QUANTUM QUANTUM MECHANICSMECHANICS

WAVES:WAVES: A wave is nothing but disturbance which is A wave is nothing but disturbance which is occurred in a medium and it is specified by its occurred in a medium and it is specified by its frequency, wavelength, phase, amplitude and frequency, wavelength, phase, amplitude and intensity.intensity.

PARTICLES:PARTICLES: A particle or matter has mass and it is located A particle or matter has mass and it is located at a some definite point and it is specified by at a some definite point and it is specified by its mass, velocity, momentum and energy.its mass, velocity, momentum and energy.

Waves and Particles : What do we mean by them?

Ball, Car, person, or point like objects called particles. They can be located at a space point at a given time.They can be at rest, moving or accelerating.

Falling Ball

Ground level

Material Objects:

Waves and Particles: What do we mean by them ?

Ripples, surf, ocean waves, sound waves, radio waves.

Need to see crests and troughs to define them.

Waves are oscillations in space and time.

Direction of travel, velocity

Up-downoscillations

Wavelength ,frequency, velocity and oscillation size defines waves

Common types of waves:

Particles and Waves: Basic difference in behaviour

When particles collide they cannot pass through each other ! They can bounce or they can shatter

Before collision After collision

Another aftercollision stateshatter

Collision of truck with ladder on top with a Car at rest ! Note the ladder continue itsMotion forward ….. Also the small care frontEnd gets smashed.

Head on collision of a car and truckCollision is inelastic – the small car is dragged along By the truck……

Waves and Particles Basic difference:

Waves can pass through each other !

As they pass through each other they can enhance or canceleach other

Later they regain their original form !

Waves and Particles:

Waves

Spread in space and time

Wavelength Frequency

Can be superposed – show interference effects

Pass through each other

Particles

Localized in space and time

Cannot pass through each other -they bounce or shatter.

A SUMMARY OF DUAL ITY OF NATUREWave particle duality of physical objects

LIGHT

Wave nature -EM wave Particle nature -photons

Optical microscope

Interference

Convert light to electric current

Photo-electric effect

PARTICLES

Wave nature

Matter waves -electronmicroscope

Particle nature

Electric currentphoton-electron collisions

Discrete (Quantum) states of confinedsystems, such as atoms.

• The physical values or motion of a macroscopic particles can be observed directly. Classical mechanics can be applied to explain that motion.

• But when we consider the motion of Microscopic particles such as electrons, protons……etc., classical mechanics fails to explain that motion.

• Quantum mechanics deals with motion of microscopic particles or quantum particles.

Classical world is Deterministic:Knowing the position and velocity ofall objects at a particular timeFuture can be predicted using known laws of forceand Newton's laws of motion.

Quantum World is Probabilistic:Impossible to know position and velocitywith certainty at a given time.

Only probability of future state can be predicted usingknown laws of force and equations of quantum mechanics.

Observer Observed Tied together

Wave Nature of Matter

Louis de Broglie in 1923 proposed thatmatter particles should exhibit wave properties just as light waves exhibitedparticle properties. These waves have very small wavelengths in most situationsso that their presence was difficult to observe

These waves were observed a few years later by Davisson andG.P. Thomson with high energy electrons. These electrons showthe same pattern when scattered from crystals as X-rays of similarwave lengths.

Electron microscopepicture of a fly

de Broglie hypothesis:

• In 1924 the scientist named de Broglie introduced electromagnetic waves behaves like particles, and the particles like electrons behave like waves called matter waves.

• He derived an expression for the wavelength of matter waves on the analogy of radiation.

• According to Planck’s radiation law

• Where ‘c’ is a velocity of light and ‘λ‘is a wave length.

• According to Einstein mass-energy relation

From 1 & 2

Where p is momentum of a photon.

)1..(..........

c

h

hE

p

hmc

h

chmc

2

)2......(2mcE

• The above relation is called de Broglile’ s matter wave equation. This equation is applicable to all atomic particles.

• If E is kinetic energy of a particle

• Hence the de Broglie’s wave length

mE

h

2

mEp

m

pE

mvE

2

2

2

1

2

2

• de Broglie wavelength associated with electrons:

• Let us consider the case of an electron of rest mass m0 and charge ‘ e ‘ being accelerated by a potential V volts.

• If ‘v ‘ is the velocity attained by the electron due to acceleration

• The de Broglie wavelength

0

20

2

2

1

m

eVv

eVvm

AV

meV

m

h

vm

h

0

00

0

26.12

2

Characteristics of Matter waves:• Lighter the particle, greater is the wavelength

associated with it.

• Lesser the velocity of the particle, longer the wavelength associated with it.

• For v = 0, λ=∞ . This means that only with moving particle matter wave is associated.

• Whether the particle is charged or not, matter wave is associated with it. This reveals that these waves are not electromagnetic but a new kind of waves .

• It can be proved that the matter waves travel faster than light.

We know that

The wave velocity (ω) is given by

Substituting for λ we get

• As the particle velocity v cannot exceed velocity of light c, ω is

greater than velocity of light.

h

mc

mch

mcE

hE

2

2

2

v

cw

mv

h

h

mcw

mv

hh

mcw

w

2

2

2

)(

)(

Experimental evidence Experimental evidence for matter wavesfor matter waves

There was two experimental evidencesThere was two experimental evidences

1.1. Davisson and Germer ’s experiment.Davisson and Germer ’s experiment.

2. G.P. Thomson Experiment.2. G.P. Thomson Experiment.

DAVISON & GERMER’S EXPERMENT:

• Davison and Germer first detected electron waves in 1927.

• They have also measured de Broglie wave lengths of slow electrons by using diffraction methods.

Principle: • Based on the concept of wave nature of matter fast

moving electrons behave like waves.

Hence accelerated electron beam can be used for diffraction studies in crystals.

High voltage

Faraday cylinder

Galvanometer

Nickel Target

Circular scale

cathode

Anode

filament G

Sc

G

Experimental arrangement:

• The electron gun G produces a fine beam of electrons.

• It consists of a heated filament F, which emits electrons due to thermo ionic emission.

• The accelerated electron beam of electrons are incident on a nickel plate, called target T.

• The target crystal can be rotated about an axis parallel to the direction of incident electron beam.

• The distribution of electrons is measured by using a detector called faraday cylinder c and which is moving along a graduated circular scale S.

• A sensitive galvanometer connected to the detector.

Results:

• When the electron beam accelerated by 54 volts was directed to strike the nickel crystal, a sharp maximum in the electron distribution occurred at an angle of 500 with the incident beam.

• For that incident beam the diffracted angle becomes 650 .

• For a nickel crystal the inter planer separation d = 0.091nm.

I

θ0

V = 54v

500

CURRENT

250

250650

Incident electron beam

Diffractedbeam

650

• According to Bragg’s law

• For a 54 volts , the de Broglie

wave length associated with the electron is given by

• This is in excellent agreement with

the experimental value.• The Davison - Germer experiment

provides a direct verification of de Broglie hypothesis of the wave nature of moving particle.

nm

nm

nd

165.0

165sin091.02

sin20

nm

A

AV

166.054

26.12

26.12

0

0

G.P THOMSON’S EXPERIMENT: • G.P Thomson's experiment proved that the diffraction pattern

observed was due to electrons but not due to electromagnetic radiation produced by the fast moving charged particles.

EXPERIMENTAL ARRANGEMENT:

• G.P Thomson experimental arrangement consists of (a) Filament or cathode C. (b) Gold foil or gold plate (c) Photographic plate (p) (d) Anode A.

• The whole apparatus is kept highly evacuated discharge tube.

G.P THOMSON EXPERIMENT:

Discharge tube

cathodeAnode Gold foil

Vacuum pump

slitPhotographic plate

Photographic film

Diffraction pattern.

If θ is very small 2θ = R / L

Tan 2θ = R / L

2θ = R / L ………. (1)

Incident electron beam

radiusGold foil

L o

R

θθA

B c

E

θ

Brage plane

• When we apply potential to cathode, the electrons are emitted and those are further accelerated by anode.

• When these electrons incident on a gold foil, those are diffracted, and resulting diffraction pattern getting on photographic film.

• After developing the photographic plate a symmetrical pattern consisting of concentric rings about a central spot is obtained.

According to Braggs law

)3....(..........

)1.....(,.

)2.......(2

1.,

)(2

.....,

sin2

r

Ld

eqfrm

d

nfor

nd

smallveryisif

nd

n

massicrelatavestaismwhere

electronforeVm

h

...,.....,

).,(2

0

0

According to de Broglie’ s wave equation

Ad

eVm

h

r

Ld

eqfrom n

0

0

08.4

)2

(

)3(.,

• The value of ‘d’ so obtained agreed well with the values using X-ray techniques.

• In the case of gold foil the values of “d” obtained by the x-ray diffraction method is 4.060A.

Heisenberg uncertainty principle:

• This principle states that the product of uncertainties in determining the both position and momentum of particle is approximately equal to h / 4Π.

where Δx is the uncertainty in determine the position and Δp is the uncertainty in determining momentum.

• This relation shows that it is impossible to determine simultaneously both the position and momentum of the particle accurately.

4

hpx

• This relation is universal and holds for all canonically conjugate physical quantities like

1. Angular momentum & angle

2. Time & energy

4

4h

Et

hj

Consequences of uncertainty principle:

• Explanation for absence of electrons in the nucleus.

• Existence of protons and neutrons inside nucleus.

• Uncertainty in the frequency of light emitted by an atom.

• Energy of an electron in an atom.

Physical significance of the wave function:

• The wave function ‘Ψ’ has no direct physical meaning. It is a complex quantity representing the variation of a Matter wave.

• The wave function Ψ( r, t ) describes the position of a particle with respect to time.

• It can be considered as ‘probability amplitude’ since it is used to find the location of the particle.

• ΨΨ* or ׀Ψ 2׀ is the probability density function. • ΨΨ* dx dy dz gives the probability of finding the electron in

the region of space between x and x + dx, y and y + dy, z and z + dz.

• The above relation shows that’s a ‘normalization condition’

of particle.

1

1

2

-

-

*

dxdydz

dxdydz

Schrödinger time independent wave equation:

• Schrödinger wave equation is a basic principle of a fundamental Quantum mechanics.

• This equation arrives at the equation stating with de Broglie’s idea of Matter wave.

• According to de Broglie, a particle of mass ‘m’ and moving with velocity ‘v’ has a wavelength ‘λ’.

)1.(..........mv

h

p

h

• According to classical physics, the displacement for a moving wave along X-direction is given by

• Where ‘A’ is a amplitude ‘x’ is a position co-ordinate

and ‘λ’ is a wave length.

• The displacement of de Broglie wave associated with a moving wave along X-direction is given by

)2

sin( xAS

)2

sin(),( xAtr

• If ‘E’ is total energy of the system E = K.E + P.E ---------(4)

• If ‘p’ is a momentum of a particle K.E = ½ mv2

= p2 / 2m • According to de Broglie’ s principle λ = h / p p = h / λ K.E = p2 / 2m • The total energy

• Where ‘V’ is potential energy.

)5....(..........2

2

..

2

2

2

2

VEm

h

m

hVE

EKEPE

• Periodic changes in ‘Ψ’ are responsible for the wave nature of a moving particle

)3(4

11

4

).2

sin(4

).2

sin(]2

[

).2

cos(2

].2

sin[)(

2

2

22

2

2

2

2

2

2

2

2

22

2

dx

d

dx

d

xAdx

d

xAdx

d

xA

xAdx

d

dx

d

][8

][8

][]4

1[

2

2

5,....

2

2

2

2

2

2

2

2

2

2

2

2

2

2

VEh

m

dx

d

VEdx

d

m

h

VEdx

d

m

h

VEm

h

eqfrm n

• This is Schrödinger time independent wave equation in one dimension.

In three dimensional way it becomes…..

0][8

2

2

2

2

VE

h

m

dx

d

0][8

2

2

2

2

2

2

2

2

VEh

m

zyx

Particle in a one dimensional potential box:

• Consider an electron of mass ‘m’ in an infinitely deep one-dimensional potential box with a width of a ‘ L’ units in which potential is constant and zero.

Lxxxv

Lxxv

&0,)(

0,0)(

X=0 X=L

V=0

+ + + +

+ + + +

+ + + +

++++

X

V

V

Nuclei

One dimensional periodic potential in crystal.

Periodic positive ion cores Inside metallic crystals.

The motion of the electron in one dimensional box can be described by the Schrödinger's equation.

d 10][2

22

2

VE

m

dx

d

Inside the box the potential V =0

)2(0][2

22

2

E

m

dx

d

22 2

.,mE

kwhere

)3(022

2

k

dx

d

)4(cossin)( kxBkxAx

The solution to equation (3) can be written as

Where A,B and K are unknown constants and to calculate them, it is necessary To apply boundary conditions.

• When x = 0 then Ψ = 0 i.e. |Ψ|2 = 0 ------1 x = L Ψ = 0 i.e. |Ψ|2 = 0 ------2

• Applying boundary condition (1) to equation (4)

A sin k(0) + B cos K(0) = 0 B = 0

• Substitute B value equation (4)

Ψ(x) = A sin kx

• Applying second boundary condition for equation (4)

• Substitute k value in equation (5)

• To calculate unknown constant A, consider

normalization condition.

L

nk

nkL

kL

kLA

kLkLA

0sin

0sin

cos)0(sin0

L

xnAx

)(sin)(

LA

LA

L

nx

n

Lx

A

dxL

xnA

dxL

xnA

dxx

L

L

o

L

o

L

/2

12

1]2

sin2

[2

1]2

cos[1[2

1

1][sin

1)(

2

0

2

2

22

0

2

Normalization condition

23

22

21

2

3213 sinsinsin)/2(

sin/2

nnnn

L

zn

L

yn

L

xnL

L

xnL

n

n

• The wave functions Ψn and the corresponding energies En which are often called Eigen functions and Eigen values, describe the quantum state of the particle.

The normalized wave function is

2

22

22

22

22

8

22

2&

.,2

2.,

mL

hnE

m

hLn

E

h

L

nk

wherem

kE

mEkwhere

• The electron wave functions ‘Ψn’ and the corresponding probability density functions ‘ |Ψn|2 ’ for the ground and first two excited states of an electron in a potential well are shown in figure.

X=0 X=L

E2=4h2/8mL2

E1=h2 / 8mL2

E3=9h2 / 8mL2

n = 1

n = 2

n = 3

L / 2

L / 2

L / 3 2L / 3

√ (2 / L)

2

22

8mL

hnEn L

xnLn

sin/2

Conclusions;1.The three integers n1,n2 and n3 called quantum

numbers are required to specify completely each energy state. since for a particle inside the box, ‘ Ψ ’ cannot be zero, no quantum number can be zero.

2.The energy ‘ E ’ depends on the sum of the squares of the quantum numbers n1,n2 and n3 and no on their individual values.

3.Several combinations of the three quantum numbers may give different wave functions, but of the same energy value. such states and energy levels are said to be degenerate.

A localized wave or wave packet:

Spread in position Spread in momentum

Superposition of wavesof different wavelengthsto make a packet

Narrower the packet , more the spread in momentumBasis of Uncertainty Principle

A moving particle in quantum theory

ILLUSTRATION OF MEASUREMENT OF ELECTRONPOSITION

Act of measurementinfluences the electron-gives it a kick and itis no longer where itwas ! Essence of uncertaintyprinciple.