ph-20101 bohr model and qt of hydrogen atom

8/8/2019 PH-20101 Bohr Model and QT of Hydrogen Atom

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Atomic ModelsHow to see

Early modelsAtomic spectraThe Bohr model

Correspondence principleDemerits


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How to seede Broglie wavelength

The probe has to interact/scatter the objectFor photons: 1 eV ~ 10 -10 m - atomic physics 1 MeV ~ 10 -15 m - nuclear physics 1 GeV ~ 10 -20 m - particle physics

Examples of probes in atomic scale

XRD Electron microscope (SEM, TEM)


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Early models of atom

Plum pudding model of 1890s

Rutherfords experiment in 1911


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Atomic spectra


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Atomic spectra: the key

Planetary model (Rutherford) of atom

proton

electron

An atomic electronshould, classically, spiralrapidly into the nucleusas it radiates energy dueto its acceleration


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The Bohr atomElectron waves & stationary states

de Broglie wavelength of e : hmv

=

Centripetal ElectricF F =2 2

2

0

14

mv er r

=04

ev

mr =

04Orbital e wavelength:r h

c m

=

This corresponds to the circumference of e orbit: 2 r

=An e can circle a nucleus only if its orbit contains

an integral number of de Broglie wavelengths

Condition for orbit stability: 2 1, 2,3,nn r n = =


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Stationary states


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Bohr orbitsCondition for orbit stability: 2 1, 2,3,nn r n = =

04 2n nr nh

r e m

=

2 20

2Orbital radii in Bohr atom: 1, 2,3,...nn hr n

me

= =

11 20 1 0Bohr radius: 5.292 10 ; na r m r n a

= = =

Angular momentum quantization (alternate approach)

h

mv = 2

nn r =

2( )( / )mr v r = mvr = L I = n=2h

n

=


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Energy levels2 2

04

mv e E KE PE

r r = + =

04

ev

mr =

2

04n

n

e E

r =

41

2 2 2 20

1 1,2,3,...8n

E me E nh n n

= = =


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12 21 1 1:l u

E Hydrogen spectrumhc n n

=


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Correspondence principleThe greater the quantum number, the closer

quantum physics approaches classical physics

At very high n we have more dense

levels which are more like continuum


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Bohr atom model: DemeritsApplicable only to hydrogen and other one-

electron ions such as He+

and Li2+

Cannot explain why some lines are moreintense than others

Cannot explain why many lines consist of several separate lines whose wavelengthsdiffer very slightly

No light on how individual atoms interactQuantum mechanics was developed(1925,1926:Schrodinger, Heisenberg, Born,Dirac & others) to overcome these shortfalls


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Quantum theory of

hydrogen atom Quantum mechanics: recap

Schrodingers eqn. for Hydrogen atom Separation of variables Quantum numbers

Electron probability density Selection rules Zeeman effect


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Quantum mechanics: recapExplores probabilities instead of asserting

Eg. Hydrogen atom: r g.s. from Bohr theory = 5.3 10-11

mQM Most probable r g.s. = 5.3 10 -11 m

Wave function itself has no physical interpretation

| | 2 probability of finding the body (+ve, real quantity)2

Normalization: 1dV

=

2

1 21

2Probability:

x

x x x

P dx= 2 2

2 2 2

1Wave equation: (same sense of II law)

y y x v t

=

2 2

2Schrodinger equation: (1D)

2i U

t m x

= +


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Schrodingers equation2 2 2

2 2 2 2

2( ) 0 (3D)

m E U

x y z

+ + + =

2

0

Electric potential energy:4

eU

r =

In spherical polar coordinates, the Schrodingers equation becomes

2 2

2 2 2 2

2 20

sin sin sin

2 sin0

4

r r r

mr e E

r

+

+ + + =


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Separation of variablesHydrogen atom wave fn.: ( , , ) ( ) ( ) ( )r R r =

Simply, if R = R dR

r r dr

= = 2 2

2 2Similarly ,d d

R Rd d

= =

Substituting above in Schrodingers eqn. and rearranging,

2 2

2 2 2 2

2 20

sinsin sin

2 sin 14

d dR d d r

dr dr d d

mr e d E r d

+

+ + = 2

22

1l

d m

d

=

if ( ) ( ), then ( ) ( ) . f x g y f x g y const = = =


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Substituting for ml and rearranging, yields

22 22

2 20

1 2 1sin

4 sin sin

lmd dR mr e d d r E R dr dr r d d

+ + =

=l(l + 1) (Again we have different variables on both sides)

22

2Equation for : 0ld

md

+ =

2

2

1Equation for : sin ( 1) 0

sin sin

lmd d l ld d

+ + =

22

2 2 20

1 2 ( 1)Equation for : 0

4d dR m e l l

R r E Rr dr dr r r

+ + + =

b


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Quantum numbersThe solution of equation for is given by ( ) lim Ae =

Exploiting the symmetry that and + 2 identify the samePlane, we have,

( ) ( ) = + ( 2 )l lim im Ae Ae + = 0, 1, 2, 3,.m =

The differential equation for has a solution provided:is an integer and 0, 1, 2,...,ll l m m l =

The final solution of radial part yields,4

12 2 2 2 20

1; 1, 2,3,... ; 132n

E me E n n ln n

= = = +

0,1,2,..., ( 1)l n =

Thus the principal ( n), orbital ( l) and magnetic ( m)quantum numbers are defined

O bi l b


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Orbital quantum number 2

22 2 2

0

1 2 ( 1)Equation for : 0

4d dR m e l l

R r E Rr dr dr r r

+ + + =

E includes electrons orbital kinetic energy also !!

radial orbital E KE KE U = + +2

04radial orbital

eKE KE

r = +

22

2 2 2

1 2 ( 1)0

2radial orbitald dR m l l

r KE KE Rr dr dr mr

+ + + =

If R(r) has to be an exclusive function of r,2

2

( 1)

2orbital

l lKE

mr

+=

212orbital orbital

KE mv= {2

22 orbital L

L mv r mr

= =

2 2

2 2

( 1)

2 2

L l l

mr mr

+ = Electron angular momentum ( 1) L l l = +

M i b


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Magnetic quantum number e - revolving around the nucleus minute current loop

Has a magnetic field like that of magnetic dipole

ml specifies the direction of L by

determining the componentof L in the field direction.

Interacts with external magnetic field B

Space quantization

0, 1, 2,..., z l l L m m l= =

U i i i l & L


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Uncertainty principle & LzWhy only L z is quantized?

L can never point any specific directionbut in cone where L z=m l

If not the uncertainty principle will beviolated

If L were in z direction, e - is confined to xy plane and hence z = 0, p z

L precesses constantly about z-axis

El b bili d i


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Electron probability density

No definite orbits

QM i f


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QM view of atoms

Th bi l


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The orbitals

S l ti l


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Selection rules

{*Allowed transitions: 0 , ,l ln l m nlmu u x y z

=Transitions not obeying above condition are forbidden transitions

Selection rules: 1l = 0, 1lm =

( )ln l m

( )l

nlm

I t ti ith ti fi ld


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Interaction with magnetic fieldThe torque on a magnetic dipole in amagnetic field of flux density B is

sin { B r F =

Potential energy 0 when / 2.mU = =

/2For other orientations mU d

=

/2sin B d

=

cos B =

IA = 2ef r = 2v r r f = =22 L mvr mfr = =

Electron magnetic moment

2e

Lm

=

Gyromagnetic ratio


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cosmU B =

2e

Lm

=

cos2me

U LBm

=

( 1) L l l= +

cos( 1)

lml l

=+

2m l

eU m B

m

=

Bohr magneton:2 Bem

=

In a magnetic field, the energy of a

particular atomic state depends on alsolm

m l BU m B =

Z ff t


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Zeeman effectIn a magnetic field, the energy of a

particular atomic state depends on alsolma level with unique ' ' splits into

different levels having different ' ' l

n

m

1 0 0

2 0

3 0 0

4Normal Zeeman

effect

4

B

B

B ev v v Bh m

v v

B ev v v Bh m

= =

=

= + = +

0, 1lm =

m l BU m B =

0 m E U E

vh h

+= =

0 = + l B B

v v m h


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A l Z ff t


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Anomalous Zeeman effect

The previous QM treatment could not explain both anomalous Zeeman effect and

fine structureTwo Dutch graduate students (Samuel Goudsmit & George Uhlenbeck) proposed in1925 that

Every e - has an intrinsic angular momentum, called spin, whose magnitude is thesame for all electrons. Associated with this angular momentum is a magneticmoment

Electron spin


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Electron spin1 1 3

Spin angular momentum: ( 1) 12 2 2

S s s = + = + =

Classical model of a spinning electron. This model gives anincorrect magnitude for the magnetic moment, incorrect

quantum numbers, and too many degrees of freedom. Spinarises from relativistic dynamics.

12 z s

S m= =

Spin magnetic moment: Se

Sm

= 2Sz Bem

= =

Stern Gerlach experiment


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Stern-Gerlach experiment

Cause for deflection: cos z SdB

F dz

=

Magneti c moment of silver atom is due to one electron

First proof of space quantization

Spin orbit coupling


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Spin-orbit couplingcos ,mU B =

cos Sz B = =

m BU B =

L S: origin of fine structure


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L.S: origin of fine structure

Vector atom model


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Vector atom modelTotal angular momentum: J L S= +

1( 1) ,2

J j j j l s l= + = + =

, , 1,..., 1, z j j j m m j j j j= = +

Precession of L S & J


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Precession of L, S & JJ is also space quantized

A relation for


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A relation for J 2

Two triangles are not similarand are not parallel

S L

LS

J

=

J L S = +

( 2 )2e

L Sm

= +

( )2e J Sm

= +

projection of on J J =

J

J

= 2e J J J Sm J

+ =

( ) ( ) L L J S J S =

2 J J S S J S= +

1( )

2 J S J J S S L L = +

1( )

22

J J J J S S L Lem J

+ + =

A relation for (contd )


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A relation for J (contd.)1

( )2

2 J

J J J J S S L Le

m J

+ + =

2 2 2 21( 1) [ ( 1) ( 1) ( 1) ]2

2 ( 1)

J J J J S S L Lem J J

+ + + + + +=

+

( 1) ( 1) ( 1)( 1) 1

2 2 ( 1)e J J S S L L

J J m J J

+ + + += + + +

( 1) J B J J J g = +

( 1) ( 1) ( 1)1

2 ( 1)

is the Lande g factor which is needed to calculate the relative

splitting of energy levels in weak magnetic fields

J

J J S S L Lg

J J

+ + + += ++

LS Coupling


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LS CouplingHow to couple angular momenta in many electron atoms

, ,i ii i

L L S S J L S= = = +

H line


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H line

Selection rule

1l =

For many e atoms

11

0

L J

S

= = =

More complications exist Relativistic effects Vacuum fluctuations, etc.

ph-20101 bohr model and qt of hydrogen atom

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