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Test: Quantum Mechanics - 1


20 Questions MCQ Test GATE Physics Mock Test Series | Test: Quantum Mechanics - 1


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Attempt Test: Quantum Mechanics - 1 | 20 questions in 40 minutes | Mock test for GATE preparation | Free important questions MCQ to study GATE Physics Mock Test Series for GATE Exam | Download free PDF with solutions
QUESTION: 1

Consider a two state system with normalized energy eigen state ψ1 & ψ2 and energy E1 < E2 what is the possible range for the expectation value of on an orbitrary linear combination of two state

Solution:



(H) > E1 ......(1)
similarly

 ......(2)
form equation (1) & (2)

 

QUESTION: 2

Suppose a wave function  and an operator is given by  is given by

Solution:

We have operator, 


Now, my operator will be 

∴  

∴ 
∴ 

QUESTION: 3

Consider the following Stem-Gerlach apparatus incoming beam consist of electron 2/3 of them having spin  and other 1/3 have spin  in the z-direction

what fraction of the incident beam are detected in the up and down outputs of the apparatus

Solution:

*Answer can only contain numeric values
QUESTION: 4

Consider a system whose initial state at r = 0 is given in term of a complete and orthonormal set of three vectors  as follows  the probability of find tlie system at tune t in state  is _______ (upto two decunalplaces)

Solution:

We have wave function at t = 0

Therefore, the wave function at t = t is given by

(where, E1, E2 and E3, are energy of difference states)

The probability of finding the system at time t in state

*Answer can only contain numeric values
QUESTION: 5

An election is confined in an infinite square well of width 10x10^-15 m. Calculate the wavelength of the electron emitted when the proton undergoes a transition from the first excited state (n=2) to the ground state (n=1).​

Solution:

The energy of the particle in the box of width L is given by



λ = 201 fm

*Answer can only contain numeric values
QUESTION: 6

Consider a tliree dimensional harmonic oscillator with Hamiltonian

The number of distinct eigenstates with energy eigenvalue 5/2 ℏω is_________
(Answer should be an integer).

Solution:

Energy expression for 3-D harmonic oscillator, 


The degneracyofnth state of an isolated 3D harmonic oscillator is given by

QUESTION: 7

Consider an operator  for a system of total angular momentum j = 1 then winch of the following state having non zero uncertainty
Solution:

Operator Jx in matrix form for J = 1

The eigenvalue equation is given by



Consider  is the eigen function of Jx corresponding the eigen value λ = 0


And according normalization condition,


⇒  
∴  
∴  
Similarly the eigen function corresponding to others two eigen values are

As  are the eigen state of operator Jx.
Therefore, uncertainty of Jx will be zero for these three states and  is not an eigen states of Jx.
So, uncertainty of Jx will be non-zero for this state.

QUESTION: 8

In the simple harmonic oscillator

what are the condition on m and m' for  to be non-zero 
Solution:




For non zero of 
⇒  m = n ± 1
And 

QUESTION: 9

Find the differential cross-section for die scattering of slow (low velocity) particle from a spherical delta poten- tial V (r) = V0 δ (r - a)
Solution:

In case, the incident particle have low velocities only S-waves, L = 0 contributes in the scattering
Therefore, scattering amplitude is given by


Differential cross-section

*Answer can only contain numeric values
QUESTION: 10

For case of  n = 2, ℓ = 1, m = 0 the value of r at which the radial probability density of the hydrogen atom reaches its maximum is ____ a0.  (answer should be an integer)

Solution:


Tlie probability density is given by

For the maximum of tlie probability density

*Answer can only contain numeric values
QUESTION: 11

Consider a system winch is mtially in the state  was measured with value -ℏ, the probability is_______(upto one decimal place) 

Solution:



for eigenvalue mL = -1, eigenfunction probability of measuring Lz in state is

*Answer can only contain numeric values
QUESTION: 12

An electron is confined in the ground state of a one dimensional harmonic oscillator such that  energy required to excite to its first excited state is _____ (MeV)

Solution:

The virial theorem states that



for ground state

for harmonic oscillator 

Energy required to excite the electron to its first excited state

QUESTION: 13

A particle of mass m coming in from the left with energy E > 0, encounters barrier potential

The wave function is given by


The constant A and B satisfy which one of the following relation?
Solution:

The barrier potential is given as


The wave function is given as

Hie quantity B in equation (i) represent the amplitude ot reflected ray in medium 1 and A in equation (ii) represent the amplitude of transmitted ray in medium III
Hence, Reflectivity R = |A|2 and transmitivity T = | B |2
smce R + T = 1
⇒ 

QUESTION: 14

Calculate the width of the probability density distribution for r(i.e Br) for hydrogen atom for the state 
Solution:

Tlie width of the probability distribution is given by 




width ot probability distribution

QUESTION: 15

At t = o, a state is given by

Where are ortlionomial stationary states of energy El & E2 respectively with E2 > El what is the shortest tune T > 0 for w hich is orthogonal to

Solution:

At t = 0, the wave function is given by 
Therefore, the wave function at later time t = T is given by

These two states ortliogonal at time T


For shortest time m = 0

QUESTION: 16

If the state of a particle moving in one dimensional harmonic oscillator is given by

Where represent the normalized nth energy eigenstate find the expectation value of  number operator 
Solution:





QUESTION: 17

A particle of mass m moves in a one dimensional potential box

Consider the V0 part as perturbation, using first order perturbation method calculate the energy of ground state.
Solution:

Tlie energies and wavefunction of a particle in flat box of length 6a




energy correction to first order perturbation 


*Answer can only contain numeric values
QUESTION: 18

A spin state precesses in a magnetic field same way as the classical magnetic dipole precesses in magnetic field with lasmor frequency given by  consider the Hamiltonian  Larrnor frequency  is (in tem is of ω0)
(answer should b e an mteger)

Solution:





From equation (i) and (ii) 

*Answer can only contain numeric values
QUESTION: 19

Consider a system of four non-interacting identical spin 1/2 particles that are in same state  and confined to move in a one-diniension infinite potential well of length a: V(x) = 0 for 0 < x < a and V(x) = ∞ forotlier values of x. The ground state energy of the system in units of is (answer should be an integers).

Solution:

The energy of the particle in one dimensional infinite potential box is given by

4 particles having same state 
ground state energy'of system is


QUESTION: 20

Consider a system whose intial state and hamiltonian are given by

find the total energy of a system
Solution:


eigenvalues of energy 

λ = -5,3, 5; E1 = - 5, E2 = 3, E3 = 5
and their correspondmg eigenvectors


expanding in terms of eigenvectors. 



Total energy of the system 


Second method:

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