Test: Quantum Mechanics - 1 - Physics MCQ

# Test: Quantum Mechanics - 1 - Physics MCQ

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## 20 Questions MCQ Test GATE Physics Mock Test Series 2025 - Test: Quantum Mechanics - 1

Test: Quantum Mechanics - 1 for Physics 2024 is part of GATE Physics Mock Test Series 2025 preparation. The Test: Quantum Mechanics - 1 questions and answers have been prepared according to the Physics exam syllabus.The Test: Quantum Mechanics - 1 MCQs are made for Physics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Quantum Mechanics - 1 below.
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Test: Quantum Mechanics - 1 - 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

Detailed Solution for Test: Quantum Mechanics - 1 - Question 1

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

......(2)
form equation (1) & (2)

Test: Quantum Mechanics - 1 - Question 2

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

Detailed Solution for Test: Quantum Mechanics - 1 - Question 2

We have operator,

Now, my operator will be

∴

∴
∴

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Test: Quantum Mechanics - 1 - 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

Detailed Solution for Test: Quantum Mechanics - 1 - Question 3

*Answer can only contain numeric values
Test: Quantum Mechanics - 1 - 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)

Detailed Solution for Test: Quantum Mechanics - 1 - Question 4

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
Test: Quantum Mechanics - 1 - 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).​

Detailed Solution for Test: Quantum Mechanics - 1 - Question 5

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

λ = 201 fm

*Answer can only contain numeric values
Test: Quantum Mechanics - 1 - 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).

Detailed Solution for Test: Quantum Mechanics - 1 - Question 6

Energy expression for 3-D harmonic oscillator,

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

Test: Quantum Mechanics - 1 - Question 7

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

Detailed Solution for Test: Quantum Mechanics - 1 - Question 7

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.

Test: Quantum Mechanics - 1 - Question 8

In the simple harmonic oscillator

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

Detailed Solution for Test: Quantum Mechanics - 1 - Question 8

For non zero of
⇒  m = n ± 1
And

Test: Quantum Mechanics - 1 - 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)

Detailed Solution for Test: Quantum Mechanics - 1 - Question 9

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
Test: Quantum Mechanics - 1 - 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)

Detailed Solution for Test: Quantum Mechanics - 1 - Question 10

Tlie probability density is given by

For the maximum of tlie probability density

*Answer can only contain numeric values
Test: Quantum Mechanics - 1 - Question 11

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

Detailed Solution for Test: Quantum Mechanics - 1 - Question 11

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

*Answer can only contain numeric values
Test: Quantum Mechanics - 1 - 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)

Detailed Solution for Test: Quantum Mechanics - 1 - Question 12

The virial theorem states that

for ground state

for harmonic oscillator

Energy required to excite the electron to its first excited state

Test: Quantum Mechanics - 1 - 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?

Detailed Solution for Test: Quantum Mechanics - 1 - Question 13

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
⇒

Test: Quantum Mechanics - 1 - Question 14

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

Detailed Solution for Test: Quantum Mechanics - 1 - Question 14

Tlie width of the probability distribution is given by

width ot probability distribution

Test: Quantum Mechanics - 1 - 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

Detailed Solution for Test: Quantum Mechanics - 1 - Question 15

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

Test: Quantum Mechanics - 1 - 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

Detailed Solution for Test: Quantum Mechanics - 1 - Question 16

Test: Quantum Mechanics - 1 - 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.

Detailed Solution for Test: Quantum Mechanics - 1 - Question 17

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
Test: Quantum Mechanics - 1 - 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)

Detailed Solution for Test: Quantum Mechanics - 1 - Question 18

From equation (i) and (ii)

*Answer can only contain numeric values
Test: Quantum Mechanics - 1 - 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).

Detailed Solution for Test: Quantum Mechanics - 1 - Question 19

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

Test: Quantum Mechanics - 1 - Question 20

Consider a system whose intial state and hamiltonian are given by

find the total energy of a system

Detailed Solution for Test: Quantum Mechanics - 1 - Question 20

eigenvalues of energy

λ = -5,3, 5; E= - 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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