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Bohr model of atom
How to see
Early modelsAtomic spectra
The Bohr model
Correspondence principleDemerits…
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Early models of atom
Plum pudding model of 1890s
Rutherford’s 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 electron
should, classically, spiral
rapidly into the nucleus
as it radiates energy due
to its acceleration
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The Bohr atom
Electron waves & stationary states
de Broglie wavelength of e : hmv
Centripetal Electric F F 2 2
2
0
1
4
mv e
r 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 orbits
Condition for orbit stability: 2 1, 2,3,nn r n
042n
n
r nhr
e m
2 20
2Orbital radii in Bohr atom: 1,2,3,...n
n hr n
me
11 2
0 1 0Bohr radius: 5.292 10 ; na r m r n a
Angular momentum quantization (alternate approach)
h
mv 2
n
n r
2( )( / )mr v r mvr L I n2
hn
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Energy levels
2 2
04
mv e E KE PE
r r
04
ev
mr
2
04n
n
e E
r
4
12 2 2 2
0
1 1,2,3,...8
n E me E nh n n
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12 2
1 1 1:l u
E Hydrogen spectrumhc n n
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Correspondence principle
The 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: Demerits
Applicable only to hydrogen and other one-
electron ions such as He+
and Li2+
Cannot explain why some lines are more
intense than others
Cannot explain why many lines consist of several separate lines whose wavelengths
differ very slightly
No light on how individual atoms interact Quantum 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
• Schrodinger’s eqn. for Hydrogen atom • Separation of variables
• Quantum numbers
• Electron probability density• Selection rules
• Zeeman effect
Q t h i
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Quantum mechanics: recap
Explores probabilities instead of asserting
Eg. Hydrogen atom: r g.s. from Bohr theory = 5.3×
10-11
m QMMost probable r g.s.= 5.3×10-11 m
Wave function
Ψ itself has no physical interpretation
|Ψ|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
2
Schrodinger equation: (1D)2
i U t m x
S h di ’ ti
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Schrodinger’s equation
2 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 Schrodinger’s equation becomes
2 2
2 2 2 2
2 2
0
sin sin sin
2 sin0
4
r r r
mr e E
r
S ti f i bl
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Separation of variablesHydrogen atom wave fn.: ( , , ) ( ) ( ) ( )r R r
Simply, if R R dRr r dr
2 2
2 2Similarly ,
d d R R
d d
Substituting above in Schrodinger’s eqn. and rearranging,
2 2
2 2 2 2
2 2
0
sinsin sin
2 sin 14
d dR d d r
dr dr d d
mr e d E r d
22
2
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
l md 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 : 0l
d m
d
2
2
1Equation for : sin ( 1) 0
sin sin
l md d l l
d d
22
2 2 2
0
1 2 ( 1)Equation for : 0
4
d dR m e l l R r E R
r dr dr r r
Q t b
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Quantum numbers
The solution of equation for is given by ( ) l im Ae
Exploiting the symmetry that and + 2 identify the same
Plane, we have,
( ) ( ) ( 2 )l l im im Ae Ae 0, 1, 2, 3,...m
The differential equation for has a solution provided:is an integer and 0, 1, 2,...,l l l m m l
The final solution of radial part yields,
4
12 2 2 2 2
0
1; 1,2,3,... ; 1
32n
E me E n n l
n n
0,1,2,..., ( 1)l n
Thus the principal (n), orbital (l ) and magnetic (m)
quantum numbers are defined
O bit l t b
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Orbital quantum number 2
2
2 2 2
0
1 2 ( 1)Equation for : 0
4
d dR m e l l R r E R
r dr dr r r
E includes electron’s orbital kinetic energy also !!
radial orbital E KE KE U 2
04
radial orbital
e KE KE
r
22
2 2 2
1 2 ( 1)0
2radial orbital
d dR m l l r KE KE R
r dr dr mr
If R(r) has to be an exclusive function of r, 2
2
( 1)
2
orbital
l l KE
mr
21
2orbital 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 ti t 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 t i t i i l & L
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Uncertainty principle & Lz
Why only L z is quantized?
L can never point any specific direction
but in cone where L z =ml
If not the uncertainty principle will be
violated
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 t b bilit d it
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Electron probability density
No definite orbits
QM i f t
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QM view of atoms
The orbitals
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The orbitals
Selection rules
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Selection rules
*Allowed transitions: 0 , ,l l n l m nlmu u x y z
Transitions not obeying above condition are forbidden transitions
Selection rules: 1l 0, 1l m
( )l n l m
( )l nlm
Interaction with magnetic field
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Interaction with magnetic fieldThe torque t on a magnetic dipole in a
magnetic field of flux density B is
sin { B r F t
Potential energy 0 when / 2.mU
/2
For other orientations mU d
t
/2
sin B d
cos B
IA 2ef r 2v r r f
22 L mvr mfr
Electron magnetic moment
2
e L
m
Gyromagnetic ratio
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cosmU B
2
e
Lm
cos2
m
eU LB
m
( 1) L l l
cos( 1)
l m
l l
2m l
eU m B
m
Bohr magneton:2
B
e
m
In a magnetic field, the energy of a
particular atomic state depends on alsol m
m l BU m B
Zeeman effect
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Zeeman effect
In a magnetic field, the energy of a
particular atomic state depends on alsol m
a level with unique ' ' splits into
different levels having different ' 'l
n
m
1 0 0
2 0
3 0 0
4 Normal Zeeman
effect
4
B
B
B ev v v Bh m
v v
B e
v v v Bh m
0, 1l m
m l BU m B
0 m E U E v
h h
0 l Bv v m B
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Anomalous Zeeman effect
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Anomalous Zeeman effect
The previous QM treatment could not explain both anomalous Zeeman effect and
fine structure
Two Dutch graduate students (Samuel Goudsmit & George Uhlenbeck) proposed in
1925 that
Every e- has an intrinsic angular momentum, called spin, whose magnitude is the
same for all electrons. Associated with this angular momentum is a magnetic
moment
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 an
incorrect magnitude for the magnetic moment, incorrect
quantum numbers, and too many degrees of freedom. Spinarises from relativistic dynamics.
1
2 z s
S m
Spin magnetic moment: S
eS
m
2Sz B
e
m
Stern Gerlach experiment
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Stern-Gerlach experiment
Cause for deflection: cos z S
dB F
dz
Magnetic 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
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Vector atom model
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Vector atom model
Total 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 & J
J is also space quantized
LS Coupling
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LS Coupling How to couple angular momenta in many electron atoms
, ,i i
i i L L S S J L S
H line
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Ha
line
Selection rule
1l
For many e atoms
11
0
L J
S
More complications exist• Relativistic effects
• Vacuum fluctuations, etc.