REVIEW OF QUANTUM MECHANICS

IMRANA ASHRAF ZAHIDDEPARTMENT OF PHYSICSQUAID-I-AZAM UNIVERSITY

ISLAMABAD, PAKISTAN

Layout• Quantum Mechanics: Revisited• Radiative Processes for isolated atoms• Transition rates: Semi-classical• Line-broadening Mechanisms–Homogeneous broadening – Inhomogeneous broadening

Quantum Mechanics: Revisited

Wave MechanicsQuantum mechanical systems ( such as atoms,

molecules, ions etc.) are given by wave function ψ(r, t).

Itself ψ(r, t) has no physical meaning- it allows to calculate the expectation values of all observables of interest.

Measurable quantities are called observable and are represented by hermition operator Ô. Expectation values is given by

.,o,o 3 trtrrd

ProbabilityThe probability of finding the system in volume

element d3r is.

If system exist, its probability of being somewhere has to equal 1.

.,, trHtrt

i

H- the Hamiltonian of the system. The energy of the unperturbed system- an atom not interacting with light is

rVmPH 2

2

The time development of wave function is determined by Schrödinger equation,

Stationary StatesStationary states of Schrödinger equation are those for which space and time dependence are separated

Time independent equation,

Eigen functions having same eigen values are normal

Completeness

The wave function

n

tinn

nn

nerUtCtrtr ,,

- Expansion coefficients nC t-constant for problems related with free Hamiltonian

-time dependent for interaction Hamiltonian

Gives the probability of finding the system in state n.

,

ˆ ˆn mi tnmn m

n m

O e OC C

3 *mˆr oUnm nO d rU r

Where

DIRAC’S NOTATION

• The wave function of wave mechanics corresponds to the state vector of the Dirac’s formulation of the quantum mechanics.

• The relation between state vector and wave function is analogous to using vectors instead of coordinates.

In Dirac Notation

yVxVV yx

A vector V can be expanded as,

yVxVV yx

X-component of a vector is obtained by

xVxV

.In Dirac Notation

yx VVyVVx and

Using these equations we can write

VyyVxxV

1x x y y

For n dimensions

VnnVn

x x y y V

n

n n I

The expectation value of the operator ^O

ˆ ˆO t O t

Hermitian

The set of eigen vectors of a hermitian operator is complete. This means that any arbitrary vector can be expressed as a sum of orthogonal eigen vectors.

nn

n xC

0

Eigen vectors are orthonormal

mnfor 0 mnfor 1

nm

nmmn XX

Completeness relation for discrete case is

n

nn IXX

State vectors obey the Schrödinger's equation

n

tin neC

Hi

n

,

Expectation value can be written as

Two-level Atomic SystemWave function

State vector

a bi t i ta bC e a C e b

Radiative Processes for Isolated Atoms

Spontaneous Emission• Atom is in state of given material.• E2 >E1 – atom will tend to decay to state • The corresponding energy difference (E2-E1) is released

by the emission of a photon.

hEE )( 12

1

2

1.Radiative: Spontaneous emission 2.Non-radiative:

12 EE

Is delivered in some other form than electromagnetic radiation e.g. it may go into kinetic energy of the surrounding molecules.

Two possible ways for atom to decay

Energy difference

22 N

dtdN

sp

22 AN

dtdN

The rate of decay of these atoms are

Let N2 number of atoms in level 2 per unit volume

Asp1

AteNN 02

atom.another by emitted that toipsrelationsh phase definite no has waveRadiative

possible. isdirection Any

.population initial theis N0

A- Transition probability for spontaneous emission or Einstein’s A coefficient.

STIMULATED EMISSION

hEE 12incident on it.Finite probability of emission of another in phase photon.Rate of transition

• Atom is initially in level 2• An electromagnetic wave of frequency υ given by

W21- stimulated transition probability or rate- (time)-1

W21- depends on intensity of electromagnetic wave.

• Atom is in level 1• Electromagnetic wave of frequency υ incident on the

material.• Finite probability that atom will be raised to level 2.• Energy of incident wave is absorbed by the atom

Process is called the stimulated absorption.

STIMULATED ABSORPTION:

N1 -number of atoms at a given time lying in level 1

112 1

. .st ab

dN W Ndt

Stimulated Transition Rate

Transition Rates

TRANSITION RATES OF STIMULATED EMISSION AND ABSORPTION

• Semi-classical Theory• Atom is quantized• Field is treated classically (using Maxwell’s equations)

tiEtiE

erutaerutatr21

2211,

System: Two-Level Atomic System

u1(r) & u2(r) - eigen -function of unperturbed Hamiltonian(H0).satisfies time-independent equation.

iii uEuH 0a1(t) - probability amplitudes.

2

ia t - probability of finding atoms in state i.

2112

0

EE

- Transition frequency0

• A monochromatic electromagnetic wave incident on atom.• Atom acquire an additional energy H’ during interaction Total Hamiltonian

HHH 0

TIME EVOLUTION OF THE SYSTEMSchrodinger equation

trHttri ,,

Putting value of ψ(r,t)

tiEtiE

tiEtiEtiEtiE

erutaerutaH

erutaerutaHerutat

ierutat

i

21

2121

2211

221102211

Multiplying with ru1 and integrating over whole space.

Using

1311 rdruru

3 30i i i i i iu r H u r d r E u r u r d r E

03212

3201 rdruruErdruHru

0321 rdruru

113

11 HrdruHru

123

21 HrdruHru

tieHtaHtai

ta 01221111

1

22212120

1 HtaetaHi

ta ti

INTERACTION HAMILTONIAN:

NLMDEQEDI HHHHH EDH -dominates

-due to the Interaction of the electric dipole moment of the atom with the electric field of the electromagnetic wave called- Electric-dipole interaction.

H

Similarly

DIPOLE APPROXIMATIONr = vector indicating the electron’s position with respect to

the nucleus

ELECTRIC FIELD IN TERMS OF PLANE WAVE

Distance of electron from the nucleus is given by Bohr radius.

011

0 5.0105 Ama

<< Optical wavelength

erp

The spatial variation of the electric field across the dimensions of the atom is very small - neglecting it for long wave-lengths - is called dipole approximation.

tE cos2 0

tE cos0

tEerH cos2 0

rdruHruH 32112

rtdErureru 3021 cos

tEp cos2 012

rdrerurup 32112

Interaction Hamiltonian

Using this

where

is the matrix element of electric dipole operator.

tEpH cos2 01111

031111 rdrerurup

ru 1 ru1

11 22 0H H

Similarly

A level does not have a dipole moment.er- has odd parity

and - has even parity

Electric dipole transition only occurs between states of opposite parity.

taeHi

ta ti2121

01

taeHi

ta ti1212

01

Two differential equations- can be solved by using initial conditions.

Let at 0t , atom is in level 1

101 a 002 a

Assume transition probability is weak--- perturbation analysis can be used.

Using these we get

(0) (2) (4)1 1 1 1a t a a a

(1) (3) (5)2 2 2 2a t a a a

1)0(1 a

As atom is initially in level 1

According to Perturbation theory

First order corresponds to the probability that atom go to level 2 from level 1.

dtaeeEp

idtta titi

tt)0(

10

021

02

00

22

Putting value of )0(1a

00021

)1(2

111 00

ie

ieEp

ia

titi

ω ~ field frequency and ω0 ~ atomic frequency

At resonance

0

0 is rapidly oscillating term. Neglecting tie 0

This is called “Rotating wave approximation (RWA)”.

ta )2(1

ta )1(2

The second order of is obtained by substituting the

21 12 01a t p E function of time

ta2 ta )3(2Then putting this in , we get the third order of

ta1

ta2

= series is even powers of E0

= series is odd powers of E0

First order is enough for transition rates.

value of

2

2

20212

22sin4

tEpta

!32

22sin

3ttt

!32

22sin

32 ttt

Expand

Probability of finding the atom in state 2 is

As t increases, the maximum in the curve moves upwards proportional to t2 and zeros of the function move in along the horizontal axis towards the origin.

For limit 0 t

22sin 24

t t

tt tt 0 tt tt

2

0

02sin

t

0

22sin

2

tt can be replaced by Dirac delta function

For the area under the curve equal to unity it can be replaced by Dirac delta function with properties

Using

02

20212

2

2

t

Epta

This is for a single atom.

Electromagnetic wave interact with an ensemble of atoms with randomly oriented dipole moment with respect to field.If θ is the angle between p and E0,

2221

20

2021 cospEEp

Take average over all the random orientations of dipole moment.

If all angles θ are equally probable, then

2 22 221 0 0 21 cosp E E p

d

d

0

0

2

2

sin

sincoscos

2cossin0

0

d

?sincos0

2

d

xcos dxd sinPutting

32

3sincos

1

1

31

1

2

0

2

xxd dx

31

23

2cos2

2 2 221 0 21 0

13

p E p E

002

212

20 3

2 tEpta

200

2 En

0

2

221212 3

4n

pW

The energy density of electromagnetic wave is

0 0

12W

At

0 00

12 0W This is physical unacceptable result

Reason:

• We have assumed that the interaction between the electromagnetic wave and system could continue coherently for an infinite time.

• There are of number of phenomena that prevent the interaction of atom with electromagnetic wave for long time. For example collision, Spontaneous emission.

• Above equations are valid only in time interval between one collision and the next.

• After each collision the relative phase between the atom’s wave-function and electric field of the wave under go a random jump.

• The problem is to find the interaction of atom with a broadband field

is valid provided the Dirac delta function - an infinitely sharp function centered at

0 and of unit area such that

10 d

0 tg

0

10 dgt

again centered at , again of unit area.

but with a finite spectral width.

12W

Is replaced by a new function

The shape of function and value of its line width depends Upon the particular broadening mechanism involved.

tgpW 2

21212 34

Stimulated emission rate is obtained by changing initial conditions.

1 0 0a 102 aand 1221 ppppp 2112 As

2112 WWW

21W

Line-broadening Mechanisms

•Line- Broadening Mechanisms

Broadening• The term is used to denote the finite

spectral width of the response of the atomic systems to the electromagnetic fields.

Two Types of Line Broadening 1. Homogeneous Broadening 2. Inhomogeneous Broadening

Homogeneous Broadening• Broadening mechanism is homogeneous

when it broadens the line of each individual atom and therefore of the whole system in the same way.

• Atoms are indistinguishable• All atoms have same transition frequency

and same energy spectrum• Examples: Collision and Natural

Broadening

Inhomogeneous broadening• A Broadening mechanism is said to be

inhomogeneous when it broadens the atomic lines by different amount for different atoms.

• In this case different atoms in an ensemble has different transition frequency and frequencies are distributed over a range.

• Example: Doppler Broadening

Collision BroadeningBroadening in Gases

Due to collision of an atom with other atoms, ions, free electrons or the walls of the container.

Broadening in SolidsDue to the interaction of with the phonons of the lattice

• It leads to the change of relative phase between atomic dipole moment and that of a incident wave

• Collision interrupt the process of coherent interaction between the atom and the incident wave

• Atom no longer sees a monochromatic wave instead a broadband field

Collision Broadening cont’dHow to deal with this?• Add all the frequencies of jump during

collision.• Use Fourier theory to handle multiple

frequencies.• Assume no collision for time interval T2 –

mean free path

dpn

dW

d

)(3

4 and between intervalfrequency in the wave theofdensity energy theBe

ddLet

02

120

2212

radiation. theofspectrumfrequency entire over the ngintergrati

by obtained isy probabilitn transitiooverall The

2/

2

We assume the distribution of the values of can be described by a probability density

1P

P is the probability that the time interval between two successive collisions lies between

and

TeT

d

2

c 20

d . T is the mean free path in which there is no collision.

Pd T

)/exp(P)P(

get we, timeaafter occurscollision ay that probabilit thebe )P( If

cd

0

212 212 2

0

The fieldE(r,t) 2E cos( ) is a monochromatic field wherecos( ) gives the phase or the mono chromalicity of the field and this field leads to unphysical result

4W ( )3

tt

pn

FOURIER THEORY:This method is used to handle the multiple frequencyThe Fourier theory allows the representation of a function in terms of its frequency or temporal characteristics and one can easily

-

-

move between the two representation

F(t) F( )e

1 F( ) F(t)e2

i t

i t

d

dt

The power spectrum W can be obtained as

the Fourier transform of the signal auto -correlation function Wiener- Kinchine theorem

CORRELATION:The method of calculating the similarity of two function is

called the correlation integral and result is correlation function.

AUTO CORRELATION FUNCTION:If the two functions are different ,the integral is calledcross correlation for same function auto correlation

1 2

T

TT

E t E t Lim dtE t E tT

2

When the functions are alinged we get maximum of Auto -correlation function

T

TT

Lim dtE tT

2

-

Power spectrum W can be obtained from the Parseval's

theoremT

T

W d C E t dt

EEeeddtE titi2

Ed

EEddttE

2

2

2

2

2

2202

12

2 / 2

o

o

ti t i t

ot

E E e dt

E SinE

function.- aget again Then we As

2

2 E

2

22

/02

2 0

20

2 2

2

2 2

12 2 1

tot

TSinE

e dT

ET

T

Times of flight are distributed according to P

Total intensity is made up of large number of time segments

22

2

2

where

1 1

1

g

gT

T

Such that

' 1g d

Function is maximum for 0

Doppler broadening• This is due to the random motions of the atoms.• It only occurs in gasses

Consider a field of frequency incident on an atom with transition frequency which is moving with a velocity v in the propagation direction of the wave. Atom will see a wave of frequency

0

1 VC

Due to Doppler effect

• The negative and positive sign applies whether the velocity is in the same or opposite direction to that of the wave. If the atom is moving in the opposite direction to that of wave the frequency observed by the atom is higher than the value observed in lab. frame.

• The absorption will occur only when the apparent frequency as seen from the atom is equal to the atomic transition frequency

0

01 vc

0

1 vc

Rewriting above equation

This is equivalent to say that atom is not moving but has a resonant frequency equal to

00

1 vc

Or

Incident field sees a shifted transition frequency of the atom. Absorption will occur when the frequency of e.m.wave is equal to If atom is moving away from the field ( same direction)

0

01 vc

0

1 vc

Incident field sees a shifted frequency of the atom andAnd absorption occur when this where0

00

1 vc

0 00

cv

We need to find the spectral function for Doppler broadening

0g g

As atoms are moving with different velocities, therefore field Sees different transition frequencies.

Using kinetic theory of gasses that an atom of mass M in a gas at temperature T has a velocity component between v and v+ dv is given by Maxwellian distribution

2 2

exp2 2M Mvp d dKT KT

gives the probability that the transition frequency lies between and

g d d

The frequency function is related to g p d

0 0 0g d p d

That is the number of atoms absorbing with in the frequency interval from to is equal to the fraction ofAtoms moving with velocity between v and v+ dv as

d

00

dv cd

20

0 20 0

exp2 2

Mcc MgKT KT

The shape of curve corresponding to this equation is calledGaussian. The maximum again occurs at 0

1

2

max0 2c Mg

KT

Find FWHM

Spontaneous Emission Rate

SPONTANEOUS EMISSION TRANSITION RATE

2..

2 ANdtdN

emsp

100

032

20

kTec

where A is called the Einstein A co-efficient or spontaneous emission transition rate.• Assume that the material is placed is blackbody cavity whose walls are kept at a constant temperature T. • Once thermodynamic equilibrium is reached, an

electromagnetic energy density with spectral distribution

Established

• Material will immersed in this radiation.

• Both stimulated emission and absorption processes will occur in addition to the spontaneous process.

• In thermal equilibrium, number of transitions per second from level 1 to level 2 must be equal to number of transition per second from level 2→1.

eee NWNWAN 1212212

02121 BW

01212 BW

kTe

e

eNN 0

1

2

120 EE 12 NN

Define

and

where B21 and B12 - called Einstein B co-efficient.From Boltzmann statistics

for

00 121

221

1

2 B

NN

BNN

A e

e

e

e

0

2

1

212 21

1

e

e

e

e

NAN

NB BN

e

e

NN

1

2Putting value of , we get

0

0 0 0

12 21 12 21

kT

kT kT

Ae A

B B e B e B

100

032

320

kTecn

For a medium,

n- refractive index of the medium

100

032

320

2112

kTkT ecn

BeB

A

BBB 2112

Comparing two values we get,

1

1

100 32

330

kTkT ecn

e

BA

3 302 3

A nB c

Probability of absorption and stimulated emission due to black body radiation are equal.

Bcn

A 32

330

Once the value of B, due to black body radiation is knownWe can find the value of A as we know

2

20

234 p

nW

This is true for monochromatic field.

d

00

2

20

234

dpn

dW

0

2

20

234

p

ndW

22 2

0

43

B pn

For black body radiation, the elemental spectral energy density of radiation whose frequency lies between and

This is same result as obtained from Quantum Electrodynamics approach because her we use Planck’s law (which is quantum electrodynamically correct). As A increases as the cube of the frequency, so the process of spontaneous emission increases rapidly with frequency.It is easy to produce infrared laser as compared to UV laser If frequency increases by a factor 1010, then A increases by 101000.

230

30

43

p nA

c

where a is Bohr radius cma 810

1810 sA

sec101 8A

t sp

Putting these values, we get

For magnetic dipole transitions, A is approximately 105 times small.

The order of magnitude of spontaneous emission at optical frequency

150 10 hz cm5105 eap

SPECTRUM OF THE SPONTANEOUS EMISSION

Line shape due to natural broadening• For any transition, the spectrum of the emitted

radiation is the same as that observed in absorption

• Assume that an ideal electromagnetic filter- transmitting only those frequencies between ω and ω + dω is placed between the material and the walls of the black body cavity.

• If the material, the filter, and the black body cavity are kept at the same temperature T, then ratio between the populations of the two levels will again be given by

kTe

e

eNN 0

1

2

The density of electromagnetic radiation

at any point inside the cavity will also be given by

1032

20

kTec

and the net exchange of the energy between the material and the cavity within the transmission bandwidth of the filter must be zero.

This means that the energy emitted by the material in the bandwidth dω around ω due to spontaneous and stimulated emission must be equal to energy absorbed.

Define a spectral co-efficient Aω such that

-Number of atoms per unit time which upon decay emit a photon of frequency between ω and ω+dω.

dAN 2

dAA

Similarly,

dNB

dNBdNBdNA eee122

represents the number of transitions per unit absorption or stimulated emission induced by black body radiation with frequency between ω and ω+dω.Equilibrium condition

Using above equations we get,

BA

BA

B

tgpn

W 2

20

234

BgB

BA

BgA

where can be obtained from

The spectrum of the radiated wave is the same as for absorption or stimulated emission.

A Ag

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