Unit 1 Laser

8/11/2019 Unit 1 Laser

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Lecture Notes on Wave Mechanics & X-Ray Diffraction - By Deepika Gupta, Lecturer-SIET Page 1

UNIT - 1

Review of Elementary Quantum Mechanics

Plancks Quantum TheoryAt the end of 17thcentury, Newton proposed Corpuscular theory of light. According to this theory, lighconsists of minute fast moving elastic particles known as corpuscles. The phenomenon of interferenc

diffraction, polarization etc. could not be explained on the basis of this theory. To explain thesphenomenons Huygen proposed the wave theory of light. According to wave theory, light travels in thform of waves in hypothetical medium ether. It successfully explained the interference, diffraction etc. was followed by Maxwells electromagnetic theory. There are certain other phenomenon likphotoelectric effect, compton effect, zeeman effect etc. which could not be explained by these theories. gave birth to the quantum theory of light.

According to Plancks hypothesis,1. Matter consists of large no. of oscillators vibrating in all possible directions.

2. Oscillator of frequency can exists in states whose energy E is an integral multiple of constant h an

frequency of oscillator.

E = nh, n= 1, 2, 3,

3. Oscillator emits energy only when it passes from higher energy level to lower energy level.

Plancks radiation law

This law can be expressed in terms of wavelength as:

Wiens FormulaFor small temperatures,T is small. So, exp(hc/kT)>>1 and 1 can be neglected.

Rayleigh-Jeans law


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For large temperaturesT is large.

PhotonsAccording to quantum theory, light is not continuous but it is discrete and consists of small bundles orpackets of energycalledphotons.

Properties of photon1. A photon always travels with the velocity of light.

2. Photons are electrically neutral. They cannot be deflected by electric or magnetic fields. They do not

ionize.

3. Rest mass of photon is zero.

4. Energy: E = h= hc/ .

5. Mass: E = mc2, m = E/c2= hc/c2 m = h/c

6. Momentum: p = mc = h/

Photoelectric Effect

The emission of electron by metal surface when illuminated by light or any other radiation of suitable wavelengthor frequency is calledphotoelectric effect.

Einsteins photoelectric equationAccording to Einsteins explanation, in photoelectric effect one photon is completely absorbed by oneelectron, which thereby gains the quantum of energy and may be emitted from metal. The energy pfphoton is used in two parts:

1. A part of its energy is used to free the electron from the atom and away from the metal surface. This

energy is known as photoelectric work function.

2. Other part of energy is used to gain the kinetic energy to the electron.

This equation is known as Einsteins photoelectric equation.


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radiation of lower frequency is called modified radiation. This phenomenon is known as ComptoEffect.

According to Compton, the phenomenon of scattering is due to an elastic collision between two particles, thphoton of incident radiation and the electron of scatterer. When the photon of energy h collides with thelectron of the scatterer at rest, it transfers some energy to the electron; the scattered photon witherefore have a smaller energy than that of incident photon. The observed change in frequency owavelength of the scattered radiation is known as Compton Effect.

Applying the law of conservation of momentum along and perpendicular to the direction of incidenphoton on the system. We haveMomentum before collision = Momentum after collision

'0 cos cos .........................(1)

'0 sin sin ..................................(2)

h hp

c c

hp

c

+ = +

=

Kinetic energy gained by the electron( ')K h h =

Multiply these equations by c and rearranging them. We get,cos 'cos ..........................(3)

sin 'sin ..................................(4)

pc h h

pc h

=

=

Squaring and adding these equations. We get,2 2 2 2( ) 2 'cos ( ') ..........................(5)p c h h h h = +

2 2 2 4

0

2

0

E p c m c

E K m c

= +

= +

2 2 2 4 2 2 2 40 0 0

2 2 2 2

0

2

( ') 2 ( ')

p c m c K m c K m c

p c h h m c h h

+ = + +

= +

2 2 2 2 2

0( ) 2 ' ( ') 2 ( ')........................(6)p c h h h h m c h h = + +

Comparing equation (5) and (6). We get,

2

02 ' 2 ( ') 2 'cosh h m c h h h h + =

Dividing this equation by 2 2h c

Scattering angle

Recoil e-

Scattered photon

Incident photon

Energy = h

Energy = h

Momentum p = h/c

Momentum p = h/c

Unmodified radiation


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0 ' ' (1 cos )m c

h c c c c

=

0 0

' (1 cos ) (1 cos )h h

m c m c = = (1 cos )c =

is called Compton Shift and c is called Compton wavelength.

At 0180= Compton shift will be maximum and its value is 0.0484 .

So, the change in wavelength due to scattering is called Compton Effect.

Direction of Recoil ElectronOn dividing eqn (4) by (3). We get,

'sin 'sintan

'cos 'cos

h

h h

= =

2

0

'(1 cos )

h

c c m c

=

2

0

1 1(1 cos )

'

h

m c

=

2

2

01 (2sin )21

'

h

m c

+

=

2

2

0

'

1 (2sin )2

h

m c

=

+

2

2

2

sin / 1 .2 sinsin2

tan

1 2 sin cos2

cos1 .2 sin2

+

= =

+

+

2

0

.................................h

m c

=

Q

( ) 2 2 2

2sin cossin 2 2tan

1 cos 2 sin 2sin 2 sin2 2 2

= =

+ +

cos cot2 2tan

(1 )sin (1 )

2

= =+

+

2

0

cot2tan

(1 )h

m c

=

+

Kinetic Energy of Recoil ElectronKinetic energy of the electron is given by

( ')K h h =


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2

2

0

'

1 (2sin )2

h

m c

=

+

Putting the value of ' . We get,

2

.

1 2 sin2

K E h h

=

+

2

2

2 sin 2.

1 2 sin2

K E h

= +

This shows the K.E of recoiled electron depends upon scattering angle.

Kinetic energy will be maximum, when

sin 12

=

2 2

= =

So, the maximum kinetic energy of recoil electron

2

0

2. ,1 2

hK E h m c

= =+

2

0

max

2

0

2

.2

1

hh

m cK E

h

m c

=

+

2 2

max

2

0 2

0

2.

21

hK E

hm c

m c

=

+

Wave-Particle duality of MatterLight when considered as a wave based on interference, diffraction and polarization prove the wavnature of radiations because they require two waves at the same position at the same time. On the othehand, it is impossible for two particles to occupy the same position at the same time. But for thexperiments like photoelectric effect or Compton Effect radiation is considered as particle and interacwith matter. So, there are two contradictory aspects:

A wave spreads out and occupies a relatively large space.

A particle occupies a definite position in space and hence occupies small space.If one has to explain the results of the experiments performed with radiation then it becomes necessar

to accept this contradictory nature of light.

De-Broglie Matter WavesIn 1924, De-Broglie said that matter has dual characteristic just like radiation. His concept was based ohis observations:

The whole universe is composed of matter and electromagnetic radiations. Since both are formof energy so can be transformed into each other.

The nature loves symmetry. As the radiation has dual nature, matter should also posses ducharacter.


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The waves associated with moving particles are matter waves or de-Broglie waves.

Wavelength of de-Broglie Matter wavesConsider a photon whose energy is given by

E = h= hc/.(1)If a photon posses mass, then according to the theory of relativity, its energy is given by

E = mc2.(2)From (1) & (2)

Mass of photon m = h/c

Therefore, Momentum of photonp = mc = (h/c).c = h/

If instead of a photon, we consider a material particle of mass m moving with velocity v, then thmomentum of the particle, p = mv. Therefore, the wavelength of the wave associated with the movinparticle is given by,

= h/mvThis is called de-Broglie wavelength.

Different Expressions:a) If E is the kinetic energy of a material, then

E = mv2/2 = P2/2m

Or P = 2mE

Therefore, the De-Broglie wavelength

2

h

mE=

b) When a charge particle carrying a charge q is accelerated by potential difference V volts, then ikinetic energy is

E qV=


2hmqV

=

c) When a material particle is in thermal equilibrium at a temperature T, then3

2E kT=

Where k is Boltzmanns constant = 231.38 10 / J K Therefore, the De-Broglie wavelength

3 32 ( )

2

h h

mkTm kT

= =

d) If the velocity of the particle is comparable with the velocity of light, then the mass of the particlis given by

0

2

21

mm

v

c

=

Where0

m is the rest mass of the particle.



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2

2

0

1v

hh c

mv m v

= =

e) De-Broglie wavelength associated with electrons:

Let us consider the case of an electron of rest mass 0m and charge e which is accelerated by

potential V volt from rest to velocity v, then

21

2

mv eV = or0

2eVv

m

=

If the relativistic variation of mass with velocity of the electron is ignored, then

0m m and

0

h

m v=

So, 0

0 02 2

h m h

m eV eVm= =

Or,34

10 31

6.625 10

2 1.632 10 9.1 10V

=

012.26AV

=

If V=100 volt, then01.226A=

Therefore, the wavelength associated with an electron accelerated to 100 volt is 1.226 .

Properties of Matter Waves

Lighter is the particle, greater is the wavelength associated with it.

Smaller is the velocity of particle, greater is the wavelength associated with it.

When 0v= then = , i. e., wave becomes indeterminate and if v= then 0= . This shows thmatter waves are generated by the motion of the particles. These waves are produced whethe

the particle are charged or uncharged. This implies that these waves are not electromagnetic buthey are a new kind of waves.

The velocity of matter waves depends upon the velocity of matter particle.

The velocity of matter waves is greater than the velocity of light..The energy of a wave of frequency is given by E h=

The energy of a particle of mass m is given by 2E mc=

2h mc= or2mc

h=

The wave velocity is =

So,

h

mv

=

, where v is the velocity of particle.

So,2c

v=

As particle velocity v cannot exceed c, is greater than velocity of light. Therefore, the velociof matter waves always greater than c. This shows that matter waves are not physical waves.

Wave Velocity & Group Velocity


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Whenever we talk about a particle, we think of a point object which occupies a definite position in spacand which has definite momentum but in quantum mechanics particle can be described by a wavpacket.A wave packet comprises a group of waves slightly differing in velocity and wavelength, with phases anamplitude such that they interfere constructively over a small region of space where the particle can be located anoutside this space they interfere destructively so that the amplitude reduces to zero.

The velocity of component waves of a wave packet is called asphase velocityof those waves.

The velocity, with which the wave packet travels, is calledgroup velocity.

Phase velocity of de-Broglie wavesAccording to de-Broglie, a particle of mass mmoving with velocity vis associated with a wave whoswavelength is given by

h

mv=

The propagation constant k of the wave is given by2 2 ( )mv

kh

= =

Let E be the energy of the particle, E h= E

h=

Therefore, angular frequency 2 2 /E h = = 22 mc

h

= ..( 2E mc=Q )

De-Broglie phase or wave velocity pv is given by

2 22

2 ( )p

mc h cv

k h mv v

= = =

As v is less than c, de-Broglie wave velocity must be greater than c. so, the de-Broglie wave traiassociated with the particle would travel much faster than the particle itself and would leave the particfar behind. This statement is nothing but collapse of the wave description of the particle.

Group velocity of de-Broglie waves

The angular frequency and propagation constant of de-Broglie wave associated with a particle of remass mocan be calculated as..According to theory of relativity,

0

2

21

mm

v

c

=

We know that

2

2 mch

= and 2 2 ( )mvkh

= =

So, on substituting the value of m2

0

2

2

2

1

m c

vh

c

=

and 02

2

2

1

m vk

vh

c

=

Differentiating these eqns. We have


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0

3/22

2

2

1

m vd

dv vh

c

=

and 03/2

2

2

2

1

mdk

dv vh

c

=

The group velocity gv of de-Broglie waves associated with the particle is given by

/

/g

d d dvv

dk dk dv

= =

Substituting the values of ( /d dv ) and ( /dk dv ), we get

gv v= Thus, de-Broglie wave group associated with moving particle travels with same velocity as thparticle.

Since,2

p

cv

v=

Therefore, wave velocity and group velocity of de-Broglie wave is related as2.pv v c=

Or2.p gv v c=

Davisson & Germers electron diffraction experimentDe-Broglie said that material particles have wave like character, & they are expected to show thphenomenon like interference and diffraction. In 1927, Davisson and Germer prove the existence of deBroglie waves. In his experiment electrons are produced by thermionic emission from a tungstefilament mounted in an electron gun. The ejected electrons are accelerated towards anode in an electrfield of known potential difference and collimated into a narrow beam. The narrow beam of electrons iallowed to fall on the surface of a nickel crystal. Nickel is used as target because its atoms are arrangein regular manner so the surface of crystal acts as a diffraction grating. The electrons scattered from thtarget are collected by a Faraday cylinder called electron detector which is connected to a sensitivgalvanometer and can be moved along a circular scale.

For different values of potential the electrons were collected at different positions. The current which the measure of the intensity of the diffracted beam, is plotted against the diffraction angle for eacaccelerating potential.

VH.T.

NickelTar et

Electrondetector

G

Scale

ElectronGun


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It is observed that at the voltage of 40 volts, a smooth curve is obtained and a bump begins to appear ithe curve for 44 volts electrons. As the potential difference is further increased the bump start shiftinupward and becomes most prominent for 54 volts electrons at = 500. Beyond 54 volts the bumgradually diminishes and become insignificant at 68 volts electrons. The peak at 54 volts and 5provided the evidence that electron were diffracted and verifies the existence of electron wave.According to de-Broglie, the wavelength associated with electron accelerated through a potential V given by

0 0 012.26 12.261.67

54A A A

V= =

From X-ray analysis, it is known that nickel crystal acts as plane diffraction grating with grating space = 0.91 . According to experiment we have diffracted electron beam at = 500. The corresponding angof incidence relative to the family of Bragg plane

0180 50 652

= =

Using Braggs equation (taking n = 1). We get0

02 sin 2(0.91)sin 65 1.65d A = = = .

This is in good agreement with the wavelengtcomputed from de-Broglie hypothesis. Hencconfirms the de-Broglie concept of matte

waves.

Heisenbergs Uncertainty PrincipleIn 1927, Heisenberg proposed a very interesting principle, which is a direct consequence of the dunature of matter, known as uncertainty principle or to determine the exact position and momentumsimultaneously Heisenberg proposed the principle of uncertainty.It is impossible to measure precisely and simultaneously both the members of pairs of certain canonicalconjugate variables that describe the behaviour of an atomic system.Canonically conjugate pairs of physical quantities are those in which measurement of one quantitaffects the capacity to measure the other.The product of uncertainties in determining the position and momentum of a particle at the same instant is of thorder of h .

Uncertainty in momentum xp , and in position x

.2

x

hp x

h

Uncertainty in angular momentum and angular position

650 500

54 V

d = 0.91


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.2

hJ

h

Uncertainty in kinetic energy and time

.2

hT t

h

The exact principle is stated as The product of uncertainties in determining the position and momentum of th

particle can never be smaller than the number of the order2

h.

Physical Significance of Heisenbergs Uncertainty Principle

This principle explains why it is possible for radiation and matter to have dual nature.

It helps in understanding many phenomenon like absence of electrons within nucleus, existencof protons & neutrons in nucleus, binding energy of an electron in atom etc.

It also states that we can only predict the probable behaviour of quantum mechanical systems not the exact behaviour.

Applications of Heisenberg Uncertainty Principle

Non-Existence of electrons in the nucleus

Radius of Bohrs First orbit Binding energy of an electron in an atom

Zero point energy of harmonic oscillator

Wave FunctionAccording to uncertainty principle, the position and momentum of a particle cannot be measureaccurately at the same time; the measurement of one quantity introduces an uncertainty into the other.In case of electromagnetic waves, the electric and magnetic fields varies periodically. In water waveheight of the water surface varies periodically. Similarly in sound waves, pressure varies periodicallySo, what varies in matter waves? Answer is ..Wave function ().

Schrodinger describes the amplitude of matter waves by a complex quantity

(x, y, z, t) known as wavfunction.

Physical significance of wave function

The value of wave function associated with a moving particle at a particular point (x, y, z) ispace at the time t is related to the possibility of finding the particle there at that time.

The physical significance of wave function is that the square of its absolute value2

at a point

proportional to the probability of finding the particle described by the wave function in a smaelement of volume d ( dxdydz ) at that point.

2

gives the probability of finding the particle at that point at any given moment.2

is calle

probability density and is called probability amplitude. Particle is necessarily somewhere i

space, therefore

21d

=

A wave function satisfying this relation called normalized wave function.

Nature of wave function

It must be finite everywhere.


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It must be single valued.

It must be continuous.

Schrodinger time independent wave equation (Method 1)According to de-Broglie theory, a particle of mass m always associated with a wave whose wavelengtis /h mv= . If particle is a wave it is expected that there should be some sort of wave equation whicdescribes the behaviour of the particle. Consider a system of stationary waves associated with a particlLet x, y, z be the co-ordinates of particle and is the wave displacement at any time t. is finite, sing

valued and periodic function. The classical differential equation of wave motion is expressed as2 2 2 2

2

2 2 2 2 .......................................(1)v

dt dx dy dz

= + +

2 2 2 22 2 2

2 2 2 2................v

dt dx dy dz

= = + +

Q .( 2 laplacian =Q operator)

Solution of eqn (1) is given by

0 0sin sin 2t t = = (2)

0(2 ) cos 2

dt t

dt

= (3)

2 2 22 2 2

02 2

4(2 ) sin 2 4

d vt t

dt

= = = ..(4)

Putting value of2

2

d

dt

in equation (1)

2 22 2

2

4 vv

=

22

2

40

+ =

( )2

2 2 2

2

40.......... / m v h mv

h

+ = =Q

If E and V be the total and potential energies of particle, then its kinetic energy

21

2mv E V =

In this case it should be remembered that potential energy is independent of time.2 2 2 ( )m v m E V =

Therefore,2

2

2

8( ) 0m E V

h

+ =

2

2

2( ) 0

mE V + =

h ( ).......... 2h =Q h

This is Schrodinger time independent wave equation.For a free particle, V=0

2

2

20

mE + =

h

Method 2The differential equation for a free particle in one dimension is given by


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2 2

2 2 2

1

x v t

=

---------------------------------(1)

Standard solution of this equation is

0

0

22

0 02 ( )( )

i t

i t

i t i t

e

di e

dt

di i e e

dt

=

=

= =

Putting the value of2

2

d

dt

in eqn (1). We get,

2 2

02 2

i tde

dx v

=

2 22 2

0 00 02 2 2 2

i t i t d de e

dx v dx v

= =

2 2 2

00 02 2 2

(2 ) 4...........( 2 , / )

dv

dx v

= = = =Q

2 2 2 2

002 2

4

...........( / )

d m v

h mvdx h

= =Q

If E and V be the total and potential energies of particle, then its kinetic energy

21

2mv E V =

In this case it should be remembered that potential energy is independent of time.2 2 2 ( )m v m E V =

So,2 2 2

00 02 2 2

8 2( ) ( )

d m mE V E V

dx h

= =

h

Or2

002 2

2( ) 0

d mE V

dx

+ =

h

In 3-D,

2

0 02

2( ) 0

mE V + =

h

Schrodinger time dependent wave equationThe differential equation for a free particle in one dimension is given by

2 2

2 2 2

1

x v t

=


2 ( )( , )

xi tvr t Ae

=

2 ( )

.......................( , )E x

i th

EAe v

h

= = =Q

2 ( )

.....................( )E px

i th h

hAe

p

= =Q

2( ) ( )

( , ) ............(1)i

i Et px Et pxhr t Ae Ae

= = h

This represents a particle moving along the x direction having total energy E and momentum p.


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Differentiating (1) wrt t.2

( )

....................(2)

i Et pxh

i iEAe E

t

E ii t t

= =

= =

h h

hh

Differentiating (1) wrt x twice2 2 22 2 2

( ) ( )2

2 2 2( )

i Et px i Et pxh h

i p pp Ae Ae

x

= = = h h h

22 2

2...........................(3)p

x

=

h

Let E and U be the total and potential energies of the particle.So, E = K + U

=2

2

pU

m+

Multiply on both sides2

2

pE U

m

= +

Substituing the values of E and 2p . We get,

2 2

22i U

t m x

= +

hh

This is time dependent Schrodinger equation in one dimension.

In 3D, time dependent Schrodinger equation is

22

2

i U

t m

= +

hh

2 2 22

2 2 2........................( )

x y z

= + +

Q

Schrodinger time independent equation from time dependent equationThe differential equation for a free particle in one dimension is given by

2 2

2 2 2

1

x v t

=


( )

0 0..................( )

i i ipx i ipxEt px Et Et

Ae Ae e e Ae

= = = =h h h h hQ

Differentiating wrt t.We get,

0

iEtd i

E edt

= h

h

Differentiating wrt x twice. We get,22

0

2 2

iEtdd

edx dx

= h

The time dependent Schrodinger eqn is,


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2 2

22i U

t m x

= +

hh

Putting the values of ,d

dt

and

2

2

d

dx

. We get

2

002 2

2( ) 0

d mE U

dx

+ =

h

The values of energy for which the time independent Schrodinger equation can be solved are calleEigen valuesand the corresponding function is called Eigen function.

Particle in one dimension box with infinitely hard wallsConsider a particle of mass m moving along x-axis between the two rigid wall A & B at x=0 & x=LParticle is free to move between the walls. The potential energy of the particle between the two walls constant because no force is acting on the particle. The constant potential energy is taken to be zero fosimplicity. The particle does not loose energy when it strikes back and forth in the potential webecause the walls are infinitely rigid.

V(x) = for x 0 and x LV(x) =0 for 0


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nK

L

=

2 22

2

nK

L

=

2 2 2 2 2 2 2

2 2 2 2 2

2

2 4 8n

mE n n h n hE

L mL mL

= = =

h

This relation shows that energy is quantized. (n

E 2n )

The Eigen function is

( ) sin sinn

x A Kx A xL

= =

To find the value of A, we can apply the normalizing condition

2

( ) 1x dx

=

Or 2 2

0

sin 1L

nA xdx

L

=

2

0

1 21 cos 1

2

Ln

A x dxL

=

2

0

2sin 1

2 2

LA L n x

xn L

=

2

12

A L=

or

2 2A

L=

or

2A

L=

2( ) sin

nx x

L L

=

Expectation Value

It is defined as the average result of a large number of measurements taken on an independent system.

Let f(r) be any function for an observable quantity associated with a moving particle. The expectationvalue of f(r) is

2( ) (r)dxdydzf r f< >=

The expectation value of a quantity in terms of position coordinates is

n=1

n=2

n=3

n=4

E1=h2/8mL2

E2=4h2/8mL2=4E1

E3=9h2/8mL2=9E1

E4=16h2/8mL2=16E1


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2

2

2

xdx

ydy

zdz

x

y

z

< >=

< >=

< >=

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Unit 1 Laser