IIT JAM > Postulates Of Quantum Mechanics MCQ Level - 2

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Postulates Of Quantum Mechanics MCQ Level - 2 - 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)

Detailed Solution for Postulates Of Quantum Mechanics MCQ Level - 2 - Question 1

For the states to be orthogonal

and for them to be normalised,

Postulates Of Quantum Mechanics MCQ Level - 2 - 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 :

Detailed Solution for Postulates Of Quantum Mechanics MCQ Level - 2 - Question 2

As wave number,

We know that group velocity given by

The correct answer is:

Postulates Of Quantum Mechanics MCQ Level - 2 - Question 3

The solutions of a free particle are :

Detailed Solution for Postulates Of Quantum Mechanics MCQ Level - 2 - Question 3

*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.

Postulates Of Quantum Mechanics MCQ Level - 2 - 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 ?

Detailed Solution for Postulates Of Quantum Mechanics MCQ Level - 2 - Question 4

The correct answer is: 13/16

Postulates Of Quantum Mechanics MCQ Level - 2 - Question 5

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

Detailed Solution for Postulates Of Quantum Mechanics MCQ Level - 2 - Question 5

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.

Postulates Of Quantum Mechanics MCQ Level - 2 - 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.

Detailed Solution for Postulates Of Quantum Mechanics MCQ Level - 2 - Question 6

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**

Detailed Solution for Postulates Of Quantum Mechanics MCQ Level - 2 - Question 7

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

Postulates Of Quantum Mechanics MCQ Level - 2 - Question 8

The wave function of the particle lies in which region?

Detailed Solution for Postulates Of Quantum Mechanics MCQ Level - 2 - Question 8

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.

Postulates Of Quantum Mechanics MCQ Level - 2 - Question 9

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

Detailed Solution for Postulates Of Quantum Mechanics MCQ Level - 2 - Question 9

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

Postulates Of Quantum Mechanics MCQ Level - 2 - Question 10

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

Detailed Solution for Postulates Of Quantum Mechanics MCQ Level - 2 - Question 10

** 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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