Quantum Mechanics In Three Dimensions MCQ


10 Questions MCQ Test Topic wise Tests for IIT JAM Physics | Quantum Mechanics In Three Dimensions MCQ


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QUESTION: 1

A three dimensional harmonic oscillator is in thermal equilibrium with a temperature reservoir at temperature T. The average total energy of the oscillator is :

Solution:

Average K.E. in one dimension is 
Average K.E. in 3 dimension is 
The correct answer is: 

QUESTION: 2

Consider the particle in a three dimensional box problem for 2 boxes. One having equal edges of length a  and the other of edges  along  xy and z  coordinates respectively.

Solution:

The correct answer is: Ground state energy of the second box is greater

QUESTION: 3

For the 3-dimensional second box:

Solution:

First excited state, (nxnynz) can be (1, 1, 2) or (1, 2, 1) or (2, 1, 1)

Now two energy levels have the same energy. Hence non degenerate
The correct answer is: The first excited state is non degenerate

QUESTION: 4

What will be quantum number ‘n’ for earth? Take the earth sun distance to be 1.496 × 1011m

Solution:

Energy levels for hydrogen atom are

Again substituting for the gravitational analog

here r is the earth sun distance
r = 1.496 × 1011m

n = 2.53 × 1074
The correct answer is: n ~ 1074

QUESTION: 5

Which of the following is true about a quantum harmonic oscillator?
(1) A spectrum of evenly spaced energy states
(2) A potential energy function that is linear in the position coordinate
(3) A ground state characterized by zero kinetic energy.
(4) A non zero probability of finding the oscillator outside the classical turning points.

Solution:

A quantum harmonic oscillator:

∴  Equally spaced

Potential Energy function     
∴   quadratic in r.

For ground state,  


A quantum Harmonic oscillator has non zero probability of being found beyond the classical turning points! This is one of the basic difference between a classical & quantum Harmonic oscillator

The correct answer is: 1 and 4 only

QUESTION: 6

Which of the following functions could represent the radial wave function for an electron in an atom? (r is the distance of the electron from the nucleus; Ab are constants)

I. Aebr
II. A sinbr
III. A/r

Solution:

The wave function of an electron should satisfy the following properties

The only function (from the ones given) that satisfies these conditions is Ae–br

The correct answer is: I only

QUESTION: 7

Suppose the earth made a transition to the next lower level (n – 1). How much energy (in Joules) would be released? What would be the wavelength of the emitted photon be?

Solution:


QUESTION: 8

Box I has each edge of length ‘a’ units. Box II has edges as a, 2a, a/2 along the xy and z axis respectively. For what values of nxnynz will the first excited state energy of box II be same as the first excited state energy of Box I

Solution:


For nx - 1, ny = 2, nz = 1
(E1) box II  = (E1) box I
The correct answer is: (nxnynz) is (1, 2, 1)

QUESTION: 9

What will be the Bohr radius of this system (work out the actual numerical value)

Solution:


In the case of hydrogen atom,

 for the gravitational analog.

 

= 2.34 × 10–138m
The correct answer is: r ~ 2 × 10–138m

QUESTION: 10

Consider the Earth-Sun system as a gravitational analog to the hydrogen atom.

What is the potential energy function. (Let m be the mass of earth and M be the mass of the sun)

Solution:

transforms hydrogen results to the gravitational analogs
(–)ve  sign for attractive force.

The correct answer is: 

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