MPH-004 Solved Assignment 2026

Description

MPH-004 Solved Assignment 2026 Available

Assignment Code: MPH-004/TMA/2026

PART A

1. a) Electrons are accelerated through a potential 150 V and incident on a crystal with interatomic spacing d = 0.2 nm. Calculate the de Broglie wavelength and the first-order Bragg diffraction angle.

b) Estimate the uncertainty in the energy of a photon localized within a distance of 0.1 nm.

c) i) Normalize the wave function:

w(x, 0) = N[sin((pi*x)/L) + sin((2pi*x)/L)], 0 <= x < L .

ii) Calculate the expectation value of x for a particle in this state.

d) A quantum mechanical particle in one dimension has the wave function:

Psi(x) = N * x ^ n / x > 0; n > 1

Use the Schrödinger equation to determine the corresponding potential V(x)

2. a) A 1.5 mA beam of electrons enters a sharply defined boundary with a velocity 3 * 10 ^ 6 * m * s ^ – 1 and then its velocity reduces to 2 * 10 ^ 6 * m * s ^ – 1 due to the difference in potential. Calculate the transmitted and reflected currents.

b) Show that for any stationary state of a symmetric potential well: \langlex ^ 2\rangle = 0 .

c) Calculate the expectation value of the potential energy for the first excited state of a simple harmonic oscillator.

d) i) Write down the eigenfunctions for the Psi_{300} and v_{310}’ states of the hydrogen atom.

ii) Write the eigenvalues of L ^ 2 and hat L_{z} for the states Y_{300} and v_{310}

iii) Calculate the most probable value of r for the hydrogen atom in the state

PART B

3. a) Consider the following state vectors: Calculate (i) the norm of | psi rangle and (ii) the inner product langle psi | phi rangle w rangle= [[1], [i], [- 1]] ; | Phi rangle= [[2], [1 – i], [i]]

b) If an operator O is Hermitian and an operator Ü is unitary, show that the operator hat U dot o hat U ^ – 1 | Hermitian.

c) The spectral representation of an operator in a two-dimensional orthonormal basis | phi_{1} rangle,| phi 2 rangle|

tilde Omega =3| phi 1 rangle \langlephi_{1}|sqrt(2)| * phi_{1}\rangle\langlephi_{2}|sqrt(2)| * phi_{2}\rangle\langlephi_{1}|2| * phi_{2}\rangle langle phi 2 |

Determine the matrix elements of Ω.

d) | 1 rangle and | 2 rangle are the orthonormal basis states of a two-dimensional Hilbert space. A Hermitian operator A in this basis is given by the spectral representation: hat A =3|1 rangle \langle1|- 1| * 2\rangle langle2|. For a normalized state given by

| psi rangle= 1 2 |1 rangle+ (sqrt(3))/2 |2 rangle

calculate

langle overline A rangle and the root mean square deviation in A.

e) Derive the Heisenberg equations of motion for hat S_{x} ; and hat s_{y} for the Hamiltonian tilde H =co tilde S Z

4. a) For the simple harmonic oscillator

i) Show that [ tilde H , hat a ]=- hbar e0 hat B

ii) ‘Calculate the matrix element langle n| overline x | n + 1 rangle

b) Write down the angular momentum states | j ,m j rangle and calculate the matrix elements of j_{x} j_{y} and j_{z} for j = 1/2

ii) For the angular momentum state | 3 ,1 rangle show that J + |3,1 rangle= pi * sqrt(10) |3,2 rangle and J – |3,1 rangle= 2h * sqrt(3) |3,0 rangle

c) Show that in the terms of the | dagger rangle and | 4 rangle basis vectors defined by hat S_{z} | dagger rangle= hbar 2 | dagger rangle; hat S z | dagger rangle=- hbar 2 | mp rangle we can write: hat S x = n/2 [| tau rangle langle+|+ rangle langle tau|]