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Test: Quantum Mechanics - 2

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Test: Quantum Mechanics - 2 for GATE 2023 is part of GATE Physics Mock Test Series preparation. The Test: Quantum Mechanics - 2 questions and answers have been prepared according to the GATE exam syllabus.The Test: Quantum Mechanics - 2 MCQs are made for GATE 2023 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Quantum Mechanics - 2 below.
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Test: Quantum Mechanics - 2 - Question 1

Consider the state  where  are the eigenstates of  Value of   is approximately ________ (Upto two decimal places)

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

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Test: Quantum Mechanics - 2 - Question 2

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)

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

Broshe wavelnegth

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Test: Quantum Mechanics - 2 - 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)

Detailed Solution for Test: Quantum Mechanics - 2 - Question 3

To find A, we have to normalize the wavefunction

Probability of finding particle between* = 0 and x = a/4

Test: Quantum Mechanics - 2 - 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.

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Test: Quantum Mechanics - 2 - Question 5

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

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Eigenvalves of the spin operator  do not depent on direction  They are always

Test: Quantum Mechanics - 2 - Question 6

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

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Test: Quantum Mechanics - 2 - Question 7

represent the total angular momentum operator. The value of  is

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Test: Quantum Mechanics - 2 - 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

Detailed Solution for Test: Quantum Mechanics - 2 - Question 8

differential cross section

Test: Quantum Mechanics - 2 - 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

Detailed Solution for Test: Quantum Mechanics - 2 - Question 9

The wavefunction of the unperturbed second energy level is

First order correct to the energy.

Test: Quantum Mechanics - 2 - 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.

Detailed Solution for Test: Quantum Mechanics - 2 - Question 10

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Test: Quantum Mechanics - 2 - 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)

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Test: Quantum Mechanics - 2 - 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

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Test: Quantum Mechanics - 2 - 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

Detailed Solution for Test: Quantum Mechanics - 2 - Question 13

The given wave function,

Therefore, energy of the given state

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Test: Quantum Mechanics - 2 - 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)

Detailed Solution for Test: Quantum Mechanics - 2 - Question 14

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Test: Quantum Mechanics - 2 - 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)

Detailed Solution for Test: Quantum Mechanics - 2 - Question 15

It is clear from the wacefunction that n = 3.

spherical harmonics are orthonormal functions, so

Test: Quantum Mechanics - 2 - 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

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Test: Quantum Mechanics - 2 - Question 17

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

The lowest energy of the system is

Detailed Solution for Test: Quantum Mechanics - 2 - Question 17

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

Test: Quantum Mechanics - 2 - 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)

Detailed Solution for Test: Quantum Mechanics - 2 - Question 18

Kinetic energy (T) = E - mc2

Test: Quantum Mechanics - 2 - 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

Detailed Solution for Test: Quantum Mechanics - 2 - Question 19

Test: Quantum Mechanics - 2 - 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?

Detailed Solution for Test: Quantum Mechanics - 2 - Question 20

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