GATE PH concept:harmonic-oscillator PYQs & Answers

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concept:harmonic-oscillator

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Questions 1–10 of 10. Imported answers require independent review unless marked reviewed.

Topic practice 10 questionsPage 1 of 1

If the perturbation

is added to the Hamiltonian of a one-dimensional harmonic oscillator, the matrix element

is/are non-zero for which of the following states? Here, the eigenstates of the harmonic oscillator are denoted by

Your answerSelect all correct options. You can choose more than one.

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The energy

and degeneracy

of the second excited state of a three-dimensional, isotropic quantum harmonic oscillator with angular frequency

are

Your answerChoose one option.

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Two identical particles with a fixed total energy

are in thermal equilibrium in a one-dimensional harmonic oscillator potential

. Let the entropy of the particles be denoted by

if they are fermions with spin

and

if they are bosons with spin 0. Then, which of the following options is correct? (

is the Boltzmann constant)

Your answerChoose one option.

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Let

and

denote the energy eigenstates of a one-dimensional simple harmonic oscillator. The position and momentum operators are

and

, respectively. The matrix element

is non-zero when

Your answerChoose one option.

only

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For a one-dimensional harmonic oscillator, the creation operator (

) acting on the

state

, where

, gives

. The matrix representation of the position operator

for the first three rows and columns is

Your answerChoose one option.

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Consider a state described by

, where

and

are respectively the second and fourth normalized harmonic oscillator wave functions and

is the angular frequency of the harmonic oscillator. The wave function

will be orthogonal to

at time

equal to

Your answerChoose one option.

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Let

and

respectively denote the lowering and raising operators of a one-dimensional simple harmonic oscillator. Let

be the energy eigenstate of the simple harmonic oscillator. Given that

is also an eigenstate of

, the corresponding eigenvalue is

Your answerChoose one option.

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The Hamiltonian for a quantum harmonic oscillator of mass

in three dimensions is

where

is the angular frequency. The expectation value of

in the first excited state of the oscillator in units of

(rounded off to one decimal place) is

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A one dimensional simple harmonic oscillator with Hamiltonian

is subjected to a small perturbation,

. The first order correction to the ground state energy is dependent on

Your answerChoose one option.

only

and

only

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The degeneracy of the third energy level of a 3-dimensional isotropic quantum harmonic oscillator is

Your answerChoose one option.

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GATE PH concept:harmonic-oscillator PYQs

How to practice concept:harmonic-oscillator

Question 26

Question 41

Question 42

Question 43

Question 45

Question 35

Question 4

Question 21

Question 19

Question 22