Chapter8. Optical spectraPart 1

EE 430.423.001 2016. 2^nd Semester

2016. 11. 24.

Changhee Lee

School of Electrical and Computer Engineering Seoul National Univ.

chlee7@snu.ac.kr

Chapter8. Optical spectra

Part 1

EE 430.423.001 2016. 2^nd Semester

8.1 General Remarks

Ground state Excited state

A₂₁: Spontaneous emission B₂₁: Stimulated emission

B₁₂

absorption

B₂₁, A₂₁

1 2

hn hn

Absorption and Emission of Light Scattering of Light

hn hn

EE 430.423.001 2016. 2^nd Semester

A. Beiser, Concepts of Modern Physics, 6th ed., McGraw-Hill, New York, USA, 2003, Chapter 3

Atomic spectra

EE 430.423.001 2016. 2^nd Semester

8.2 Elementary theory of atomic spectra

Assumptions of the Bohr model (1913)

1. The electrons of an atom can occupy only certain discrete quantized states or orbits.

2. The electrons can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency ν determined by the energy difference of the levels according to the Planck relation:

http://en.wikipedia.org/wiki/Bohr_model

n h E

E

E   



3. The allowed orbits depend on quantized (discrete) values of orbital angular momentum, L.

...

, 3 2 1 number

quantum principal

2 )

(

, , n

n h mur

rp L







 

...) , 3 2 1 (

)

(

2 n n , ,

p n h

r    



EE 430.423.001 2016. 2^nd Semester

The Bohr atom

r mu r

e

o

2 2

2

4

1 



The electron velocity is related with its orbit radius

+ r

Assume a circular electron orbit

-e

mr u e



 4

Total energy of the electron

r e r

mu e V

T E

o

o



 8 4

2

1













Condition for orbit stability ^{o n}

r

m r e

nh mu

n   nh  4   2 

Orbit radii in the Bohr atom ₂ ²

( 1 2 3 , ...)

2 2

, , n

n me a

h

r

 n









A 529 . 0

radius Bohr

o 1



 r a

Energy levels

constant) (Rydberg

J 10 18

. 2 eV 6 . 8 13

...

3, 2, 1, n

1 )

8 ( 8

18 2 2

4

2 2 2

2 4 2























h R me

n R h n

me r

E e

o

n o o n



 

EE 430.423.001 2016. 2^nd Semester

The Bohr atom

731 , 973 , 10

Hz 10

29 . 3 /

1 ) ( 1

1 15

2 2 2

1 1

2

 











 



ch m R h R

n h n

R h

E

n

n₁=1 n₁=2, 3, 4, . . . Lyman series (far UV)

n₁=2 n₁=3, 4, 5, . . . Balmer series

(visible and near UV) n₁=3 n₁=4, 5, 6, . . . Paschen series

(near IR)

n₁=4 n₁=5, 6, 7, . . . Brackett series (IR) n₁=5 n₁=6, 7, 8, . . . Pfund series (IR)

EE 430.423.001 2016. 2^nd Semester

The Bohr atom

Effect of a finite nuclear mass

, (reduced mass) 8

4

M m

mM h

R e

o

H

   





Spectra of the alkali metals

defects quantum

,

) (

1 )

(

1

2 1

2 2 2

2 1 1







n 



 





 

 

n n

h R

The principal series is the most intense in emission and is also the one giving the strongest absorption lines when white light is passed through the vapor of the metal.

In the fundamental series, the quantum defects are very small. As a consequence the frequencies of the lines of this series are very nearly the same as those of the corresponding series in hydrogen.

EE 430.423.001 2016. 2^nd Semester

8.3 Quantum mechanics

The quantity whose variations make up matter waves is called the wave function, solutions of the Schrödinger’s wave equation.

) , ( t r 



The probability of experimentally finding the body described by the wave function  at the point x, y, z at the time t is proportional to the value of ||

there at t.

(Max Born, 1926)

density y

probabilit )

,

( ² 

 r  t

functions normalized

, integrable lly

quadratica

*

1

  













  dxdydz 

All state functions satisfy the time-dependent Schrödinger equation:

) , ( )

,

( r t H r t

i t  

   





EE 430.423.001 2016. 2^nd Semester

Stationary states

) ( )

( )]

2 (

[

2

r E

r r

m V







     



2

2

( )

) ,

( r t

r

n



  



Probability distribution for an eigenstate is independent of time

t i E n

n

e

z y x t

r 





 ( , ) ( , , )

coherent states

t i E t

i E

e c

e

c





  

h E E

E E

e c

c e

c c c

c c

c

1 2

1 2

1

* 2 1

* 2 2

* 1 2

* 1 2

* 2 2

* 2 1

* 1 1

* 1

*

or



 















n





















8.3 Quantum mechanics

EE 430.423.001 2016. 2^nd Semester

8.4 The Schrödinger wave equation

0 )

8 (

) ( )

( )]

2 ( [

2 2 2

2 2

















 





 V m E

r E

r r

m V











relation) Broglie

(de h k

p   



equation) (Planck

 n  

 h E

particle) free

a of equation n

(dispersio 2

)

(

m

 k

  





 r  t A k  e

d k  )

2 ( ) 1

,

(









 



 











 

 







 

 





















2 2 )

( 2 2

) ( 2 2 2

2

2 , 0 2 )

)(

2 ( 1

2 ) )(

2 ( ) 1

, ( 2 )

(

m i t

k d m e

k k A

k d m e

i t k A t

m r i t

t kx i

t kx i

 

 

 

 

 



 





 



EE 430.423.001 2016. 2^nd Semester

8.5 Quantum mechanics of the hydrogen atom

r r e

V

V h E

r r r r

r E r

r m V

r H



 

 





 























) 4 (

0 )

8 ( sin

) 1 sin (sin

) 1 1 (

) , , ( )

, , ( ) 2 (

) , , (

2

2 2 2

2 2

2 2

2 2









 



 







 







 









 



 



   

 

1 sin

) , ,

(



 

^{r   r dr  d  d }

EE 430.423.001 2016. 2^nd Semester

8.5 Ground state of the H atom

r H

H r

o

H o

o

r

o r

r

e a

a C

dr r e

C

h E e

a h

e

h e h

E

r e h

e e h

E

Ce





 



 











 

 





 

 





 





 

 



















 



 



 

 



 



 





100 3

3 2 0

2 2

2 4 2

2

2 2 2

2 2

2 2 2

2 2

1 1

1 4

8

1 2 2 8 ,

0 2 2

8

EE 430.423.001 2016. 2^nd Semester

Excited state of the H atom

0 )

8 ( sin

) sin (

) sin (sin

1

) ( ) ( ) ( )

, , (

2 2 2 2

2 2

2

2

    

 











 

 



 

















V h E

r dr

r dR dr

d R

d d d

d d

d

r R r

number quantum

magnetic .

. . 3, 2,

1, 0,

) 1 ( )

8 ( ) 1 (

) 1 ( )

sin (sin 1 sin

1

2 2 2 2

2 2

2 2

2























 

 



 



m e

l l V

h E r dr

r dR dr

d R

l d l

d d

d m

d m d

im

 



 









r r e

V



) 4 (





EE 430.423.001 2016. 2^nd Semester

Eigenfunction of angular momentum

) 2 (

) 1 ,

( 

 



m

l

e

Y  

0 ) ( sin ]

) 1 ( [ ) sin (sin

1

2

2

 





 

 

 





l m d l

d d

d

The regularity in the solution Θ at the poles of the sphere, where θ=0, π, forces the eigenvalue to be of the form ℓ(ℓ+1) for some non-negative integer with ℓ ≥ |m| .

The associated Legendre polynomials are the canonical solutions of the general Legendre equation.

) 1 1

(

cos  

 - x

x  ^{] 0}

) 1 1 ( [ ] )

1

[(

2

2

 

 



 

 x

l m dx l

x d dx

d

formula) '

(Rodrigues s

polynomial Legendre

: ) 1

! ( 2 ) 1 (

) ) (

1 ( ) 1 ( ) (

2

2 / 2

l l

l l l

m l m m

m m

l

dx x d x l

P

dx x P x d

x P











The ( − 1)

factor is known as the Condon–Shortley phase and some authors omit it.

EE 430.423.001 2016. 2^nd Semester

Properties of the Legendre polynomials

Liboff, Chapter 9

EE 430.423.001 2016. 2^nd Semester

L

L ˆ and ˆ of

ions eigenfunct

common

, 2 , 1 , 0

, ) 1

ˆ

Y 

l ( l  Y l     L



, , 1 , 0 , , ) 1 ( ,

ˆ Y mY , m l - l- l

L

 

       

Eigenfunction of angular momentum

).

( )

( )

,

(  





m

Y

  



 



m

l

P e

m l

m l

Y l (cos )

)!

(

)!

( 4

1 ) 2

, (

2 / 1

 

 









 

' ' 1

1

2 0

* ' 4 '

* '

'

) cos ( )

(

m

l

Y d d d Y Y

Y 

  



    



)

( ) 1 ( ) ,

(

m

l

Y

Y

   

spherical harmonics

 

 4

1 ) 2

,

(



 





Y l

l

l m

m l

EE 430.423.001 2016. 2^nd Semester

Eigenfunction of radial equation

The radial differential equation can be solved by the standard method of power series expansion. Its solutions are well known ‘‘associated Laguerre functions,’.

2 ) (

2

4 )

(

4 2

) (

1 ) (

1 )

(

1

) (

) (

) (

constant.

ion normalizat a

is

and 2

where ),

( )

(

2 2

2 0

2 1

2 0 1 1 1 0 0

1 1

2 1

2

1 2 2

/























 





 





 

 



 



 



 











































 

 













L L L L L L

d e e d d

L d

A na r

L e

A r

R

l n n

l l

l n

nl H

l l n l nl

nl

EE 430.423.001 2016. 2^nd Semester

) , ( ) ( )

, ,

( r    R

r Y

 



Schrödinger equation for the H atom

1 ) 8 (

8

4 2

h n e r

E e

n o o

n





 ^{ }





Each of the bound states is specified by three quantum numbers:

, l l

, . . . , , . . . ,

l l, m

n , . . . , ,

, , l

, . . . , ,

, n

l

( 1 ) 0 ( 1 )

number quantum

nagnetic

) 1 (

3 2 1 0

number quantum

orbital

3 2 1 number

quantum principal

























1



0

)

1 2 (

n

l

n l

Each energy level corresponding to the principal quantum number n has a total of n² fold orbital degeneracy.

EE 430.423.001 2016. 2^nd Semester

Excited state of the hydrogen atom

R. A. Serway, C. J. Moses, C. A. Moyer, Modern Physics, 3^rd ed., Thomson Learning, Inc., USA, 2005



 





r

o o

e a e

r a

Y r

R

 



  ) sin

( 8 ) 1

, ( )

(

3 1

1 21

1 , 21

 





 ) cos

( 4 2

) 1 , ( )

(

3 0

1 21

210

ao

r

o o

a e r a

Y r R





EE 430.423.001 2016. 2^nd Semester

Hydrogen atom wavefunctions

EE 430.423.001 2016. 2^nd Semester

Radiative transitions and selection rules

EE 430.423.001 2016. 2^nd Semester

Radiative transitions and selection rules

rule) selection

- ( 1 , 0

rule) selection

- ( 1

*

m m

l l

dxdydz e

A AB













  ^{ r }

M