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*Answer can only contain numeric values

QUESTION: 1

Consider the state where are the eigenstates of Value of is approximately ________ (Upto two decimal places)

Solution:

*Answer can only contain numeric values

QUESTION: 2

An electron with energy 'E' is coming from far left to a potnetial step at x = 0. If E/V then ratio of wavelength (λ_{1}/λ_{2}) of electron in region I and II is (Upto two decimal places)

Solution:

Broshe wavelnegth

*Answer can only contain numeric values

QUESTION: 3

Wavefunction of a particle iii some one dimension potential is given by,

where k is the positive real constant. Probability of:finding the particle between x = 0 and x = a/4 is _____ %. (Upto two decimal places)

Solution:

To find A, we have to normalize the wavefunction

Probability of finding particle between* = 0 and x = a/4

QUESTION: 4

Consider the spin state of a system in reresentation

What is the average value of the operator corresponding to the x-component of the spin.

Solution:

QUESTION: 5

Find the eigenvalues of the component of spin operator of an electron in direction of a unit vector with spherical coordinates

Solution:

Eigenvalves of the spin operator do not depent on direction They are always

QUESTION: 6

An operator operates on a state vector such that is known to be existed, then

Solution:

QUESTION: 7

represent the total angular momentum operator. The value of is

Solution:

QUESTION: 8

A particle of mass m is being scattered by a potential given by V(r) = The differential cross-section calculated by Born's approximation is

Solution:

differential cross section

QUESTION: 9

Consider a particle in the 1-D potential well given by

Now, a small perturbation is added to the well given as

The first-order correction to the energy of second energy level, is

Solution:

The wavefunction of the unperturbed second energy level is

First order correct to the energy.

QUESTION: 10

In a two-dimensional vector space spanned by basis vectors Two vectors in this space are given as

An operator is represented by matrix.

Solution:

*Answer can only contain numeric values

QUESTION: 11

Wave function of a particle is given as If energy of particle in this state is and form of potential is then magnitude of the vector is _____ (upto two decimal places)

Solution:

*Answer can only contain numeric values

QUESTION: 12

An electron is placed in an uniform magnetic field where points in the direction of vector The Hamiltonian of the system is H = is a constant and are pauli matrices Energy of ground state in terms of kB_{0}

Solution:

QUESTION: 13

A particle of mass 'm' with in box of length L, the wave function of the state is given by

then the energy of the particle of the state is

Solution:

The given wave function,

Therefore, energy of the given state

*Answer can only contain numeric values

QUESTION: 14

A linear harmonic oscillator is perturbed by the potential V' (x) = ax^{2}-bx^{4}; a and b constants. If first order energy correction to the second excited energy level is zero then in units of is (Upto two decimal places)

Solution:

*Answer can only contain numeric values

QUESTION: 15

The wave function of a hydrogen atom is given by where 'a_{0}' is Bohr radius and is spherical harmonic function for l = 2 and m = 0. If probability of finding the electron is maximum at 'r_{mp}' and average distance between electron and proton is <r> then ratio of the two is (upto two decimal places)

Solution:

It is clear from the wacefunction that n = 3.

spherical harmonics are orthonormal functions, so

QUESTION: 16

Angular frequency for a wave propagating inside a waveguide is given in terms of the wave number K and the width of the guide b as

If group velocity is 3/4^{th} of the velocity then k is

Solution:

QUESTION: 17

Seven spin -3/2 identical non-interacting particle are subjected to the potential given as

The lowest energy of the system is

Solution:

It is anisotropic harmonic oscillator

because of spin 3/2, maximum four particles can be filled in one energy level. Energies of two lowest energy levels are

QUESTION: 18

A particle of mass 'm' in 1-D harmonic oscillator potential In the non-relativistic limit, where the kinetic energy T and momentum P are related as T = p^{2}/2m. The ground state energy is known to be For the relativestic corrections in relation between T and P, calculate energy shift in ground state to the order of 1/c^{2} (c is speed of light in vacuum)

Solution:

Kinetic energy (T) = E - mc^{2}

QUESTION: 19

The wave function of a one-dimensional system is given as

where A, n and a are positive constants. The form of the potential if V(x) → 0

Solution:

QUESTION: 20

The wavefunction of a hydrogen atom is given by

where a_{0} is Bohr radius and is spherical harmonic function for l = 1 and m = 0. What is the dispersion in the measurement of distance of electron from nucleus?

Solution:

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