🧐Functional Analysis Unit 12 – Differential Equations & Quantum Mechanics

Key Concepts and Definitions

• Quantum mechanics mathematical framework describes the behavior of matter and energy at the atomic and subatomic scales
• Hilbert spaces abstract vector spaces with inner products that allow the representation of quantum states and operators
• Operators linear transformations acting on elements of a Hilbert space, representing physical observables (position, momentum, energy)
• Eigenvalues and eigenvectors characteristic values and vectors associated with an operator, determining the possible outcomes of measurements
  • Eigenvalues represent the possible values of a physical observable
  • Eigenvectors represent the corresponding quantum states
• Commutators measure of the non-commutativity of two operators, crucial for understanding the uncertainty principle and incompatible observables
• Bra-ket notation compact and convenient way to represent quantum states and operators in Hilbert spaces
  • Bra $\langle \psi |$ represents a dual vector (complex conjugate transpose) of a ket vector
  • Ket $| \psi \rangle$ represents a quantum state vector in a Hilbert space

Fundamental Principles

• Wave-particle duality matter and energy exhibit both wave-like and particle-like properties, depending on the context and measurement
• Heisenberg's uncertainty principle impossibility of simultaneously measuring certain pairs of physical observables (position and momentum) with arbitrary precision
• Schrödinger equation fundamental equation in quantum mechanics, describing the time evolution of a quantum system's wave function
  • Time-dependent Schrödinger equation $i\hbar \frac{\partial}{\partial t} |\psi(t)\rangle = \hat{H} |\psi(t)\rangle$
  • Time-independent Schrödinger equation $\hat{H} |\psi\rangle = E |\psi\rangle$, an eigenvalue problem for stationary states
• Born's probabilistic interpretation square modulus of a wave function $|\psi(x)|^2$ represents the probability density of finding a particle at position $x$
• Superposition principle quantum systems can exist in a linear combination of multiple states simultaneously until a measurement is made
• Collapse of the wave function measurement of a quantum system causes the wave function to instantaneously reduce to one of the eigenstates of the measured observable

Mathematical Foundations

• Linear algebra foundation for the mathematical formulation of quantum mechanics, dealing with vector spaces, linear transformations, and matrices
• Functional analysis branch of mathematics that extends the concepts of linear algebra to infinite-dimensional spaces, essential for quantum mechanics
• Hilbert spaces complete inner product spaces, serving as the mathematical arena for quantum mechanics
  • Completeness ensures that limits of Cauchy sequences converge within the space
  • Inner product allows the definition of orthogonality and the computation of probabilities
• Operators linear transformations acting on elements of a Hilbert space, representing physical observables and symmetry transformations
  • Hermitian operators have real eigenvalues and orthogonal eigenvectors, representing observable quantities
  • Unitary operators preserve inner products and represent symmetry transformations or time evolution
• Spectral theory study of the eigenvalues and eigenvectors of operators, essential for understanding the possible outcomes of measurements
• Tensor products mathematical operation that combines two or more Hilbert spaces to create a larger composite space, used to describe multi-particle systems

Differential Equations in Quantum Mechanics

• Schrödinger equation central differential equation in quantum mechanics, describing the time evolution of a quantum system
  • Time-dependent Schrödinger equation $i\hbar \frac{\partial}{\partial t} |\psi(t)\rangle = \hat{H} |\psi(t)\rangle$, where $\hat{H}$ is the Hamiltonian operator
  • Time-independent Schrödinger equation $\hat{H} |\psi\rangle = E |\psi\rangle$, an eigenvalue problem for stationary states with energy eigenvalues $E$
• Hamiltonian operator represents the total energy of a quantum system, consisting of kinetic and potential energy terms
  • For a single particle in one dimension, $\hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x)$, where $m$ is the particle's mass and $V(x)$ is the potential energy
• Boundary conditions constraints on the wave function at the boundaries of a system, essential for solving differential equations and determining eigenstates
• Stationary states solutions to the time-independent Schrödinger equation, representing quantum states with well-defined energy
• Expectation values average values of physical observables in a given quantum state, calculated using the inner product between the state and the observable's operator
• Ehrenfest's theorem relates the time evolution of expectation values to the commutator of the observable with the Hamiltonian

Operator Theory and Hilbert Spaces

• Adjoint operators generalization of the Hermitian conjugate for infinite-dimensional operators, satisfying $\langle \hat{A}^\dagger \psi | \phi \rangle = \langle \psi | \hat{A} \phi \rangle$
• Self-adjoint operators operators equal to their adjoints, ensuring real eigenvalues and orthogonal eigenvectors
• Compact operators class of operators that can be approximated by finite-rank operators, important for spectral theory and the study of integral equations
• Spectral theorem states that self-adjoint operators have a unique spectral decomposition in terms of their eigenvalues and eigenvectors
  • Spectral decomposition $\hat{A} = \sum_n a_n |\psi_n\rangle\langle\psi_n|$, where $a_n$ are the eigenvalues and $|\psi_n\rangle$ are the eigenvectors
• Functional calculus allows the definition of functions of operators, extending the notion of matrix functions to infinite-dimensional operators
• Unbounded operators operators not defined on the entire Hilbert space, requiring a careful treatment of their domains and self-adjointness
  • Position and momentum operators are examples of unbounded operators in quantum mechanics
• Spectral measures generalization of projection operators, allowing the representation of observables with continuous spectra

Applications in Physics

• Quantum harmonic oscillator model system describing a particle in a quadratic potential, used to study vibrations in molecules and lattices
  • Hamiltonian $\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2\hat{x}^2$, where $\omega$ is the angular frequency
  • Energy eigenvalues $E_n = \hbar\omega(n + \frac{1}{2})$, where $n = 0, 1, 2, \ldots$
• Hydrogen atom simplest atomic system, consisting of a proton and an electron, serving as a prototype for more complex atoms and molecules
  • Schrödinger equation in spherical coordinates, with a Coulomb potential $V(r) = -\frac{e^2}{4\pi\varepsilon_0 r}$
  • Energy levels $E_n = -\frac{13.6 \text{ eV}}{n^2}$, where $n = 1, 2, 3, \ldots$
• Angular momentum operators represent the intrinsic and orbital angular momentum of particles, with eigenvalues quantized in units of $\hbar$
  • Commutation relations $[\hat{L}_i, \hat{L}_j] = i\hbar \varepsilon_{ijk} \hat{L}_k$, where $\varepsilon_{ijk}$ is the Levi-Civita symbol
• Spin quantum number intrinsic angular momentum of particles, with half-integer values for fermions and integer values for bosons
  • Pauli matrices represent spin-1/2 operators, satisfying the SU(2) algebra
• Perturbation theory method for approximating the solutions to the Schrödinger equation when the Hamiltonian consists of a solvable part and a small perturbation
  • Time-independent perturbation theory uses the unperturbed eigenstates as a basis for expanding the perturbed eigenstates and energies
  • Time-dependent perturbation theory describes the evolution of a quantum system under the influence of a time-varying perturbation

Problem-Solving Techniques

• Separation of variables method for solving partial differential equations by assuming that the solution can be written as a product of functions, each depending on a single variable
  • Schrödinger equation in Cartesian coordinates $\psi(x, y, z) = X(x)Y(y)Z(z)$
• Fourier transforms mathematical tool for transforming functions between position and momentum representations
  • Momentum representation wave function $\psi(p) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \psi(x) e^{-ipx/\hbar} dx$
• Variational method approximation technique for finding upper bounds on the ground state energy of a quantum system by minimizing the expectation value of the Hamiltonian over a set of trial wave functions
  • Variational principle $E_0 \leq \frac{\langle \psi | \hat{H} | \psi \rangle}{\langle \psi | \psi \rangle}$ for any normalized trial wave function $|\psi\rangle$
• WKB approximation semiclassical method for obtaining approximate solutions to the Schrödinger equation in the limit of small wavelengths
  • WKB wave function $\psi(x) \approx \frac{C}{\sqrt{p(x)}} \exp\left(\pm \frac{i}{\hbar} \int^x p(x') dx'\right)$, where $p(x) = \sqrt{2m(E - V(x))}$
• Numerical methods computational techniques for solving the Schrödinger equation when analytical solutions are not available
  • Finite difference methods discretize the spatial derivatives in the Schrödinger equation, leading to a matrix eigenvalue problem
  • Spectral methods expand the wave function in terms of a set of basis functions, converting the Schrödinger equation into a matrix equation

Advanced Topics and Extensions

• Relativistic quantum mechanics incorporates the principles of special relativity into quantum mechanics, leading to the Klein-Gordon and Dirac equations
  • Klein-Gordon equation $(\Box + m^2)\phi = 0$, describing spin-0 particles
  • Dirac equation $(i\gamma^\mu \partial_\mu - m)\psi = 0$, describing spin-1/2 particles
• Quantum field theory unifies quantum mechanics and special relativity, describing particles as excitations of underlying fields
  • Creation and annihilation operators $\hat{a}^\dagger_{\mathbf{k}}$ and $\hat{a}_{\mathbf{k}}$ create and destroy particles with momentum $\mathbf{k}$
  • Fock space Hilbert space representation of quantum states in terms of occupation numbers of different particle states
• Path integral formulation alternative formulation of quantum mechanics, expressing the transition amplitude between two states as a sum over all possible paths connecting them
  • Feynman path integral $\langle x_f | e^{-i\hat{H}t/\hbar} | x_i \rangle = \int \mathcal{D}x(t) \exp\left(\frac{i}{\hbar} \int_{t_i}^{t_f} L(x, \dot{x}, t) dt\right)$, where $L$ is the Lagrangian
• Quantum entanglement phenomenon in which the quantum states of two or more particles are correlated, even when the particles are separated by large distances
  • Bell's theorem demonstrates that entangled states violate the assumptions of local realism, confirming the non-local nature of quantum mechanics
  • Quantum teleportation protocol for transferring the quantum state of a particle using entanglement and classical communication
• Quantum computing paradigm that exploits quantum-mechanical phenomena (superposition and entanglement) to perform computations more efficiently than classical computers
  • Qubits quantum analog of classical bits, represented by two-level quantum systems (spin-1/2 particles, two-level atoms)
  • Quantum gates unitary operations acting on qubits, analogous to classical logic gates
  • Quantum algorithms (Shor's algorithm, Grover's search) demonstrate the potential speedup of quantum computers over classical ones for certain problems