Test: Quantum Mechanics - 2 - Physics MCQ

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.

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

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

 represent the total angular momentum operator. The value of  is

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  

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

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

An operator  is represented by matrix.

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)

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

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 

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)

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 'r_mp' and average distance between electron and proton is <r> then ratio of the two  is (upto two decimal places)

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 

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

The lowest energy of the system is 

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)

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 

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?