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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

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, (** n_{x}, n_{y}, n_{z}**) 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 × 10^{11}m

Solution:

Energy levels for hydrogen atom are

Again substituting for the gravitational analog

here ** r** is the earth sun distance

The correct answer is:

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;

**I. ***Ae*^{–}^{br}

**II. **A sin*br*

**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

Solution:

QUESTION: 8

Box I has each edge of length ‘*a*’ units. Box II has edges as a, 2a, a/2 along the *x*, *y* and z axis respectively. For what values of *n _{x}*,

Solution:

For ** n_{x}** - 1,

(

The correct answer is: (

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^{–138}*m*

The correct answer is: ** r** ~ 2 × 10

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

Solution:

transforms hydrogen results to the gravitational analogs

(–)*ve* sign for attractive force.

The correct answer is:

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