FI 3103 Quantum Physics Schrodinger Equation in 3D

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FI 3103 Quantum Physics

Alexander A. Iskandar

Physics of Magnetism and Photonics Research Group Institut Teknologi Bandung

Schrodinger Equation in 3D

The Central Potential Hydrogenic Atom

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Schrödinger equation in 3D

▪ For a 3D problem, we extend the representation of the wave function and potential function of the 1D case into 3D as

▪ And the 3D linear momentum operator becomes

▪ So that the Time-independent Schrödinger equation in 3D is given as

Alexander A. Iskandar Schrodinger Equation in 3D 3

     

, , , ,

x x y z

V x V V x y z

  

 

r r

2 2 2

2 2

2 2 2 2 2 2 2

2

2 2 2 2

2 2 2

x y z

x p p p

p

m m m

m x m m x y z

 

 

 

   

           p

       

2 2

2 V E

m   

  r  r r  r ^H^ˆ

   

^r ^ ^r ^^E^

 

^r

Schrödinger equation in 3D

▪ Interpretation of wavefunction:

probability of finding particle in a volume element centred on r probability density at r

(probability per unit volume) )2

, (r t



3 2

) , ( t d r r

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Hamiltonian for a hydrogenic atom

▪ H-atom is our first example of the 3D Schrödinger equation.

▪ In a hydrogenic atom or ion with nuclear charge +Ze there is the Coulomb attraction between electron and nucleus. This has spherical symmetry –potential only depends on r. This is known as a CENTRAL POTENTIAL.

▪ The Hamiltonian operator is

2 2

2

0

ˆ

2 _e 4

H Ze

m  r

   

2

0

( ) 4

V r Ze

 r

  r

+Ze -e

Hamiltonian for a hydrogenic atom

▪ The natural coordinate system is spherical polars. In this case the Laplacian operator becomes

▪ Then the Hamiltonian becomes

▪ And TISE for H-like atom is

or (m = m_efrom now on)

 

² ² ² ² 2 ² ² 2 ²

0 0

ˆ ˆ

2 _e 4 2 _e 2 _e 4

Ze L Ze

H r

m  r m r r r m r  r

   

          

r

2 2

2

2 1 ˆ

r L r r

r

r 



 







 



2 2 2

2

2 2

0

( ) ˆ ( )

( ) ( )

2 2 4

L Ze

r E

mr r r mr r

   



   

      

r r

     

Hˆ r  r E r

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The angular wavefunction

▪ In the spherical coordinate system, this suggests we look for separated solutions of the form

▪ The angular part are the eigenfunctions of the total angular momentum operator . These are the spherical harmonics, so we already know the corresponding eigenvalues and eigenfunctions

▪ Note: this argument works for any spherically-symmetric potential V(r), not just the Coulomb potential.

   

     

2 2

ˆ , ,

ˆ , 1 ,

z lm lm

lm lm

L Y m Y

L Y l l Y

   



 

l= orbital quantum number.

m= magnetic quantum number (2l + 1possible values).

( ) (r ) R r Y( ) _lm( )

 r        ˆ2

L

The radial equation

▪ Substitute separated solution into the time-independent Schrödinger equation

▪ Using separation of variables method, we obtain the equation for the radial part as

or,

( , , )r R r Y( ) _lm( , )

     

2 2 2

2

2 2

0

ˆ

( ) ( )

2 2 4

L Ze

r E

mr r r mr r

   



   

      r  r

2 2 2

2

2 2

0

( 1)

2 2 4

d dR l l Ze

r R R ER

mr dr dr mr  r

  

    

 

0 ) 2 (

) 1 ( 4

2 2

0 2 2

2 2

 



 



 



 



 





 R r

mr l l r E Ze

m dr

d r dr

d 

 

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The radial equation

▪ Introduce the parameter

▪ The above equation could be recast as

where,

▪ The solution of the above equation is obtained like previous example, i.e. by first finding the asymptotic behaviour (large r as well as small r).

Er m

2

8

  r

0 ) 4 (

1 )

1 ( 2

2 2

2

 



 



 



 



 

 



 r

r

 r

r r

r ^R

l l d

d d

d

E m Ze

2 4 ₀

2

 

 

The radial equation

▪ The behavior of R(r)if r>> :

▪ Thus, we write the solution for R(r) as

 

r e^^r ² or e^^r²

   

⁰ R 4

1

2 2



 R r d

R d

r r

0 ) 4 (

1 )

1 ( 2

2 2

2  



 



 

 



 

 



 r

r

 r

r r

r ^R

l l d

d d

d

 

r e ^r G

 

r R  ^ ²

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The radial equation

▪ With , yields the equation for G(r)as

▪ The behavior asr<< is governed by the equation

▪ With solutions

 

2

 

1



1

  

0

1 ₂

2 2

 



 



 

 

 



 



 

 r

r r

 r

r r

r

r G

d dG d

G

d 

0 ) 4 (

1 )

1 ( 2

2 2

2

 



 



 

 



 

 



 r

r

 r

r r

r ^R

l l d

d d

d

 

r e ^r G

 

r R  ^ ²

 

2

  

1

  

0

2 2

2

 





 r

r r

r r r

r G

d dG d

G

d  

 

r r^or r^^^¹ G

The radial equation

▪ Thus the general solution forG(r)is written as

▪ Substituting, yields the governing equation for H(r)as

▪ Whose solution can be obtained as a series solution.

▪ Further, to have a convergent solution, (= n_r+ l+ 1) has to be an integer, and the solution is known as the Laugerre polynomials.

 

r r H

 

r G  ^

 

₂ ² ² ¹

 

¹

 

⁰

2

 

 



 



  

 r

r

 r

r r

r

r H

d dH d

H

d  

 

2

 

1



1

  

0

1 ₂

2

2  



 



   

 



 



 

 r

r r

 r

r r

r

r G

d dG d

G

d 

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The radial solution

▪ R(r)depends on nand lbut not on m

▪ The associated Laugerre polynomials

) 2

( )

(r G r e^^r R

) 2

( )

(r H r r e^^r

R_l ^l

2 )

1 2 (

1( ) )

(r L _^_ r r e^^r R_nl _n^l_l ^l

The radial solution

▪ R_nl(r)depends on nand lbut not on m

▪ For atomic units set a₀= 1

3/ 2

10 0

0 3/ 2

21

0 0 0

3/ 2

20

0 0 0

3/ 2 2

32

0 0 0

3/

31

0

( ) 2 exp( / )

( ) 1 exp

2 2

3

( ) 2 1 exp

2 2 2

( ) 4 exp

3 3

27 10 ( ) 4 2

9 3

R r Z Zr a

a

Z Zr Zr

R r

a a a

Z Zr Zr

R r

a a a

Z Zr Zr

R r

a a a

R r Z

a

 

   

 

     

      

     

     

       

     

     

      

     

 

  

 

2

0 0 0

3/ 2 2 2

30 2

0 0 0 0

1 exp

6 3

2 2

( ) 2 1 exp

3 3 27 3

Zr Zr Zr

a a a

Z Zr Z r Zr

R r

a a a a

    

     

    

     

        

     

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Hydrogenic Solution

▪ The wave function solution is obtained as with,

▪ And the energy associated with this wave function is

where,

is called the fine structure constant.

2 )

1 2 (

1( ) )

(r L _^_ r r e^^r R_nl _n^l_l ^l

) , ( ) ( ) , ,

(    

 r R_nl r Y_lm

2 2 2

0

2 ( )

) 2( 1 2

4 



  Z

mc E E

m

Ze  

 

137 1 4 ₀

2



 c

e

 



Energy Spectrum and Degeneracy

▪ Recall that we found for a Hydrogenic atom, the energy is given by

where we introduced the principle quantum number nas with

▪ Thus, the ground state, n= 1has only one possibility

▪ While, the first excited state, n= 2has two possibilities

2 2 2 2

1 ( )

n Z E_n mc 





1



n l n _r

 

 1 1,2,

, 1 , 0

, 2 , 1 ,

0     







 n n l

k n l

r r

l n_r 0

0 , 1 and

1 ,

0   

 l n l

n_r _r

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Energy Spectrum and Degeneracy

n= 1 n= 2 n= 3

n= 2