question & answer of quantum mechanics

8/14/2019 Question & Answer of Quantum Mechanics

1/31

REMEMBER THESE

1. The emisison of free electrons from the metal surface when exposed to light is called photoelectric effect.

2. Plancks quantum theory states that the emission or absorption of energy are not continuous process.

3. Einstein s photoelectric equation is given by

4. From the dual nature of radiation De Broglie proposed the wave nature of matter.

5. As per De Broglies hypothesis, a material particle of mass m moving with velocity v has a wave associated

with it

6. Wavelength of De Broglie wave is ratio of Plancks constant to momentum of particle

7. The De Broglie wavelength of a charged particle accelerated by a potential is inversely proportional to squareroot of accelerating potential.

8. The matter wave representation is symbolic only. It is a wave of probability.

9. The wavelike behavior of the material particle is observable forth microscopic particles for macroscopic bodie

Newtonian mechanics applies.

10. To represent localized particle instead of a single wave, a wave packet or wave group is used.

11. For the De Broglie waves the phase velocity or the wave velocity is more than velocity of light and group

velocity is equal to particle velocity

12. Existence of matter waves was verified by observing electron diffraction

13. For macroscopic bodies the physical variables like position and momentum can be determined

simultaneously with accuracy but such simultaneous measurements are not possible in microscopic world

14. Heisenbergs uncertainty principle states that position and momentum cannot be measured simultaneously

with accuracy.

15. The wave function is the wave variable that characteristics the matter waves.

16. is a function of x, y, z and t.

17 Condition for normalization is


2/31

18 . According to Born represents the probability density.

19. It should always be a well behaved function i it is to represent a moving particle.

20. The Schrdingers time independent wave equation is

21. Schrdingers time dependent Wave equation is

22. Schrdingers time independent equation as applied to a free particle in a rigid box gives the energy eigen

values as

23. The probability of finding the particle is different at different points in the rigid box.

24. The energy of a harmonic oscillator is quantized and is given by

25. An operator is an expression that acts on a function within some definite domain to produce new values

within that domain.

26. The values of the energy E for which we can solve the Schrdinger wave equation are called eigen values are

corresponding wave functions are called eigen wave functions.


3/31

SHORT ANSWER TYPE QUESTIONS

Q 1. Define Eigen values and Eigen functions. What do you understand by eigen values and eigen functions?

Ans.Eigen Values and Eigen functions

If there is a class of functions which when operated by an op.erator a, are marely multiplied

by some constant , i.e. if

then function f (x) are called elgen functions of operator a, and various possible values are called eigen values of

operator.

Then sin 3x is elgen function and 9 is elgen value of operator

In quantive mechanics,

The allowed eigen functions are continuous finite and single valued.

Q2 Define photo-electric effect.

Ans. The phenomenon of emission of electrons from a substance, when exposed to

electromagnetic radiations of suitable wavelength or frequency is called photo-electric effect.

Q 3. Explain the meaning of Compton shift.

Ans. Compton shift give the difference in the wave length of scattered and incident Xrays photons by the relatio

i e, (Compton shift) gives the increase in wavelength, which is independent of the incident wave length as

well as the nature of the scattering substance but depends an angle of scattering angle only Compton shift is

minimum for scattering angle of 0 and maximum for 1800.


4/31

Q 4. Distinguish between phase and group velocity.

Ans A moving particle is not equivalent to a single wave, but equivalent to wave packet A wave packet consists

of a group of waves and is formed by the superposition of a number of ways of slightly different wavelengths.

The velocity with which a definite phase (e g crest, trough etc) of the wave propagates is called phase velocity It

is denoted by

The velocity with which the wave packet of de-Broglie waves travels is known as group velocity It is denoted

by

Q5 What do you mean by matter waves.

AnsA moving particle such as an electron or proton can be described as a matter h

wave its wavelength is given by =

where P Is the momentum of that particle Matter waves propagate in the form of wave packet with group

velocity.

Q 6. In Compton scattering, the Compton shift is maximum when the angle of scattering is

(i) 900 (ii) 180 (iii) 0 (iv) 450

Ans.1800

Q 7. State de-Brogile hypothesis Or Explain de-broglie concept of matter waves.

Ans.According to De-Broglie hypothesis a material particle of mass m, moving with

velocity V has a wave associated with it. This wave t known as matter wave or De-Brogile wavelength.

h h

The wavelength of such a wave is given by

where h is Plancks constant and p is momentum of particle.

Q 8. What are orthogonal wave-functions?

Ans.The two wave-functions Ni and are said to be orthogonal if they satisfy the

conditions


5/31

Q9. What voltage must be applied to an electron microscope to produce electrons

of wavelength 0.5 A.

Q 10. What is the physical significance of wave function?

Ans.The wave-function N is assigned three basic properties:

1. It can interfere with itself i.e. it explains the phenomenon of electron diffraction.

2. It is large in magnitude where the particle or photon is likely to be located and small elsewhere.

3. The wave function describes the behaviour of a single particle or photon and r the statistical distribution of a

number of such quanta.

Q 11. What is Compton effect? OR What is Compton effect? Also give the expression for A?

Ans.When highfrequency radiations falls on matter a part of it is scattered without only change in wavelength.

This is known as classical scattering. Compton discovered that it addition to this, the secondary radiations

contains radiation of lower frequency or longer wavelength than those of incident beam. This is known as

compton effect

Q 12. What is uncertainty principle?

Ans.It states that in any simultaneous determination of the position and momentum of a particle, the product o

uncertainity is equal to or greater than plancks constant h


6/31

Q 13. Give the Born interpretation of wave-function. OR What is Borns probability interpretation of wave

function?

Ans. Max Born interpreted the relation between wave function and location of particle x by drawing an analogy

between intensity of light or photon beam and intensity of electron beam.

Consider a small element volume dV defined by co-ordinates

(x, x + dx) ; (y, y + dy) and (z,zdz)or

Then probability of finding the particle existing with in this element of volume dV is given by

For the motion of a particle in one dimension the quantity

is the probability that particle will found over small

distance dx at time t and

is the probability per unit distance.

Q 14. What is the need of quantum mechanics?

Ans. Newtons laws of motion provide information about the motion of macroscopic bodies. These laws cannot

be applied on the microscopic bodies like electrons, protons etc. from where matters are composed of. So, new

concepts were developed to deal with the motion of microscopic particles known as Quanfum Mechanics.

Q 15. What is the significanc of zero point energy?

Ans.Energy of oscillator

A comparison with the result obtained by old quantum theory shows that onlydifference is that all the equally spaced energy levels are shifted upward by an amount equal

1

to half the separation of energy levels i.e. to

, called zero point energy. Thus, it is clear


7/31

that even in the lowest state, the harmonic oscillator has energy greater than that it would have if it were at rest

in its equilibrium position. The existence of zero point energy is in agreement with experiment and is important

feature of quantum mechanics.

Q 16. What are the limitations of wave function ?

Ans.Limitation of ;

1. must be finite for all values of x, y, z.2. must be single valued function.

3. must be continuous in all regions except in those regions where the potential energy is infinite.

Q 17. What do you mean by normalisation of wave function ?

Ans.A function which satisfies the following condition is said to be normalised

LONG ANSWER TYPE QUESTIONS

Q 1. Give the significance of Compton Effect. Find expression for:(a) Compton Shift.

(b) K.E. of recoil electron.

Ans. Significance of Comptom Effect : The comptom effect gives indirect proof for

the following:

(i) The particle nature of light

(ii) The validity of relatinistic mass variation formula

(iii) The correctness of energy andmomentum expression of photon.

(iv) The laws of conservation ofenergy and momentum.

(a) Expression for Compton shift Consider an X-ray photon of frequency v and energy hv. Let it collide withan electron at rest in the target. A part of its energy hv is imparted to the electron


8/31

which gets ejected with a velocity v in a direction making an angle 0 with that of incident Xphoton.

The remaining energy is given out in form of scattered X-ray photon of lower frequen which moves in the

direction making an angle (p with that of incident photon.

From theory of relativity, mass of electron moving with velocity v is given by

.(1)

(2)

.(3)

Applying law of conservation of momentum, along x-axis

.(4)

Applying law of conservation of momentum along y-axi

(5)


9/31

(6)

(b)Kinetic Energy of Recoil electron

K.E of scattered electron

This is equal to decrease in energy of incident photon i.e.


10/31


11/31

..(4


12/31

.(5)

This is the reduced form of the wave equation which can have easy soktion V Asymptotic Solution The wave

function dies out at a large distance and becomes

finite at X=0

The asymptotic from of the equation af large y when ? is negligible in comparison with

Y2 gives

.(6)

The solution of is of the form

However the finiteness of wave function even at

large distance eliminates the possibilities of and the negative sign only of the exponential solution is

permitted at large y.


13/31

Recursion formula The general solution at any distance can be written as

where H (y) is afinite Hermite polynomial in y and can be written in a series form as

From (7), we have

..(8)

..(9)

Substituting equation (9) in equation (5), we get

..(10)

By substituting the series form H (y) into equation (10), we can obtain a recursion krmula connecting the

coefficients of the polynomial H (y).

.(11)


14/31

Here y is reduced in its power by one.

.(12)

..(13)

For a power series to be equal to zero, all its coefficients must be identically equal to zero and the summation

can be removed.

.(14)

This is called recursion formula.

Elgen values of harmonic Oscillator: The polynomials terminate at some value of h, so that the subsequent

coefficients etc. are identically equal to zero.


15/31

.(15)

The allowed integral values of n lead to certain discrete values of energy given by equation 15 known as eign

values of harmonic oscillator. It can be observed that corresponding to

, the oscillator has an energy

This is called zero point energy. The

equation 15 shows that the energy levels of harmonic oscillator are equally spaced. Significance of zero point

energy : A comparison with the result obtained

by old quantum theory shows that the only difference is that all the equally spaced energy

1

levels are upward by an amount equal to half the seperation of energy levels i.e. to

called the zero point energy. Thus it is clear that even in the lowest state, the harmonic oscillator has energy

greater than that it would have if it were at rest in its equilibrium position. The existence of zero point energy is

in agreement with experiment and is important feature of quantum mechanics.

The energy of levels of the harmonic oscillator

according to wave mechanics are represented in Fig.

Wave function of the oscillator: We can derive

the wave functions the oscillator in the different states

using recursion formula.


16/31

2/2

and the wave function has the simple form

The first excited state has n = 1

and

and a3, a5, a7 etc. = 0 as can be seen from recursion relation. Even coefficients will also be zero

and hence

The recusion relation shows that the terms in Hermites polynomials are either all odd

or all even.

Hence a., = 0 and the second excited state will be given by

The first few wave functions are shown in. a graphical manner in the following figures.


17/31

The wave functions can be found to have a finite amplitude and probability even beyond the range of oscillatorpotential well. Moreover, the wave amplitudes N2 for large n are found to increase towards the side of the

potential well similar to a classical oscillator as shown in fig. This shows that an oscillator spends maximum time

at turning points at the extreme points of displacement. The dotted curve in fig. (d) gives the classical probability

distribution which is proportional to the reciprocal of velocity. For large n, there will be a correspondence

between classical and quantum behaviour as the probabilities in both the cases tend to be equal for large n

values as shown in Fig. above.

Q 3. Prove Heisenbergs uncertainty principle

Ans.According to Heisenbergs uncertainty principle we cannot measure the exact position and momentum of

particle simultaneously.

Now group velocity of a wave packet is given by


18/31

.(1)

..(2)

Here At is the time required for one complete wave packet to cross the reference time. Now frequency Al) is

related to At by relation

Q 4. If the energy of the particle is zero, then prove using quantum mechanics that it cannot exist in a box.

Ans.According to Schroedinger eq.

Consider a particle trapped in box of length L. Box is having infinite hard walls e. at

walls

Inside the box V = 0

Its solution is

According to boundary conditions


19/31

Boundary condition at x = L gives

i.e. Wave function is zero which means particle is not present inside the box.

So E = 0 is not possible.

Q5. It T is the relativistic kinetic energy of a particle of rest mass , then show

that .


20/31

Q6. Discuss harmonic oscillator in quantum mechanism. Define energy eigen values for it. Does it explain the

tunnelling phenomona for a particle in a box?

Ans. Schrodinger one-dimensional time independent equation is


21/31

gives eigen values. Harmonic oscillator explains the tunneling phenomena for a particle in a box.

Q7. A particle is constrained to move on the X-axis between X = 0 and X = a. Write down the Schrodinger wave

equation for the particle. Solve it to obtain the normalised wave function and energy eigen values for the

particle OR Write Schrodinger wave function for a particle in a box and solve it to obtain the eigen values and

eigen functions.

Ans.Consider a free particle of mass m confined to a one-dimensional box:


22/31


23/31


24/31

Q 9. The energy of a linear harmonic oscillator in its third excited state is 0.1 eV.

Calculate the frequency of vibration.

Ans.

Q10. Discuss compton scattering and derive a relation for change in wavelength of scattered photon.

Ans.When high frequency radiations fall on matter a part is scattered without any change in wavelength. This is

known as classical scattering. Compton discovered that in addition to the classical scattering in which the

wavelength of the radiations remains unchanged the secondary radiations contains radiations of lower frequenc

or longer wavelength than those of the incident beam. This is known as compton effect.

Applying principle of conservation of energy, we have


25/31

Applying principle of conservation of momentum, we have

(2)

in the direction of incident photon Le. along X-axis and


26/31

Q 11. What is the need of quantum mechanics? Discuss Borns interpretation and normalization of wave-function. At a certain time, the normalized wave-function of a particle moving along x-axis has the form given

by

Ans. Dynamics of macroscopic bodies can be explained on the basis of Newtons laws of classical physics. But

experimental results related to microscopic bodies cannot be explained on the basis of Newtons law or classical

physics. So. quantum mechanics was brought forward.My Rnrn interoreted the relation between the wave-function and the location of the particle x by drawing an

analogy between intensity of light or nion uam zrci r1; 1.

electron beam.

The process of integration over all possible locations to give unity is called normalisation.


27/31

Q 12. For a particle in one-dimensional box, show the value of uncertainty product


28/31


29/31

Q13. State the various conditions for acceptability of a wave function and

normalize the following wave function

Ans.Conditions for acceptability of a wave-function

(i) It must be finite for eiery value of x, y, z

(ii) It must be single valued for each value of x, y, z.(iii) It must be continuous.

d d d

(iv) The first order derivative of wave function i.e.

should be continuous. Given wave function is similar to the wave-function for a particle in a box

According to Normalization condition


30/31


31/31

question & answer of quantum mechanics

Documents