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QUESTION: 1

Consider a set of wave functions ψ_{i}(x). Which of the following condition guarantees that the functions are normalized and mutually orthogonal? (i, j take values from 1 to n)

Solution:

For the states to be orthogonal

and for them to be normalised,

QUESTION: 2

The phase velocity of ripples on a liquid surface is where * S* is the surface tension and ρ the density of the liquid. The group velocity of the ripples is given by :

Solution:

As wave number,

We know that group velocity given by

The correct answer is:

QUESTION: 3

The solutions of a free particle are :

Solution:

*V = 0* for a free particle

∴

Schrödinger equation.

(General solution)

∴ * Ae^{ikx}* and

The correct answer is: Energy and momentum eigen functions with both positive and negative momentum eigenvalues.

QUESTION: 4

The wave function for a particle constrained to move in 1*D* is shown in the graph below.

What is the probability that the particle would be found between x = 2 and x = 4 ?

Solution:

The correct answer is: 13/16

QUESTION: 5

The probability current density is given as Which of the following is true?

Solution:

From the continuity equation

i.e. ** p** is constant in time

Probability current density

The correct answer is: If then probability density is constant in time.

QUESTION: 6

Smallest possible uncertainty in position of the electron moving with velocity 3 × 10^{7} m/s. Given, h = 6.63 × 10^{–34 }Js, m_{0} = 9.1 × 10^{–31} kg.

Solution:

Given, **v** = 3 × 10^{7} **m/s**

Let Δx_{min} be the minimum uncertainty in position of the electron and Δp the maximum uncertainty in the momentum of the electron.

Thus, we have,

= 0.03867 × 0.9949 × 10^{–10} **m**

= 3.8 × 10^{–12} **m**

The correct answer is: 3.8 × 10^{–12} **m**

QUESTION: 7

Choose the correct statement

Solution:

The correct answer is: All the energy eigenvalues are always real.

QUESTION: 8

The wave function of the particle lies in which region?

Solution:

The particle cannot exist outside the box, as it cannot have infinite amount of energy. Thus, it’s wave function is between 0 and L, where L is the length of the side of the box.

QUESTION: 9

A linear harmonic oscillator of mass * m* oscillates with a frequency where

Solution:

The energy of the linear harmonic oscillator is

This is a constant of motion. We can represent the constant value of * E* by means of averages of the kinetic and potential over a cycle of motion by writing.

The average value of

Since from the uncertainty principle To determine the minimum energy of the oscillator, we put

The minimum energy is

QUESTION: 10

Choose the correct statement for a free particle with ψ(x) = Ae^{ikx}

Solution:

** V = 0** for a free particle

∴

Schrödinger equation.

∴ * Ae^{ikx}* and

∴ are solution to the free particle

The correct answer is: Energy eigenvalue is and momentum is

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