Test: Quantum Mechanics - 2
20 Questions MCQ Test GATE Physics Mock Test Series | Test: Quantum Mechanics - 2
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Consider the state 
where 
are the eigenstates of 
Value of 
is approximately ________ (Upto two decimal places)
where are the eigenstates of Value of is approximately ________ (Upto two decimal places)
An electron with energy 'E' is coming from far left to a potnetial step at x = 0. If E/V =9/8 then ratio of wavelength (λ1/λ2) of electron in region I and II is (Upto two decimal places)
Broshe wavelnegth 
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)
where k is the positive real constant. Probability of:finding the particle between x = 0 and x = a/4 is _____ %. (Upto two decimal places)
To find A, we have to normalize the wavefunction 
Probability of finding particle between* = 0 and x = a/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.
Find the eigenvalues of the component of spin operator 
of an electron in direction of a unit vector with spherical coordinates 
Eigenvalves of the spin operator 
do not depent on direction 
They are always 
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
differential cross section 
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
The wavefunction of the unperturbed second energy level is
First order correct to the energy.
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
The given wave function,
Therefore, energy of the given state
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 'rmp' and average distance between electron and proton is <r> then ratio of the two 
is (upto two decimal places)
It is clear from the wacefunction that n = 3.
spherical harmonics are orthonormal functions, so
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
Seven spin -3/2 identical non-interacting particle are subjected to the potential given as
The lowest energy of the system is
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
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)
Kinetic energy (T) = E - mc2
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?
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