MSc Physics (Electronics I) SEM I 2023 2024 Feb 2024 MATHEMATICAL PHYSICS Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate full marks

1. Q1 (a) Attempt any one:
   • (i) | Evaluate the contour integral where c is the circle |z| = 10
   • (ii) Obtain the Taylor or Laurent series which represents the function 7
   • (ii) |z| > 2
   • (b) Attempt any one:
   • (i) the imaginary part of the analytic function whose real part is 3
   • (ii) Show that the function e*(cos y + isin y) is an analytic function, find its 3
2. Q2 (a) Attempt any one:
   • (i) Determine the eigenvalues and eigenvector of 7
   • (ii) The Pauli spin matrices in quantum mechanics are 7 Show that = 0,” =a unit matrix Also show that any two of these matrices anti-commute
   • (b) Attempt any one:
   • (i) Prove that the matrix | 1 is unitary. 3 (li) Using Levi-Civita symbol, show that AX (B ) (A. C ) (A. B) 3
3. Q3 (a) Attempt any one: the Legendre’s equation (1 — x?) +n(n+1)y = 0 and 7
   • (ii) Derive Bessel’s equation from Legendre’s Equation. 7
   • (b) Attempt any one:
   • (i) Show = 3 the general solution + 4y = sin3x 3
4. Q4 (a) Attempt any one:
   • (i) Fourier transform is represented in three dimensions? Hence Find the 7 Fourier transform of Yukawa potential f(r) =
   • (ii) Solve the differential equation using Laplace transform 7 + 4y = sin 2t subject to the initial conditions yp = 10, yg = 0
   • (b) Attempt any one:
   • (i) State and prove Fourier convolution theorem. 3 Find f(t) if Laplace transform F(s) = 3
5. Q5 Attempt any five:
   • (a) Show that e*(x cos y — y sin y) is a harmonic function. 2
   • (b) Find the singularity and its type of f(z) = sin — 2
   • (c) the values of x, y, z and ‘a’ which satisfy the matrix equation. 2
   • (d) Prove that = 6 2
   • (f) Show that H,(—x) = 2
   • (g) Find Laplace transform of f(t) = t using definition. 2
   • (h) Find inverse Laplace transform of 2

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