3D Schrodinger Equation
• Simply substitute momentum operator
• do particle in box and H atom
• added dimensions give more quantum numbers. Can have
degeneracies (more than 1 state with same energy). Added
complexity.


2
p  i or x 2  2 
2
2m
2  ( x, y , z , t )  V ( x, y , z , t )  
i
t
• Solve by separating variables
( x, y, z, t )   ( x, y, z ) (t )
 2
2m
 2  V ( x, y, z )  E
P460 - 3D S.E.
1
• If V well-behaved can separate further: V(r) or Vx(x)+Vy(y)+Vz(z).
Looking at second one:
 2
2m
 2  (Vx ( x )  V y ( y )  Vz ( z ))  E
a ssu m e ( x , y , z )  
2
2m
2
2m
( x 2 
2
2
y 2
x
( x )
)  (Vx  V y )
y
( y ) z ( z )

2
2m
 x y z
( x  y ) x y
 (Vx  V y ) 
 x y
2
2
 2
z 2
2
2
• LHS depends on x,y
2
2m
 ( E  Vz )
 x y z
 2 z
 E  Vz
 z z 2
RHS depends on z
 2 d 2 z

 ( E  Vz )  S
 z dz 2
2m
 2
2
2
1
(

) x y  Vx  V y  S
 x y
x 2
y 2
2m
• S = separation constant. Repeat for x and y
P460 - 3D S.E.
2
2
2 d  x
2 m  x dx 2
 Vx  S '  E x
2
2 d  y
2 m  y dy 2
 Vy  S  S '  E y
2
2 d  z
2 m  z dz 2
 Vz  E  S  E z
E x  E y  E z  S '( S  S ' )  ( E  S )  E
• Example: 2D (~same as 3D) particle in a Square Box
V  
V  0
x  0, x  a , y  0, y  a
in sid e b o x
sa tisfies
V  Vx ( x )  V y ( y )
 ( x , y )   x ( x ) y ( y )
• solve 2 differential equations and get
E  Ex  E y 
 2 2
2 ma 2
( n x2  n y2 )
• symmetry as square. “broken” if rectangle
P460 - 3D S.E.
3
E  Ex  E y 
 ( x, y )  A sin

 2 2
2 ma 2
( nx2  n y2 )
n x x
a
|  |2 dxdy  1
sin
n y y
a
nx , n y  1,2..
no rm alizat
io n
• 2D gives 2 quantum numbers.
Level
nx
ny
1-1
1
1
1-2
1
2
2-1
2
1
2-2
2
2
P460 - 3D S.E.
Energy
2E0
5E0
5E0
8E0
4
• for degenerate levels, wave functions can mix (unless “something”
breaks degeneracy: external or internal B/E field, deformation….)
y
 12  A sin ax sin 2
a
 21  A sin 2ax sin ay
 mix   12   21
2   2 1
• this still satisfies S.E. with E=5E0
P460 - 3D S.E.
5
Spherical Coordinates
• Can solve S.E. if V(r) function only of radial coordinate
 2
2M
 2  V ( r )  E ( r ,  ,  )
2
2M
[ r 2r ( r 2
r
2
r2
1
si n 
1
si n 2




r
) 
(sin 
2
 2


) 
] ( r ,  ,  )  V ( r )  E
• volume element is
d (vol)  dr(rd )(r sin d )
P460 - 3D S.E.
6
Spherical Coordinates
• solve by separation of variables
 ( r ,  ,  )  R ( r ) ( ) ( )
( E V ) R 2 M
2
( r 2r

r2
r 2
r
1
sin 2


r

2
2
 2
1
sin 2



sin  

) R
R
• multiply each side by
 r 2 sin 2 
R
P460 - 3D S.E.
7
Spherical Coordinates-Phi
• Look at phi equation first. Have separation constant
1

d2
d 2
 ( )  f ( r ,  )   ml2
• constant (knowing answer allows form)
• must be single valued
( )  eiml
(  2 )  ( )
eiml (  2 )  eiml  ml  0,1,2.......
• the theta equation will add a constraint on the m quantum number
P460 - 3D S.E.
8
Spherical Coordinates-Theta
• Take phi equation, plug into (theta,r) and rearrange. Have second
separation constant
1
R
d
dr
ml2
(
 sin 2 
r 2 dR
dr

)
1
 sin 
2 M 2r2
2
d
d
[ E  V ( r )] 
( sin dd )
 l (l  1)
• knowing answer gives form of constant. Gives theta equation which
depends on 2 quantum numbers.
d
1
sin  d
(
sin  d
d
)
ml2 
sin 2 
P460 - 3D S.E.
 l (l  1)
9
Spherical Coordinates-Theta
d
1
sin  d
(
sin  d
d
)
ml2 
sin 2 
 l (l  1)
• Associated Legendre equation. Can use either analytical (calculus) or
algebraic (group theory) to solve. Do analytical. Start with Legendre
equation
(1  z )
2
d 2 Pl
dz 2
z  cos
 2z
dPl
dz
 l (l  1) Pl  0
Pl  Legendre function
P460 - 3D S.E.
10
Spherical Coordinates-Theta
• Get associated Legendre functions by taking the derivative of the
Legendre function. Prove by substitution into Legendre equation
lml  (1  z
2
)
|ml |/ 2
d |ml | Pl
dz |ml
|
20  P2
21  (1  z
2
)
22  (1  z 2 )
1
2
dPl
dz
d
2
 2 1
Pl
dz 2
• Note that power of P determines how many derivatives one can do.
• Solve Legendre equation by series solution
(1  z
Pl 
d 2P
dz 2
2
)
d
2
 2z
Pl
dz 2
dPl
dz

a
k 0

k
z

a
k 2
k
k
dP
dz
 l (l  1) Pl  0


a
k 1
k
kz k 1
k ( k  1) z k  2
P460 - 3D S.E.
11
Solving Legendre Equation
• Plug series terms into Legendre equation
k 2
k
{
k
(
k

1
)
a
z

[
k
(
k

1
)

l
(
l

1
)]
a
z
} 0

k
k
• let k=j+2 in first part and k=j in second (think of it as having two
independent sums). Combine all terms with same power
{( j  2)( j  1)a j 2  [ j ( j  1)  l (l  1)]a j }z
• gives recursion relationship
a j 2 
j ( j 1) l (l 1)
( j  2)( j 1)
j
aj
• series ends if a value equals 0  L=j=integer
a j 2  0  j ( j  1)  l (l  1)
• end up with odd/even (Parity) series
a1  0, aeven  0 or a0  0, aodd  0
P460 - 3D S.E.
0
12
Solving Legendre Equation
• Can start making Legendre polynomials. Be in ascending power order
l  0, a0  1, a1  0  P0  1
l  1, a0  0, a1  1  P1  z
l  2, a0  1, a1  0, a2 

06
21
j ( j 1)  l ( l 1)
( j  2 )( j 1)
 3  P2  1  3 z 2
• can now form associated Legendre polynomials. Can only have l
derivatives of each Legendre polynomial. Gives constraint on m (theta
solution constrains phi solution)
lml  (1  z )
2

|ml |/ 2
d |m l |
dz |ml |
Pl
| ml | l
P460 - 3D S.E.
13
Spherical Harmonics
 00  1
z  cos 
10  z
1, 1  (1  z
2
)
1
2
 20  1  3 z 2
 2 , 1  (1  z
2
)
1
2
z
 2 , 2  (1  z 2 )
• The product of the theta and phi terms are called Spherical
Harmonics. Also occur in E&M. See Table on page 127 in book
• They hold whenever V is function of only r. Saw related to angular
momentum
Ylm  lm m
 spherical harm onics
P460 - 3D S.E.
14
3D Schr. Eqn.-Radial Eqn.
• For V function of radius only. Look at radial equation. L comes in
from theta equation (separation constant)
1 d  r 2d R 
2


V ( r )  E ) R

2

r dr 
d
r



l (l  1)

R
r2
• can be rewritten as (usually much, much better...)
 2 d 2u
 2 l (l  1)

 (V 
)u  Eu
2
2
2  dr
2
r
u ( r )  rR ( r )
• and then have probability
P ( r )  4R 2 r 2 dr
 4u 2 dr
P460 - 3D S.E.
15
3D Schr. Eqn.-Radial Eqn.
 2 d 2u
 2 l (l  1)

 (V 
)u  Eu
2
2
2  dr
2
r
u ( r )  rR ( r )
• note L(L+1) term. Angular momentum. Acts like repulsive potential
and goes to infinity at r=0 (ala classical mechanics)
• energy eigenvalues typically depend on 2 quantum numbers (n and L).
Only 1/r potentials depend only on n (and true for hydrogen atom only
in first order. After adding perturbations due to spin and relativity,
depends on n and j=L+s.
P460 - 3D S.E.
16
Particle in spherical box
u ( r )  rR ( r )
• Good first model for nuclei
V (r)  0
V (r)  
r a
r a
• plug into radial equation. Can guess solutions
 2 d 2u
 2 l ( l  1)

 (V 
)u  Eu
2  d r2
2
r2
 2 d 2u
 2 l ( l  1)


u  El u
2
2
2 d r
2
r
• look first at l=0
d 2u
dr 2

  k 2u
with
k 
2 ME

u  A sin(kr )  B cos(kr )
P460 - 3D S.E.
17
Particle in spherical box
•
l=0
d 2u
dr 2

  k 2u
with
k 
2 ME

u  A sin (kr )  B co s(kr )
• boundary conditions. R=u/r and must be finite at r=0. Gives B=0. For
continuity, must have R=u=0 at r=a. gives sin(ka)=0 and
Enlm  En 00 
 n 00 
1
2a
n 2 2  2
2 Ma 2
n  1,2....
sin( nr / a )
r
• note “plane” wave solution. Supplement 8-B discusses scattering,
 
phase shifts. General terms are
 ik  r
R( r ) 
P460 - 3D S.E.
e
r
18
Particle in spherical box
• ForLl>0 solutions are Bessel functions. Often arises in scattering off
spherically symmetric potentials (like nuclei…..). Can guess shape
(also can guess finite well)
• energy will depend on both quantum numbers
Enl  E10 E11 E12 E20 E21 E22 .....
• and so 1s 1p 1d 2s 2p 2d 3s 3d …………….and ordering (except
higher E for higher n,l) depending on details
• gives what nuclei (what Z or N) have filled (sub)shells being different
than what atoms have filled electronic shells. In atoms:
Z 2
1S
• in nuclei (with j subshells)
Z 2
6
8
14
2
( He  C  O  Si  S )
16
1s 1 p 3 1 p 1 1d 5
2
4 10 ( He  Be  Ne)
2S 2 P
2
2s 1
2
P460 - 3D S.E.
19
H Atom Radial Function
• For V =a/r get (use reduced mass)

1 d  r 2 dR 
2m 
Ze2
l (l  1)







E
R

R
2 
2



r dr  dr 
  4 0 r
r

• Laguerre equation. Solutions are Laguerre polynomials. Solve using
series solution (after pulling out an exponential factor), get recursion
relation, get eigenvalues by having the series end……n is any integer
> 0 and L<n. Energy doesn’t depend on L quantum number.
En 
 MZ 2 e 4
( 4 0 ) 2 2  2 n 2

 me c 2 2 Z 2
2n2

13.6 eVZ 2
n2
• Where fine structure constant alpha = 1/137 used. Same as Bohr
model energy
P460 - 3D S.E.
20
H Atom Radial Function
• Energy doesn’t depend on L quantum number but range of L restricted
by n quantum number.
l<n  n=1 only l=0 1S
n=2
l=0,1
2S 2P
n=3
l=0,1,2
3S 3P 3D
2 2 2
2
En 
 me c  Z
2 n2

13.6 eVZ
n2
• eigenfunctions depend on both n,L quantum numbers. First few:
R10  e
 Zr / a0
R20  ( 2 
R21 
Zr
a0
Zr
a0
a0 
4 0  2
me e 2

C

 0.5 A
)e  Zr / 2 a0
e  Zr / 2 a0
P460 - 3D S.E.
21
H Atom Wave Functions
P460 - 3D S.E.
22
H Atom Degeneracy
• As energy only depends on n, more than one state with same energy
for n>1 (only first order)
n
l
m
D
• ignore spin for now Energy
-13.6 eV
1
0(S)
0
1
-3.4 eV
2
0
0
1
1 Ground State
4 First excited states
9 second excited states
-1.5 eV
1(P)
-1,0,1
3
0
0
1
3
1
D  n2
2(D)
P460 - 3D S.E.
-1,0,1
3
-2,-1,0,1,2
5
23
Probability Density
|  |2  probability
2
|

|
dVolum e 1  norm alization



2
   | |
0
0
0

1
or 
0
2
r 2 sin  d d dr
2
  | |
1
2
r 2 d d cos dr
0
• P is radial probability density P(r )  r | Rnl |
• small r naturally suppressed by phase space (no volume)
• can get average, most probable radius, and width (in r) from P(r).
(Supplement 8-A)
dP
2
2
m o st p ro b a b le
dr
0
a vera g e  r   r
wid th  r 
P460 - 3D S.E.
r2  r 2
24
Most probable radius
• For 1S state
P ( r )  A r 2 | R |2  A r 2 e 2 r / a0
dP
dr
 0  2 re  2 r / a0 
r  a0
2r2
a0
e  2 r / a0
(" p ea k" )

r 
 rP ( r )d r 
3
2
a0
0
(
n 2 a0
Z
r2 
r 
[1 
r
2
1
2
(1 
l ( l 1)
n2
)]in g en era l)
Ar 2 e  2 r / a0 d r  3a02
3a02 
9
4
a02  0.8 7a0
•
Bohr radius (scaled for different levels) is a good approximation of
the average or most probable value---depends on n and L
• but electron probability “spread out” with width about the same size
P460 - 3D S.E.
25
Radial Probability Density
P460 - 3D S.E.
26
Radial Probability Density
note #
nodes
P460 - 3D S.E.
27
Angular Probabilities
P ( ,  ) |  ( ) |2 | ( ) |2 sin ( )
 m  eim |  |2  1
•
no phi dependence. If (arbitrarily) have phi be angle around z-axis,
this means no x,y dependence to wave function. We’ll see in angular
momentum quantization
 00 "1"
S sta tes
10  A co s
11 
A
2
sin 
P sta tes
2
2
 10
 12, 1  11
"1"
• L=0 states are spherically symmetric. For L>0, individual states are
“squished” but in arbitrary direction (unless broken by an external
field)
• Add up probabilities for all m subshells for a given L get a spherically
symmetric probability distribution
P460 - 3D S.E.
28
Orthogonality
n lm nl m 
  nn ' ll ' mm '
  2

*
2


r
sin  d rd d
nlm
n
'
l
'
m
'

0 0
0
with   Rn lm m
• each individual eigenfunction is also orthogonal.
• Many relationships between spherical harmonics. Important in, e.g.,
matrix element calculations. Or use raising and lowering operators
• example

E  const ant in zˆ
V | E | r cos
note
cos is Legendre polynom ial10
 nlm | r cos | n' l ' m'  
 mm ' l ( l ' 1) f ( r )  0
m  m'
P460 - 3D S.E.
l  l '1
29
Wave functions
• build up wavefunctions from eigenfunctions.
• example
 ( r,  ,  , t ) 
1
( 100 e iE1t /   2 211e iE2t /   211e iE2t /  )
6
• what are the expectation values for the energy and the total and zcomponents of the angular momentum?
E
  H |

*
  H dvol 
*
  i

 dvol
t
• have wavefunction in eigenfunction components
E
L2
Lz
1
1
5
9
( E1  4 E2  E2 ) 
( E1 
E1 ) 
E1
6
6
4
24
1

(l0 ( l0  1)  4  l1 (l1  1)  l1 ( l1  1))
6
1
10

( 0( 0  1)  4  1(1  1)  1(1  1)) 
6
6
1
1
3

( Lz 0  4 Lz1  Lz 1 ) 
( 0  4  1) 
6
6
6

P460 - 3D S.E.
30