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Required Mathematics

  1. Sep 20, 2005 #1


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    I'm beginning my education in quantum mechanics next semester and I'm not sure whether I should take Introduction to Analysis, which would cover "the real number system. Sequences, limits, and continuous functions in R and 'Rn'. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral" or Numerical Analysis, which would cover "programming for numerical calculations, round-off error, approximation and interpolation, numerical quadrature, and solution of ordinary differential equations. Practice on the computer." I'm going to talk to an advisor about it but I was curious what you guys thought.
  2. jcsd
  3. Sep 20, 2005 #2


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    For an undergraduate/introductory text, maybe using Griffiths' text, the first course (real analysis) is essential. I would read some complex analysis, too.

  4. Sep 20, 2005 #3


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    Definitely do the real analysis -- gives you a behind the scenes look at the math for QM.

    Reilly Atkinson
  5. Sep 20, 2005 #4


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    If one looks at the text "Mathematics of Classical and Quantum Physics," by Frederick W. Byron, Jr. and Robert W. Fuller, it will give a good idea of the mathematics involved. The however is considered complementary to graduate-level physics texts.

    Table of Contents - Book has both Volumes bound as one.

    1 Vectors in Classical Physics
    1.1 Geometric and Algebraic Definitions of a Vector
    1.2 The Resolution of a Vector into Components
    1.3 The Scalar Product
    1.4 Rotation of the Coordinate System: Orthogonal Transformations
    1.5 The Vector Product
    1.6 A Vector Treatment of Classical Orbit Theory
    1.7 Differential Operations on Scalar and Vector Fields
    1.8 Cartesian-Tensors
    2 Calculus of Variations
    2.1 Some Famous Problems
    2.2 The Euler-Lagrange Equation
    2.3 Some Famous Solutions
    2.4 Isoperimetric Problems - Constraints
    2.5 Application to Classical Mechanics
    2.6 Extremization of Multiple Integrals
    2.7 Invariance Principles and Noether's Theorem
    3 Vectors and Matrics
    3.1 "Groups, Fields, and Vector Spaces"
    3.2 Linear Independence
    3.3 Bases and Dimensionality
    3.4 Ismorphisms
    3.5 Linear Transformations
    3.6 The Inverse of a Linear Transformation
    3.7 Matrices
    3.8 Determinants
    3.9 Similarity Transformations
    3.10 Eigenvalues and Eigenvectors
    3.11 The Kronecker Product
    4. Vector Spaces in Physics
    4.1 The Inner Product
    4.2 Orthogonality and Completeness
    4.3 Complete Ortonormal Sets
    4.4 Self-Adjoint (Hermitian and Symmetric) Transformations
    4.5 Isometries-Unitary and Orthogonal Transformations
    4.6 The Eigenvalues and Eigenvectors of Self-Adjoint and Isometric Transformations
    4.7 Diagonalization
    4.8 On The Solvability of Linear Equations
    4.9 Minimum Principles
    4.10 Normal Modes
    4.11 Peturbation Theory-Nondegenerate Case
    4.12 Peturbation Theory-Degenerate Case
    5. Hilbert Space-Complete Orthonormal Sets of Functions
    5.1 Function Space and Hilbert Space
    5.2 Complete Orthonormal Sets of Functions
    5.3 The Dirac d-Function
    5.4 Weirstrass's Theorem: Approximation by Polynomials
    5.5 Legendre Polynomials
    5.6 Fourier Series
    5.7 Fourier Integrals
    5.8 Sphereical Harmonics and Associated Legendre Functions
    5.9 Hermite Polynomials
    5.10 Sturm-Liouville Systems-Orthogaonal Polynomials
    5.11 A Mathematical Formulation of Quantum Mechanics
    6 Elements and Applications of the Theory of Analytic Functions
    6.1 Analytic Functions-The Cauchy-Riemann Conditions
    6.2 Some Basic Analytic Functions
    6.3 Complex Integration-The Cauchy-Goursat Theorem
    6.4 Consequences of Cauchy's Theorem
    6.5 Hilbert Transforms and the Cauchy Principal Value
    6.6 An Introduction to Dispersion Relations
    6.7 The Expansion of an Analytic Function in a Power Series
    6.8 Residue Theory-Evaluation of Real Definite Integrals and Summation of Series
    6.9 Applications to Special Functions and Integral Representations
    7 Green's Function
    7.1 A New Way to Solve Differential Equations
    7.2 Green's Functions and Delta Functions
    7.3 Green's Functions in One Dimension
    7.4 Green's Functions in Three Dimensions
    7.5 Radial Green's Functions
    7.6 An Application to the Theory of Diffraction
    7.7 Time-dependent Green's Functions: First Order
    7.8 The Wave Equation
    8 Introduction to Integral Equations
    8.1 Iterative Techniques-Linear Integral Operators
    8.2 Norms of Operators
    8.3 Iterative Techniques in a Banach Space
    8.4 Iterative Techniques for Nonlinear Equations
    8.5 Separable Kernels
    8.6 General Kernels of Finite Rank
    8.7 Completely Continuous Operators
    9 Integral Equations in Hilbert Space
    9.1 Completely Continuous Hermitian Operators
    9.2 Linear Equations and Peturbation Theory
    9.3 Finite-Rank Techniques for Eigenvalue Problems
    9.4 the Fredholm Alternative for Completely Continuous Operators
    9.5 The Numerical Solutions of Linear Equations
    9.6 Unitary Transformations
    10 Introduction to Group Theory
    10.1 An Inductive Approach
    10.2 The Symmetric Groups
    10.3 "Cosets, Classes, and Invariant Subgroups"
    10.4 Symmetry and Group Representations
    10.5 Irreducible Representations
    10.6 "Unitary Representations, Schur's Lemmas, and Orthogonality Relations"
    10.7 The Determination of Group Representations
    10.8 Group Theory in Physical Problems
    General Bibliography
  6. Sep 20, 2005 #5
    depends on what your looking to do ...if your looking to do simulations...thatn numericals is better...but if your down right aiming for pure theory...analysis.
  7. Sep 21, 2005 #6

    George Jones

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    Here's another way to look at it.

    When I was a student I took courses in both analysis and numerical analysis. In my opinion, numerical analysis is easier for someone to pick up on their own than is analysis if the persom starts fron (almost) scratch in both.

    This is a very subjective opinion, and so is probably not true for everyone.

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