⚛️Quantum Mechanics Unit 2 – Mathematical Foundations

Key Concepts and Terminology

• Quantum mechanics describes the behavior of matter and energy at the atomic and subatomic scales
• Fundamental concepts include wave-particle duality, superposition, and entanglement
• Observables represent measurable quantities in a quantum system (position, momentum, energy)
• Operators are mathematical tools used to manipulate and extract information from quantum states
• Hilbert space provides a mathematical framework for representing quantum states and operators
  • Infinite-dimensional complex vector space
  • Equipped with an inner product for calculating probabilities and expectation values
• Dirac notation (bra-ket notation) offers a convenient way to represent quantum states and operators
• Eigenvalues and eigenvectors play a crucial role in determining the possible measurement outcomes and the corresponding probabilities

Linear Algebra Essentials

• Linear algebra serves as the mathematical foundation for quantum mechanics
• Vectors represent physical states or observables in a quantum system
  • Vectors are elements of a vector space and are typically denoted using ket notation $|\psi\rangle$
• Matrices represent linear operators that act on vectors to transform them
• Inner products calculate the overlap between two vectors and are denoted as $\langle\phi|\psi\rangle$
• Outer products create matrices from two vectors, expressed as $|\psi\rangle\langle\phi|$
• Linear operators must satisfy linearity conditions
  • $\hat{A}(|\psi_1\rangle + |\psi_2\rangle) = \hat{A}|\psi_1\rangle + \hat{A}|\psi_2\rangle$
  • $\hat{A}(c|\psi\rangle) = c\hat{A}|\psi\rangle$, where $c$ is a scalar
• Hermitian operators have real eigenvalues and orthogonal eigenvectors, making them suitable for representing observables

Vector Spaces and Hilbert Spaces

• Vector spaces are mathematical structures consisting of a set of vectors and defined operations (addition and scalar multiplication)
• Hilbert spaces are complete, infinite-dimensional vector spaces equipped with an inner product
  • Completeness ensures that limits of sequences of vectors always exist within the space
• Quantum states are represented as vectors in a Hilbert space
• The inner product in a Hilbert space allows for the calculation of probabilities and expectation values
• Orthonormal bases provide a convenient way to express quantum states as linear combinations of basis vectors
  • Example: The computational basis $\{|0\rangle, |1\rangle\}$ for a qubit
• Tensor products allow for the construction of composite Hilbert spaces describing multi-particle systems

Operators and Matrices

• Operators are mathematical objects that act on vectors to produce new vectors
• In quantum mechanics, operators represent observables and transformations on quantum states
• Hermitian operators have real eigenvalues and orthogonal eigenvectors
  • Examples: Position operator $\hat{x}$, momentum operator $\hat{p}$, Hamiltonian operator $\hat{H}$
• Unitary operators preserve the inner product and probability amplitudes
  • Examples: Time-evolution operator $\hat{U}$, quantum gates in quantum computing
• Commutators measure the non-commutativity of two operators
  • Defined as $[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$
  • Non-commuting operators lead to the Heisenberg uncertainty principle
• Matrices provide a concrete representation of operators in a chosen basis
  • Matrix elements are calculated using the inner product: $A_{ij} = \langle i|\hat{A}|j\rangle$

Eigenvalues and Eigenvectors

• Eigenvalues are scalar values associated with an operator, representing the possible measurement outcomes
• Eigenvectors are non-zero vectors that, when acted upon by an operator, result in a scalar multiple of themselves
  • Satisfying the eigenvalue equation: $\hat{A}|a\rangle = a|a\rangle$, where $a$ is the eigenvalue
• Eigenvectors corresponding to different eigenvalues of a Hermitian operator are orthogonal
• The spectral decomposition expresses an operator in terms of its eigenvalues and eigenvectors
  • $\hat{A} = \sum_i a_i |a_i\rangle\langle a_i|$, where $a_i$ are the eigenvalues and $|a_i\rangle$ are the corresponding eigenvectors
• Measuring an observable collapses the quantum state onto one of its eigenvectors, with the probability given by the square of the projection onto that eigenvector

Dirac Notation and Bra-Ket Formalism

• Dirac notation (bra-ket notation) provides a compact and convenient way to represent quantum states and operators
• Kets $|\psi\rangle$ represent column vectors (quantum states)
• Bras $\langle\psi|$ represent row vectors (dual vectors)
• The inner product between two states is written as $\langle\phi|\psi\rangle$, which is a complex scalar
• Outer products of two states create operators, denoted as $|\psi\rangle\langle\phi|$
• Operators acting on states are written as $\hat{A}|\psi\rangle$, representing the transformation of the state
• Expectation values of observables are calculated using the inner product: $\langle\hat{A}\rangle = \langle\psi|\hat{A}|\psi\rangle$
• Completeness relation: $\sum_i |i\rangle\langle i| = \hat{I}$, where $\{|i\rangle\}$ is an orthonormal basis and $\hat{I}$ is the identity operator

Fourier Analysis and Transforms

• Fourier analysis decomposes functions into a sum of sinusoidal components with different frequencies
• Fourier transforms map functions between the time/position domain and the frequency/momentum domain
  • Examples: Continuous Fourier transform, discrete Fourier transform, fast Fourier transform (FFT)
• In quantum mechanics, Fourier transforms relate wave functions in position space to wave functions in momentum space
  • $\psi(p) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \psi(x) e^{-ipx/\hbar} dx$
  • $\psi(x) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \psi(p) e^{ipx/\hbar} dp$
• Parseval's theorem states that the total probability is conserved under Fourier transforms
• Convolution in one domain corresponds to multiplication in the other domain, and vice versa
• Fourier transforms are essential for studying the dynamics and evolution of quantum systems

Probability Theory in Quantum Mechanics

• Probability theory plays a fundamental role in quantum mechanics due to the inherent uncertainties in measurements
• Quantum states are represented by wave functions $\psi(x)$, whose squared modulus $|\psi(x)|^2$ gives the probability density of finding the particle at position $x$
• Born's rule: The probability of measuring an observable $A$ and obtaining a specific eigenvalue $a_i$ is given by $P(a_i) = |\langle a_i|\psi\rangle|^2$
• Expectation values of observables are calculated as the weighted average of the eigenvalues, with weights given by the probabilities
  • $\langle\hat{A}\rangle = \sum_i a_i P(a_i) = \langle\psi|\hat{A}|\psi\rangle$
• The uncertainty principle states that the product of the uncertainties in the measurements of two non-commuting observables is bounded from below
  • Example: Heisenberg's uncertainty principle for position and momentum: $\Delta x \Delta p \geq \frac{\hbar}{2}$
• Quantum measurements are probabilistic and cause the collapse of the wave function onto an eigenstate of the measured observable
• The probabilistic nature of quantum mechanics gives rise to phenomena such as superposition and entanglement