Test: Quantum Mechanics - 2


20 Questions MCQ Test GATE Physics Mock Test Series | Test: Quantum Mechanics - 2


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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 (λ12) 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 kB0


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) = ax2-bx4; 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 'a0' is Bohr radius and  is spherical harmonic function for l = 2 and m = 0. If probability of finding the electron is maximum at 'rmp' 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 3/4th 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 = p2/2m.  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/c2 (c is speed of light in vacuum)

Solution:

Kinetic energy (T) = E - mc2


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 a0 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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