C}^{m \times k} and B∈Ck×n\mathbf{B} \in \mathbb{C}^{k \times n} be two matrices with complex entries. The product matrix C=AB∈Cm×n\mathbf{C} = \mathbf{A} \mathbf{B} \in \mathbb{C}^{m \times n} is defined by its entries cijc_{ij} as:\n\nC=AB  ⟹  cij=∑l=1kailblj\mathbf{C} = \mathbf{A} \mathbf{B} \implies c_{ij} = \sum_{l=1}^{k} a_{il} b_{lj}
Let H\mathcal{H} be a separable Hilbert space, and let H^:H→H\hat{H}: \mathcal{H} \to \mathcal{H} be a self-adjoint, bounded linear operator (the Hamiltonian). The eigenvalue problem is defined by the equation: H^∣ψ⟩=E∣ψ⟩ \hat{H} |\psi\rangle = E |\psi\rangle where ∣ψ⟩∈H|\psi\rangle \in \mathcal{H} is the eigenvector (or eigenstate), and E∈RE \in \mathbb{R} is the corresponding eigenvalue. The existence and properties of these solutions are guaranteed by the Spectral Theorem, which states that H^\hat{H} can be represented by a spectral measure μE \mu_E such that H^=∫σ(H^)EdμE(E)\hat{H} = \int_{\sigma(\hat{H})} E d\mu_E(E), where σ(H^)\sigma(\hat{H}) is the spectrum of H^\hat{H}.
Let H\mathcal{H} be a separable Hilbert space, and let ∣Ψ(t)⟩∈H|\Psi(t)\rangle \in \mathcal{H} be the state vector describing the system at time tt. Define the Hamiltonian operator H:H→HH: \mathcal{H} \to \mathcal{H} as the generator of time translations, such that HH is self-adjoint (H=H†H = H^{\dagger}). The time evolution of the state vector is governed by the differential equation:\nddt∣Ψ(t)⟩=−iℏH∣Ψ(t)⟩\frac{d}{dt}|\Psi(t)\rangle = -\frac{i}{\hbar} H |\Psi(t)\rangle
Let H\mathcal{H} be a Hilbert space defined over the configuration space x∈Rn\mathbf{x} \in \mathbb{R}^n. Define the Hamiltonian operator H^:D→H\hat{H}: \mathcal{D} \rightarrow \mathcal{H} acting on a state vector ∣ψ⟩∈D⊂H|\psi\rangle \in \mathcal{D} \subset \mathcal{H} as: H^=12m∑i=1np^i2+V(x)⋅I \hat{H} = \frac{1}{2m} \sum_{i=1}^{n} \hat{p}_i^2 + V(\mathbf{x}) \cdot \mathbf{I} where p^i=−iℏ∂∂xi\hat{p}_i = -i\hbar \frac{\partial}{\partial x_i} is the canonical momentum operator, and V(x)V(\mathbf{x}) is the potential energy function. The time evolution of the state is governed by the Schrödinger equation: iℏ∂∂t∣ψ(t)⟩=H^∣ψ(t)⟩ i\hbar \frac{\partial}{\partial t} |\psi(t)\rangle = \hat{H} |\psi(t)\rangle Furthermore, H^\hat{H} is self-adjoint, ensuring that the expectation value of the energy, ⟨H^⟩=⟨ψ∣H^∣ψ⟩\langle\hat{H}\rangle = \langle\psi|\hat{H}|\psi\rangle, is real.
Let H\mathcal{H} be a finite-dimensional Hilbert space, and let A^,B^∈L(H)\hat{A}, \hat{B} \in \mathcal{L}(\mathcal{H}) be Hermitian operators representing observables. The state of the system is represented by a normalized state vector ∣ψ(t)⟩∈H|\psi(t)\rangle \in \mathcal{H}. The time evolution is governed by the Hamiltonian operator H^∈L(H)\hat{H} \in \mathcal{L}(\mathcal{H}). The dynamics are defined by the matrix Schrödinger equation:\n\nddt∣ψ(t)⟩=−i1ℏH^∣ψ(t)⟩\frac{d}{dt} |\psi(t)\rangle = -i\frac{1}{\hbar} \hat{H} |\psi(t)\rangle\n\nFurthermore, the operators satisfy the canonical commutation relations, which must be preserved in the matrix representation: \n\n[X^,P^]=X^P^−P^X^=iℏI^[\hat{X}, \hat{P}] = \hat{X}\hat{P} - \hat{P}\hat{X} = i\hbar \hat{I} \n\nwhere X^\hat{X} and P^\hat{P} are the position and momentum operators, respectively, and I^\hat{I} is the identity matrix. The expectation value of any observable A^\hat{A} is given by ⟨A^⟩=⟨ψ(t)∣A^∣ψ(t)⟩\langle \hat{A} \rangle = \langle \psi(t)| \hat{A} |\psi(t)\rangle.
Let H\mathcal{H} be a separable Hilbert space, and let X^\hat{X} and P^\hat{P} be self-adjoint operators defined on H\mathcal{H} representing position and momentum, respectively. The commutation relation is defined by the commutator [X^,P^]≡X^P^−P^X^[\hat{X}, \hat{P}] \equiv \hat{X}\hat{P} - \hat{P}\hat{X}. The canonical commutation relation (CCR) asserts that for the standard basis representation, the commutator yields: [X^,P^]=iℏI^[\hat{X}, \hat{P}] = i\hbar \hat{I} where ℏ\hbar is the reduced Planck constant and I^\hat{I} is the identity operator on H\mathcal{H}. Furthermore, for any two observables A^\hat{A} and B^\hat{B}, the expectation value of the commutator satisfies ⟨ψ∣[A^,B^]∣ψ⟩=iℏ⟨ψ∣∂A^∂q∂B^∂p−∂B^∂q∂A^∂p∣ψ⟩\langle \psi | [\hat{A}, \hat{B}] | \psi \rangle = i\hbar \langle \psi | \frac{\partial \hat{A}}{\partial q} \frac{\partial \hat{B}}{\partial p} - \frac{\partial \hat{B}}{\partial q} \frac{\partial \hat{A}}{\partial p} | \psi \rangle. This structure dictates the uncertainty principle ΔAΔB≥12∣⟨[A^,B^]⟩∣\Delta A \Delta B \ge \frac{1}{2} |\langle [\hat{A}, \hat{B}] \rangle|.