Radial Wave Functions and Energy Levels for the Coulomb Potential in Quantum Physics, Study notes of Quantum Mechanics

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Quantum Physics 2005

Notes-

Three-dimensional Schrodinger Equation

The Schrodinger Equation

2 2

2

ˆ ˆ

( , )

2

H E

V x t i

m x t

! =!

" "# $ # + # =

" "

h h

Here’s what we have been working with:

Here’s what the 3D equation is:

2 2

ˆ ˆ

( , )

2

H E

V r t i

m t

! =!

"# $ % # + # =

"

h r r h

Cylindrical

Converting to Cartesian coordinates:

! Line element:

! Length ds:

! Volume element:

'ˆ, &ˆ, z ˆ

x y y x

x y z z

( ) , tan /

cos , sin ,

2 2 1 / 2 = + =

= = =

' &

' & ' &

= ' ˆ^ ' + ' & ˆ^ & + ˆ

r dl d u d u dzu z

= + & +

2 2 2 2 2 ds dr r d dz

dV = ' d ' d & dz

2 2 2 2 2 2

1 ˆ ˆ ˆ

1 1

u u u z z

z

' &

" " " % = + +

" " "

" " " " % = + +

" " " "

r

Spherical

2 2 2 2 2 2 2 2

1 1

1 1 1

ˆ ˆ ˆ sin

sin sin sin

& ( & & (

& & & & & (

" " " % = + + " " "

" (^)  " (^)  " (^)  " (^)  " % = (^)   + (^)  + " (^)  " (^)  " (^)  " (^)  "

r u r u u r r r

r r r r r r

x = r sin & cos (; y = r sin & sin (; z = r cos&

2 2 2 2 2 ; tan / ; tan

x y r x y z y x z

( &

= + + = =

dl = drr ˆ^ + rd &&^ ˆ + r sin& d ((ˆ

r

2 dV = r sin & drd d & (

5-11 The 3-D Box



















 



 

 ) $

 



 

 $

 



 

 $

=

otherwise in

in

in

2

, 2

2

, 2

2

, 2

0

( , , )

c c x

b b y

a a x

V x y z

The 3D Box problem is a straightforward extension of the

1D infinite well or 1D box problem where















) >

 



 

 $

=

2

2

, 2

0

( ) a x

a a x

V x

in

1- D x

P

V ( x )

x i x

P "

"

h

t

E i "

"

• h

V x E m

Px

• ( )= 2

2





















t

V x i

m

i x

h

h

(^2 ) ( ) 2 2

i V x m (^) x t

"^ " $ # + # = #

" "

h h

Notes 8 Quantum Physics F2005 10

1- D

• Volume element d + = dx
• Probability of finding a particle in d + at time t

∫

) P x t = (^) $) # x t dx

2 ( , ) ( , )

• Normalization condition

1

2 ∫ (^ , ) =

) $) #^ x t dx

• Stationary State wave function

h

iEt

xt x e

$

( , ) = !( )

• Time independentSchro&&dinger equation

! V x! E!

m (^) x

• = "

" $ ( ) 2

2

2

h^2

• Eigenfunction

( ) x

n

!

Notes 8 Quantum Physics F2005 11

3 - D

• Volume d + = dx dy dz
• Probabilityof findingaparticlein d + at time t

P x y z t x y z t dx dy dz

2 ( , , , )=#( , , , )

• Normalization conditon

( , , , ) 1

2 ∫ ∫ ∫ # =

) $)

) $)

) $) dx dy dz x y z t

• Stationary state wave function

h

iEt

x y z t x y z e

$

( , , , ) = !( , , )

Eigenfunction ( x ) n

-!

• Time independent Schrodin&& ger equation

! x y z V x y z! E!

m

$ %^2 ( , , )+ ( , , ) = 2

h^2

( )

3

( ),

2

( ),

1

Eigenfunction z n

y n

x n

-!!!

3D

( )

3

( )

2

( )

1

( , , )

1 2 3

z n

x n

x n

x y z n n n

! =!!!

2

, 2

, 2

c z

b y

a x , , ,

x a

n

a

x y z n n n

! 1

cos

sin

2 ( , , )

1 2 3



 





 



y b

n

b

cos 2 sin

2 

 





 



z c

n

c

cos 3 sin

2 

 





 



2

, 2

, 2

c z

b y

a x > > >

( , , ) 0

1 2 3

x y z = n n n

!

c m

n

b

n

a

n

n n n

E 2

2 2

2

3

2

2

2

1

1 2 3

h -









































































































= + +

• Symmetry hereisinterchangecoordinates x & y.

See Fig. 5-

112 211 , 121

$ $

a = b. c boxcase

E.g. 2 Fig. 5-

a = b = c boxcase$ 211 , 121 , 112

See Figure 5 - 27 a. b. c box

121

211

112

$

$

$

• Degeneracy here is two DISTINCT eigenfunctions

having the SAME energy value.

The ( equation

( solution for central potential

2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2

1 1 1 1 1 1 1 2 sin sin sin

1 1 2 ( ( )) 1 sin sin sin sin

Because the wavefunction must

l

l

im

mE r r r r r r

R

R m E V r r r m R r r

e

(

& & & & & (

& & & & & & ( ±

" (^)  "# (^)  " (^)  "# (^)  " # % # = (^)   + + = $

# " " # " ^ "  # "

   

= /

" (^)  " (^)  " (^)  "/ (^)  $ " 0 $ (^)   $ $ = = $ " " # " ^ "  0 "    

0 =

h

h

be single valued ml must be an integer.