🧮Mathematical Methods in Classical and Quantum Mechanics Unit 1 – Vector Spaces and Linear Algebra Foundations

Key Concepts and Definitions

• Vector spaces consist of a set of vectors and two operations (addition and scalar multiplication) that satisfy certain axioms
• Linear independence means a set of vectors cannot be expressed as linear combinations of each other
• Basis is a linearly independent set of vectors that spans the entire vector space
• Dimension of a vector space equals the number of vectors in its basis
• Linear transformations map vectors from one vector space to another while preserving vector addition and scalar multiplication
• Matrices represent linear transformations and enable computations in finite-dimensional vector spaces
• Eigenvalues are scalars $\lambda$ that satisfy the equation $Av = \lambda v$ for a square matrix $A$ and non-zero vector $v$
  • Eigenvectors are the corresponding non-zero vectors $v$ satisfying this equation
• Inner product spaces have an additional operation (inner product) that generalizes the dot product and allows for notions of length and orthogonality

Vector Space Fundamentals

• Vector spaces are defined over a field (real numbers $\mathbb{R}$ or complex numbers $\mathbb{C}$) and must satisfy closure, associativity, commutativity, identity, inverse, and distributivity axioms
• Examples of vector spaces include $\mathbb{R}^n$, $\mathbb{C}^n$, polynomials, and functions
• Subspaces are subsets of a vector space that form a vector space under the same operations
  • Must contain the zero vector and be closed under addition and scalar multiplication
• Linear combinations express vectors as weighted sums of other vectors using scalar coefficients
• Span of a set of vectors includes all possible linear combinations of those vectors
• Linearly dependent sets have vectors that can be expressed as linear combinations of the others
  • Linearly independent sets have no such redundancy
• Basis vectors span the entire space and are linearly independent, providing a minimal generating set for the vector space
• Coordinates of a vector with respect to a basis are the coefficients in its unique linear combination representation

Linear Transformations and Matrices

• Linear transformations preserve vector addition ($T(u+v) = T(u) + T(v)$) and scalar multiplication ($T(cu) = cT(u)$)
• Kernel (null space) of a linear transformation is the set of vectors that map to the zero vector
  • Dimension of the kernel is the nullity
• Range (image) of a linear transformation is the set of all vectors that can be obtained as outputs
  • Dimension of the range is the rank
• Rank-nullity theorem states that the dimension of the domain equals the rank plus the nullity
• Matrices represent linear transformations by organizing the coefficients of the transformed basis vectors
  • Matrix-vector multiplication computes the action of the transformation on a vector
• Matrix addition and multiplication correspond to adding and composing linear transformations, respectively
• Invertible matrices represent bijective linear transformations and have non-zero determinants
  • Inverse matrices can be computed using Gaussian elimination or Cramer's rule

Eigenvalues and Eigenvectors

• Eigenvalues and eigenvectors capture the stretching or shrinking behavior of a linear transformation in certain directions
• Characteristic equation $\det(A - \lambda I) = 0$ determines the eigenvalues of a matrix $A$
  • Polynomial degree equals the dimension of the matrix
• Algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial
• Geometric multiplicity of an eigenvalue is the dimension of its corresponding eigenspace (number of linearly independent eigenvectors)
  • Always less than or equal to the algebraic multiplicity
• Diagonalizable matrices have a basis of eigenvectors and can be decomposed as $A = PDP^{-1}$, where $D$ is a diagonal matrix of eigenvalues and $P$ contains the eigenvectors as columns
• Spectral theorem states that real symmetric matrices (or complex Hermitian matrices) have an orthonormal basis of eigenvectors with real eigenvalues

Inner Product Spaces

• Inner product $\langle u, v \rangle$ is a generalization of the dot product that maps pairs of vectors to a scalar value
  • Must satisfy conjugate symmetry, linearity in the second argument, and positive-definiteness
• Norm (length) of a vector $v$ is defined as $\|v\| = \sqrt{\langle v, v \rangle}$
• Cauchy-Schwarz inequality bounds the absolute value of the inner product: $|\langle u, v \rangle| \leq \|u\| \|v\|$
• Orthogonality of vectors $u$ and $v$ means their inner product is zero: $\langle u, v \rangle = 0$
• Orthonormal basis is a basis of unit vectors that are mutually orthogonal
  • Simplifies computations and provides a convenient coordinate system
• Gram-Schmidt process constructs an orthonormal basis from a linearly independent set of vectors
• Orthogonal complement of a subspace $W$ is the set of all vectors orthogonal to every vector in $W$
  • Direct sum of a subspace and its orthogonal complement equals the entire vector space

Applications in Classical Mechanics

• Configuration space of a classical system is a vector space representing all possible positions or states
  • Velocity and acceleration are vectors in the tangent space
• Generalized coordinates (e.g., Cartesian, polar, or spherical) provide different bases for describing the system
• Lagrangian mechanics uses generalized coordinates and velocities to formulate equations of motion
  • Kinetic and potential energy are expressed as functions on the configuration space
• Hamiltonian mechanics uses generalized coordinates and momenta (cotangent bundle) to describe the system
  • Symplectic structure captures the geometry of phase space and Hamiltonian dynamics
• Poisson brackets are a bilinear operation on functions that encodes the structure of Hamiltonian dynamics
  • Relate to commutators in quantum mechanics via the correspondence principle
• Noether's theorem connects symmetries of the Lagrangian or Hamiltonian to conserved quantities (e.g., energy, momentum, angular momentum)
  • Conserved quantities correspond to generators of the symmetry transformations

Quantum Mechanical Connections

• Hilbert spaces are infinite-dimensional vector spaces with an inner product, used to describe quantum states
  • Square-integrable functions (wavefunctions) form a Hilbert space
• Observables are represented by Hermitian operators acting on the Hilbert space
  • Eigenvalues correspond to possible measurement outcomes, and eigenvectors represent the associated states
• Commutator $[A, B] = AB - BA$ captures the non-commutativity of quantum observables
  • Canonical commutation relations (e.g., $[x, p] = i\hbar$) encode the uncertainty principle
• Unitary operators represent symmetry transformations and time evolution in quantum mechanics
  • Generated by Hermitian operators via the exponential map
• Tensor products allow for the description of composite quantum systems
  • Entanglement arises when a state cannot be written as a tensor product of individual subsystem states
• Density matrices provide a description of mixed states and capture statistical ensembles
  • Positive semidefinite operators with unit trace
• Quantum operations and measurements are represented by positive operator-valued measures (POVMs)
  • Generalize projective measurements and capture the effects of noise and decoherence

Problem-Solving Strategies

• Identify the vector space and its key properties (field, dimension, basis, inner product)
• Determine the type of problem (e.g., linear independence, transformation, eigenvalues, orthogonality)
• Use the defining properties and axioms of vector spaces to guide your approach
  • E.g., closure, linearity, independence, spanning
• Represent linear transformations using matrices and perform computations using matrix operations
  • Gaussian elimination, eigenvalue decomposition, singular value decomposition
• Exploit special structures or symmetries to simplify the problem
  • E.g., orthogonality, diagonalizability, Hermitian or unitary operators
• Apply relevant theorems or techniques based on the problem type
  • E.g., rank-nullity theorem, spectral theorem, Gram-Schmidt process, Noether's theorem
• Interpret the results in the context of the original problem and physical system
  • Relate mathematical quantities to physical observables or conserved quantities
• Verify your solution by checking consistency with known properties or limiting cases
  • E.g., dimensional analysis, symmetry, correspondence principle