MSc Physics (Electronics I) SEM I 2023 2024 Feb 2024 QUANTUM MECHANICS 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) slit experiment, prove that when two non-interacting beams with intensity 7 and combine in same region of space, the resultant intensity is = + if beam is composed of particle, while resultant intensity is ] = I, + In + cos(@, — if beam is composed of waves. Here — is phase difference between two beams Interpret the result
   • (ii) Wavefunction for a system of particle confined to a region x [0, L] is given by 7
   • (a) Find value of normalization constant @ in the wavefunction
   • (b) Calculate probability of finding particle in range to
   • (c) Assume particle behaves like free particle inside the region x [0, L] and show that energy eigenvalue for particle is
   • (b) Attempt any one:
   • (i) Evaluate following commutator relations: 3
   • (ii) Find linear momentum expectation value for following wavefunction 3
2. Q2 (a) Attempt any one:
   • (i) (a) Check if the following operators are Hermitian 7
   • (b) For Hermitian operator, prove that all of its eigenvalues are real and the eigenvectors corresponding to different eigenvalues are orthogonal
   • (ii) (a) Write a note on Schrodinger Picture 7
   • (b) Consider two states = + — where and are
   • (b) Attempt any one:
   • (i) How operators transform under unitary transformation? Show that if operator A is Hermitian 3 then its transform A’ is also Hermitian
   • (ii) State any 3 properties of Hilbert space. 3
3. Q3 (a) Attempt any one:
   • (i) Show that the energy and total momentum of an isolated system are constants of the motion. 7
   • (ii) Derive an expression for one dimensional harmonic oscillator and show in which domain 7 the wave function is (a) Oscillatory (b) Non-Oscillatory
   • (b) Attempt any one:
   • (i) Show that in the Eigen state of the harmonic oscillator, the average kinetic energy <T> 3 is equal to the potential energy <V>
   • (ii) Show that T+R=1 for all one-dimensional barrier problems. 3
4. Q4 (a) Attempt any one:
   • (i) operator form of Lz in spherical polar coordinates. 7
   • (i) The Schrodinger equation for hydrogen atom can be defined as + u=
5. Q0 Solve this equation when (a) p is very large i.e. p — and (b) is in neighborhood of
   • (b) Attempt any one: Show that [Lx, Ly] =ih Lz 3
   • (ii) Ground state of hydrogen atom is given by = the value of r for 3 which radial probability density is maximum
6. Q5 Attempt any five:
   • (a) Explain the concept of de Broglie wavelength. 2
   • (b) What are observables? Give 2 examples of observables. 2
   • (c) Consider a Matrix A which represents operator A, a ket |y) and a bra 2
   • (d) Define Hermitian operator and state its properties. 2
   • (e) Write down the Schrodinger equation for free particle of mass mand show the kinetic energy 2 of the particle is
   • (f) Under what conditions is the expectation of an operator A is constant in time? 2
   • (g) Evaluate the minimum value of ALy ALz 2
   • (h) Evaluate [Lz, 2

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