Transcript 3D Schrodinger Equation

3D Schr. Eqn.:Radial Eqn.
• For V funtion of radius only. Look at radial
equation
r 2 dr
1 d
( r drdR )  2M2 [V (r )  E ]R  l (rl21) R
2
• often rewritten as
u (r )  rR (r )
 2M
2
dr 2
d 2u
 [V  2M
2
]
u

Eu
l ( l 1)
r2
• note l(l+1) term. Angular momentum. Acts like
repulsive potential (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. II
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Particle in spherical Box
• Griffiths Example 4.1. E&R section 15-8. Good
V (r )   r  a
first model for nuclei
V (r )  0 r  a
• plugdr into radial equation
r
 2M
2
2M

[
V


u
2
d2
2
• look first at l=0
2
l ( l 1)
d 2u
dr 2
]u  Eu u (r )  rR (r )
  k 2u

with
k
2 ME

u  A sin(kr)  B cos(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
En 0 
n 2 2  2
2 Ma 2
 n 00 
n  1,2,3....
sin( nr / a )
1
r
2a
• note plane wave solution are
e
 
 ik  r
r
P460 - 3D S.E. II
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Particle in spherical Box
• For l>0 solutions are Bessel functions (see
Griffiths). Often arises in scattering off spherically
symmetric potentials (like nuclei…..)
• 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 nucleii (what Z or N) have filled
(sub)shells being different than what atoms have
filled electronic shells. In atoms:
Z  2 4 10 ( He  Be  Ne)
1S 2S 2P
• in nuclei (with j subshells)
Z  2 6 8 14 16 ( He  C  O  Si  S )
1s 1 p 3 1 p 1 1d 5 2s 1
2
2
2
2
P460 - 3D S.E. II
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