Modified QM

8/3/2019 Modified QM

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Quantum Mechanics


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WAVES

A wave is nothing butdisturbance which isoccurred in a medium and it is specified by its

frequency, wavelength, phase, amplitude and

intensity.

PARTICLES

A particle or matter has mass and it is located at a

some definite point and it is specified by its mass,

velocity, momentum and energy.


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The physical values or motion of a macroscopic

particles can be observed directly. Classicalmechanics can be applied to explain that motion.

But when we consider the motion ofMicroscopicparticles such as electrons, protonsetc.,

classical mechanicsfails to explain that motion.

Quantum mechanics deals with motion of

microscopicparticles orquantumparticles.


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de Broglie hypothesis

In 1924 the scientist namedde Broglie

introducedelectromagnetic waves behaves

like particles, and the particles like electronsbehave like waves called matter waves.

He derivedan expression for the wavelengthof matter waves on the analogy of radiation.


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According to Plancks radiation law

Where c is a velocity of light and is a wave length.

According to Einstein mass-energy relation

From 1 & 2

Where p is momentum of a photon.

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

.

ch

hE

!

!

p

hmc

h

chmc

!

!

!

P

P

P

2

)2......(2

mcE!


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The above relation is called de Broglie s matter waveequation. This equation is applicable to all atomic

particles.

If E is kinetic energy of a particle

Hence the de Broglies wave lengthmE

h

2!P

mEp

mpE

mvE

2

2

2

1

2

2

!

!

!


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de Brog lie wavelength associated with

electrons:

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

If v is the velocity attained by the electron due to

acceleration

The de Broglie wavelength0

2

0

2

2

1

m

eVv

eVvm

!

!

AV

m

eVm

h

vm

h

0

0

0

0

26.12

2

!

!!

P

PP


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Characteristics of Matter waves

Lighter the particle,greateris the wavelength

associatedwith it.

Lesserthe velocity of the particle, longer the wavelength

associatedwith it.

ForV= 0, =. This means that only with moving

particle matter wave is associated.

Whether the particle is chargedor not, matter wave is

associatedwith it. This reveals that these waves are not

electromagnetic but a new kin

dof waves .


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It can be proved that the matter waves travel faster thanlight. We know that

The wave velocity () is given by

As the particle velocity v cannotexceed velocity of light c,

is greater than velocity of light.

h

mcmch

mcE

hE

2

2

2

!p!

!

!

..

.

v

cw

mv

h

h

mcwhere

mv

h

h

mcw

w

2

2

2

&

))((

!

!!

!

!

P.

.P


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Experimentalevidence for matter waves

1. Davisson and Germer s Experiment.

2. G.P. Thomson Experiment.


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DAVISON & GERMERSEXPERMENT

Davison andGermer firstdetectedelectron waves in 1927.

They have also measure

dd

e Broglie wave

lengths ofs

low

electrons by usingdiffraction methods.

Principle:

Basedon the concept ofwave nature of matterslow

moving electrons behave like waves. Hence accelerated

electron beam can be used fordiffraction studies in

crystals.


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Experimentalarrangement

The electron gun G produces afine beam of electrons.

It consists of a heatedfilament F, which emits electrons due to thermionic emission

The acceleratedelectron beam of electrons are incident on a nickelplate, called target T. The target crystalcan be rotated about an axisperpendicularto the direction of incident electron beam.

The distribution of electrons is measuredby using a detector calledfaraday cylinderc and which is moving along a graduated

circular scaleS.

A sensitivegalvanometerconnected to the detector.


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High voltage

Faradaycylinder

Galvanometer

NickelTarget

Circular scale

cathode

Anode

filamentG

S

c

G


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ResultsResults When an electron beam accelerated by 54 volts was

directed to strike the nickelcrystal, a sharp maximum inthe electron distribution occurred at scattered angle of

500

with the incident beam.

For that scattered beam of electrons the diffracted anglebecomes 650 .

For a nickel crystal the inter planer separation is

d = 0.091nm.


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I

0

V = 54v

500

C

U

R

R

E

N

T

250

250650

Incident electron beam

Diffracted

beam650


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nm

nm

nd

165.0

165sin091.02

sin2

0

!

v!vv

!

P

P

PU

nm

A

AV

166.0

54

26.12

26.12

0

0

!

!

!

P

P

P

According to Braggs law

For a 54 volts , the de Broglie wavelength associated with the electron is

given by

This is in excellent agreement with theexperimental value.

The Davison - Germer experimentprovides a direct verification of deBroglie hypothesis of the wave natureof moving particle.


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G.P THOMSONSEXPERIMENT

G.PThomson's experiment proved that the diffractionpatterobserved was due to electrons but not due to electromagneticradiation produced by the fast moving charged particles.

EXPERIMENTAL ARRANGEMENT

G.PThomson experimental arrangement consists of

(a) Filament or cathode C

(b) Gold foil or gold plate

(c) Photographic plate(d) Anode A.

The whole apparatus is kept highly evacuateddischarge tube.


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W

hen we apply potentia

lto catho

de, the e

lectrons areemittedand those are further acceleratedby anode.

When these electrons incident on a goldfoil, those are

diffracted, and resulting diffraction pattern getting onphotographic film.

After developing the photographic plate a symmetrical

pattern consisting ofconcentric rings about a centralspot

is obtained.


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G.P THOMSON EXPERIMENT

Discharge tube

cathode

Anode Gold foil

Vacuum pump

slitPhoto

graphicplate


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Photographic film

Diffraction pattern.


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If is very small 2 = r /L

Tan 2 = r /L

2= r / L . (1)

E

Incidentelectron

beamradius

Gold foil

Lo

r

AB c

Brage plane


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According to Braggs law

2

20

0

1

2

)(2

)(2

)1,(sin2

c

v

mm

eVm

h

r

Ld

L

rdd

nd

nandnd

!

!

!p!p!

!

!!

P

PP

U

P

PU

UPU

According to de Broglie s waveequation

Where m0

is a relativistic mass of an electron


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Ad

eVm

h

r

Ld

r

Ld

eqfrom n

0

0

08.4

)2

(

)3(.,

!

!p! P

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 4.080A.


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Heisenberg uncertainty principle

This principle states that the product of uncertainties indetermining the both position and momentum ofparticle is approximately equal to h / 4.

Where x is the uncertainty in determine the positionandp is the uncertainty in determining momentum.

This relation shows that it is impossible to determinesimultaneously both the position and momentum of theparticle accurately.

T4

h

pxu((


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This relation is universal and holds for all canonically conjugatephysical quantities like

1.Angular momentum & angle2. Time & Energy

T

T

U

4

4

hEt

hj

u((

u((

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.


26/47

Physicalsignificance of the wave function

The wave function has no direct physicalmeaning.

It is a complex quantity representing the variation of a

Matter wave.

The wave function ( r, t ) describes theposition of a

particle with respect to time.

It can be consideredas probability amplitudesince it is

used to find the location of the particle.


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* or 2 is theprobability densityfunction.

* 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 thats a normalization

conditionof particle.

1

1

2

-

-

*

!

!

g

g

g

g

dxdydz

dxdydz

]

]]


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Schrdinger wave Equation

Schrdinger wave equation is a basic principle of a

fundamental Quantum mechanics. Consider an electron of mass m and moving with a

velocity v .

We know that electrons exhibit wave nature and for such an

undamped, harmonic electron wave equation, the generalsolution is given by

)}(exp{v

xtiAy ! [

Where y represents the displacement at time t

at a position x ,

w is the angular frequency and v is the velocity.


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Quantum mechanically we represent the wave associated

with a moving particle by a wave function .

YP

TY[

PYT]

[]

!!

!

!

v

where

xtitr

v

xtiAtr

2

.,

)1.()}........(2exp{),(

)}(exp{),(


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Now we are going to derive an expression for energy of

electron.

total energy of the electron is the sum of its K.E. and P.E.

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

)(2

)(2

1

..

.,

2

2

2

xVm

pE

xVm

pE

xVmvE

EPEKE

energytotal

!

!

!

!

]]


31/47

T

]

T

]

T]

PY

PYT]

2.,

)3()}...,(exp{),(

)}(2

exp{),(

)}(2exp{),(

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

.,

)2.()}........(2exp{),(

hwhere

pxEti

Atx

pxEth

iAtx

phxt

hEiAtx

eqvaluesthesengsubstituti

p

hhE

xtitr

n

!

!

!

!

!!

!

J

J


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Now we have to get expressions for E and p2 fromequation 3.

Differentiating the equation (3) with respect to t .,

)4.........(..........

)}(exp{)(

)}](exp{[

ti

E

Eti

Ei

t

pxEti

AEi

t

pxEti

Att

x

x!

!x

x

!x

x

!x

x

x

x!

x

x

]]

]]

]]

]

]

J

J

J

JJ

J


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Differentiating theequation (3) with respect to time x

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

)}(exp{)(

)}(exp{)(

)}](exp{[

2

2

22

2

2

2

2

2

2

2

xp

p

x

pxEti

pi

x

pxEti

Api

x

pxEtiAxx

x

x!

!x

x

!x

x

!x

x

x

x!x

x

]]

]]

]

]

]

J

J

JJ

JJ

J


34/47

Substituting the values of E and p2 from 4 and 5 in

equation 2 we get

)6.......(}{2.,.

}{2

2

22

2

22

tiVxmei

Vxmti

x

x

!x

x

x

x!

x

x

]

]

]

]]]

JJ

JJ

This is known as on e dimensional Schrdinger's time dependent wave

equation. Extending this to three dimensionalproblem.

tiV

zyxm x

x!

x

x

x

x

x

x ]]

]]] JJ}{

22

2

2

2

2

22


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For our purpose, it is sufficient to consider situations in

which the potentialenergy of the particle doesntdepend

explicitly on time. The forces that act on it, and hence V, then

vary only with the position of the particle.In this case equation 6 reduces to the time independent form.

)exp()(

)exp()(

.,

)7)........(()(),(

)exp()exp(),(

)}(exp{),(

Eti

t

ipxx

where

txAtx

ipxEt

iAtx

pxEti

Atx

J

J

JJ

J

!

!

!

!

!

J

]

J]]

]

]


36/47

0)(2

2

)exp()exp()exp(2

).........6(.,...,

)exp(

)exp(

)exp(

)exp(),(

22

2

2

22

2

22

2

2

2

2

!x

x

!x

x

!x

x

x

x!

x

x

x

x!

x

x

!x

x

!

]]

]]]

]]]

]]

]]

]]

]]

VEm

x

EVxm

iEtAEi

i

iEtVA

x

iEtA

m

eqinngsubstituti

x

iEtAx

x

iEtA

x

iEtAEi

t

iEtAtx

n

J

J

JJ

J

JJ

J

J

J

JJ

J


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This equation is referred to as time-independent

Schrdinger equation in one dimension.

In three dimensions time independentSchrdinger

equation can be written as

0)(2

.,.

0)(2)(

2

2

22

2

2

2

2

2

!

!x

xx

xx

x

]]

]]]]

VEm

ei

VEm

zyx

J

J


38/47

Particle in a one dimensionalpotentialbox

Consider an electron of mass min an infinitely deepone-dimensional potential box with a width of a Lunits

in which potential is constantandzero.

Lxxxv

Lxxv

ueg!

!

&0,)(

0,0)(

X=0 X=L

V=0


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The motion of the electron in one dimensional box canbe described by the Schrdinger's equation.

0][2

22

2

! ]]

VEm

dx

d

J

Inside the box the potential V =0

Em

kwherek

dx

d

Em

dx

d

2

22

2

2

22

2

2,,0

0][2

J

J

!p!

!

]]

]]

kxBkxAx cossin)( !]

The solution to above equation can be written as


40/47

Where A,B and K are unknown constants and tocalculate them, it is necessary to apply boundary

conditions.

When X = 0 then = 0 i.e. ||2 = 0 . aX = L = 0 i.e. ||2 = 0 b

Applying boundary condition ( a ) to equation ( 1 )

A Sin K(0) + B Cos K(0) = 0 B = 0

Substitute B value equation (1)

(x) = A Sin Kx


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Applying second boundary condition for equation (1)

Substitute B & K value in equation (1)

To calculate unknown constantA, consider normalization condition.

L

nk

nkL

kL

kLA

kLkLA

T

T

!

!

!

!

!

0sin

0sin

cos)0(sin0

L

xnAx

)(sin)(

T

] !


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LA

LA

L

nx

n

Lx

A

dxLxnA

dxL

xnA

dxx

L

L

o

L

o

L

/2

12

1]2

sin

2

[

2

1]2

cos[1[21

1][sin

1)(

2

0

2

2

22

0

2

!

!

!

!

!

!

T

T

T

T

]Normalization condition


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2

22

22

22

2

2

8

2&,.

2

2

2

2

sin/2

mL

hnE

h

L

nkwhere

m

h

L

n

E

m

kE

mEk

x

L

nLn

!

!!

!

!p!

!

T

T

T

T

T]

J

J

J

The wave functions n and the corresponding energies Enwhich are called Eigen functions and Eigen values, of thequantum particle.

The normalized wave function is


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The particleWave functions & their energyEigen valuesin a one dimensional square well potential are shown infigure.

L

zn

L

yn

L

xnLn

TTT

]3213

sinsinsin)/2(!

Normalized Wave function in three dimensions is given by


45/47

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

T

] sin/2!


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Conclusions

1.The three integers n1,n2 and n3 calledQuantum numbersare required to specify completely each energy state.

2.The energy E depends on the sum of the squares of the

quantum numbers n1,n2 and n3 but not on theirindividualvalues.

3.Several combinations of the three quantum numbers may

give differentwave functions, but not of the same energyvalue. Such states and energy levels are said to bedegenerate.


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