GATE Physics Mathematical Physics Syllabus

Mathematical Physics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematical Physics in GATE Physics, not a summary of it.

1. 1

   Vector Calculus

   7 topics

   • Linear Vector Space
     • Basis
     • Orthogonality
     • Completeness
   • Matrices
     • Similarity Transformations
     • Diagonalization
     • Eigen Values
     • Eigen Vectors
   • Linear Differential Equations
     • Second Order Linear Differential Equations
     • Solutions Involving Special Functions
   • Complex Analysis
     • Cauchy-Riemann Conditions
     • Cauchy's Theorem
     • Singularities
     • Residue Theorem
     • Applications
   • Laplace Transform
   • Fourier Analysis
   • Elementary Ideas about Tensors
     • Covariant Tensors
     • Contravariant Tensors

Mathematical Physics flashcards for GATE Physics

21 of 50 cards from the Mathematical Physics deck — real questions with worked answers.

1. What defines a linear vector space (over a field $F$)?

   A set $V$ with vector addition and scalar multiplication satisfying closure, associativity and commutativity of addition, existence of a zero vector $\vec{0}$ and additive inverses, plus the scalar axioms $a(\vec{u}+\vec{v})=a\vec{u}+a\vec{v}$, $(a+b)\vec{v}=a\vec{v}+b\vec{v}$, $a(b\vec{v})=(ab)\vec{v}$, and $1\cdot\vec{v}=\vec{v}$ for all $\vec{u},\vec{v}\in V$ and $a,b\in F$.

2. What does it mean for vectors $\vec{v}_1,\dots,\vec{v}_n$ to be linearly independent?

   The only solution to $\sum_{i=1}^{n} c_i\vec{v}_i = \vec{0}$ is $c_1=c_2=\cdots=c_n=0$; no vector can be written as a linear combination of the others.

3. Define a basis of a vector space and its relation to dimension.

   A basis is a set of vectors that is both linearly independent and spans the space, so every vector has a unique expansion in it. The number of basis vectors equals the dimension of the space.

4. What is the standard inner product (scalar product) for two complex vectors in $\mathbb{C}^n$?

   $\langle \vec{u}|\vec{v}\rangle = \sum_{i=1}^{n} u_i^{*} v_i$, where $u_i^{*}$ denotes complex conjugation. It is linear in the second argument and conjugate-linear in the first.

5. When are two vectors orthogonal, and what is an orthonormal set?

   Two vectors are orthogonal when $\langle \vec{u}|\vec{v}\rangle = 0$. A set $\{\vec{e}_i\}$ is orthonormal if $\langle \vec{e}_i|\vec{e}_j\rangle = \delta_{ij}$ (orthogonal and each normalized to unit length).

6. State the completeness (closure) relation for an orthonormal basis $\{|e_i\rangle\}$.

   $\sum_{i} |e_i\rangle\langle e_i| = \mathbb{1}$ (the identity operator). It guarantees any vector can be expanded as $|\psi\rangle = \sum_i |e_i\rangle\langle e_i|\psi\rangle$.

7. State the Cauchy-Schwarz inequality for an inner product space.

   $|\langle \vec{u}|\vec{v}\rangle|^{2} \leq \langle \vec{u}|\vec{u}\rangle\,\langle \vec{v}|\vec{v}\rangle$, with equality if and only if $\vec{u}$ and $\vec{v}$ are linearly dependent.

8. What is the purpose of the Gram-Schmidt procedure?

   It converts a linearly independent set $\{\vec{v}_i\}$ into an orthonormal set $\{\vec{e}_i\}$ spanning the same subspace, by successively subtracting projections: $\vec{u}_k = \vec{v}_k - \sum_{j<k}\langle \vec{e}_j|\vec{v}_k\rangle\vec{e}_j$, then normalizing.

9. Define a Hermitian (self-adjoint) matrix and give its key spectral properties.

   A matrix with $A^{\dagger}=A$, i.e. $A_{ij}=A_{ji}^{*}$. Its eigenvalues are real, and eigenvectors belonging to distinct eigenvalues are orthogonal; it can be diagonalized by a unitary matrix.

10. Define a unitary matrix and state its defining property.

    A matrix $U$ with $U^{\dagger}U = UU^{\dagger} = \mathbb{1}$, so $U^{-1}=U^{\dagger}$. It preserves inner products and has eigenvalues of unit modulus ($|\lambda|=1$).

11. What is an orthogonal matrix and what is its determinant?

    A real matrix $O$ with $O^{T}O = \mathbb{1}$, so $O^{-1}=O^{T}$. Its rows/columns form an orthonormal set, and $\det O = \pm 1$.

12. How is the trace of a matrix defined, and what is its key invariance?

    $\operatorname{Tr}(A)=\sum_i A_{ii}$, the sum of diagonal elements. It is invariant under similarity (cyclic): $\operatorname{Tr}(ABC)=\operatorname{Tr}(BCA)$ and equals the sum of the eigenvalues.

13. Write the eigenvalue equation for a matrix $A$ and the condition determining the eigenvalues.

    $A\vec{v}=\lambda\vec{v}$ with $\vec{v}\neq\vec{0}$. Nontrivial solutions require the characteristic equation $\det(A-\lambda\mathbb{1})=0$.

14. How are the trace and determinant of a matrix related to its eigenvalues $\lambda_i$?

    $\operatorname{Tr}(A)=\sum_i \lambda_i$ and $\det(A)=\prod_i \lambda_i$.

15. Define a similarity transformation and state which quantities it preserves.

    $A' = S^{-1}AS$ for an invertible matrix $S$. It preserves the eigenvalues, trace, determinant, and characteristic polynomial; $A$ and $A'$ represent the same operator in different bases.

16. What is the condition for a matrix to be diagonalizable?

    An $n\times n$ matrix is diagonalizable iff it has $n$ linearly independent eigenvectors (i.e. the geometric multiplicity equals the algebraic multiplicity for every eigenvalue). Then $D=S^{-1}AS$ with columns of $S$ being the eigenvectors and $D$ holding the eigenvalues.

17. By what kind of matrix can a Hermitian matrix be diagonalized, and what is the form?

    By a unitary matrix $U$: $U^{\dagger}AU = D$, where $D$ is real diagonal containing the eigenvalues and the columns of $U$ are the orthonormal eigenvectors.

18. State the Cayley-Hamilton theorem.

    Every square matrix satisfies its own characteristic equation: if $p(\lambda)=\det(A-\lambda\mathbb{1})$, then $p(A)=0$.

19. How do you compute a function $f(A)$ of a diagonalizable matrix, e.g. $e^{A}$?

    Diagonalize $A=SDS^{-1}$; then $f(A)=S\,f(D)\,S^{-1}$, where $f(D)$ applies $f$ to each diagonal eigenvalue. For example $e^{A}=S\,\mathrm{diag}(e^{\lambda_1},\dots,e^{\lambda_n})\,S^{-1}$.

20. What is the general form of an $n$th-order linear ordinary differential equation?

    $a_n(x)\frac{d^{n}y}{dx^{n}}+a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdots+a_1(x)\frac{dy}{dx}+a_0(x)y = g(x)$. It is homogeneous if $g(x)=0$, inhomogeneous otherwise.

21. How is the general solution of an inhomogeneous linear ODE structured?

    $y = y_c + y_p$, where $y_c$ (complementary function) is the general solution of the associated homogeneous equation and $y_p$ is any particular solution of the full equation.

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