MSc Physics (Electronics I) SEM I 2022 2023 Jan 2023 PHYSICS QUANTUM MECHANICS I Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate full marks

1. Q1 (a) Attempt any one: - 8 marks
   • (i) The Hamiltonian operator and two other observables A and B for a certain physical system are represented by matrices Where a and b are real numbers A state is given by |w) = + + cz where And C2, C3 are constants
2. Q1 Find the relationship between and c3 such that |w) is normalized
3. Q2 Find the expectation values of and A
4. Q3 What are the possible values of energies that can describe by vector
   • (ii) Write the answers to the following questions
5. Q1 Define a Hermitian conjugate of a general operator A and state the condition for it to be Hermitian
6. Q2 Show that the eigenvalues of Hermitian operators are real
   • (b) Attempt any one: - 4
   • (i) Evaluate
7. Q1 Ap, AE
8. Q2 Ax AE
   • (ii) A linear harmonic oscillator was initially in the state = 2 do(x) +
9. Q1 Normalize
10. Q2 (a) Attempt any one: - 8 marks
    • (i) Fora linear harmonic oscillator evaluate Ax and Ap, A particle with energy E > 0 is incident along x — axis on a potential barrier which is given by Vix) =V Obtain transmission coefficient for the case E < Vo
    • (b) Attempt any one: - 4
    • (i) Evaluate [a, H]
    • (ii) A Particle of mass m,which move freely inside an infinite potential well of length a, is initially in the state w(x, 0) = (=) + sin (=) find
11. Q1 at any later time, t
12. Q2 Calculate the probability densityp(x, t)
13. Q3 (a) Attempt any one: - 8 marks
    • (i) The Schrodinger equation for hydrogen atom can be defined as + u = 0. Solve this equation when (a) p is very large i.e. p and (b) p is in neighborhood of origin i.e. p > 0
    • (ii) Write down Schrodinger equation for two particle system. Redefine the Schrodinger equation in terms of center of mass coordinate R and relative coordinate 7. By using separation of variable technique, derive and solve equation for wavefunction corresponding to center of mass R of the system
    • (b) Attempt any one: - 4
    • (ii) Evaluate: and where and are x-component and y-component of
14. Q4 (a) Attempt any one: - 8 marks
    • (i) Derive Jy and J, matrices corresponding to angular momentum state j = 1
    • (ii) For an electron with a spin state 7 = calculate the probability that on measurement the electron will be found in
    • (1) Spin down state along y-direction
    • (2) Spin up state along x-direction
    • (b) Attempt any one: - 4
    • (i) Note down all coupled and uncoupled representations for j; = and jz = =
    • (ii) Evaluate: and [J,, J_]
15. Q5 Attempt any four: - 12 marks
    • (a) Show that unitary transformations preserve length of vectors consider the operator A= |i O O
16. Q1 show that A is Hermitian
17. Q2 Find its eigenvalues
    • (c) Show that (x) = 0 and (p) = 0 using properties of annihilation and creation operator
    • (d) Show that Hamiltonian for harmonic oscillator is
    • (e) Evaluate: AL,
    • (f) wavefunction = + Y2 , calculate the expectation value of L, operator
    • (g) Prove that all Pauli matrices follow 6? = 1 where is a 2 x 2 identity matrix
    • (h) Evaluate: j = m=

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