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Solving the radial Equation
L2 
2
2
l
 l

1 d2
L2 
ER  
rR   V  r  
R
2 
2
2 r dr
2 r 

2
We want bound states for a given potential
•These will depend on the potential V(r)
•We will only do hydrogen-like atoms
•Also depend on L2 = 2(l2 + l)
•We expect discrete bound states, n = 1, 2, 3, …
2
kZe
dr
V  r     Fdr 
 r2
2
kZe
V r   
r
2
2   E 
d
rR 
 rR 
2 
2
dr
R exp   r a 
Large-r behavior
•Bound states have E < 0
•The V(r) and L2 terms become less important
•Guess: An exponential times a polynomial
•Do TONS of work
R  r   P  r  exp   r
•Note that l < n
P  r   an1r n1  an2r n2  an3r n3 
•Normalize everything
a
 al r l
Rnl  r 
The Radial Wave Functions
•The energy works out to exactly the same formula as the Bohr model
2
•Interestingly, independent of l
2 2 4
13.6
eV
Z


k Z e
•The wave functions are complicated
En  

2 2
2
2 n
n
•They depend on both n and l
•Large r behavior depends on n:
n =1
All have l =0
•Goes like e-r/na
•Roughly, r ~ n2a
n =2
r 2 R2 r
•Small r behavior depends on l
n =3
n =4
•Goes like rl
•Only small l is non-zero
near the origin
l=2
l=3
l=1
2
l=0
All
have
n
=
4
a
 
kZe2 
a0 0.0529 nm
 
Z
Z
r 2 R2  r 
The full wave function (no spin)
 n,l ,m  r, ,   Rn,l  r  Yl .m  , 
The full wave function is given by:
•n, the principal quantum number, tells you the energy
n  1, 2,3,
•l tells you the total angular momentum
l  0,1, 2, , n  1
•m tells you the angular momentum around the z-axis
An electron in hydrogen is in the state 3,2,1
m  l , l  1, , l
•What is the wave function of the electron?
•Where is the particle least likely to be?
  R3,2  r  Y2,1  , 
•Where is the particle most likely to be?
 
1
81  a
2  r 3a
7
r e
sin  cos  e
Least likely: When  = 0:
•At r = 0, sin = 0, or cos = 0
• = 0, ½, 
•On  z-axis and in xy-plane
i
R3,2 
2 2r 2
81 15a
7
e
 r 3a
15
Y2,1  
sin  cos  ei
8
Sample
Problem
Most likely: When || biggest
•|ei| = 1, so ignore this
•Maximize with respect to 
•Maximize with respect to r
•Where is the particle most likely to be?
 
1
81  a 7
r 2 e  r 3a sin  cos  ei
z
   r 2 e r 3a 
2
2
0

cos


sin



  81  a 7 
  14  or
cos  2   0
45º
3
4

   sin  cos    r 3a r r 3a 
0

2re
 e



7
r
3
a

 81  a  
r
2
 0 r  6a
3a
2
6a
x
6a
45º
y
Spin
There is another kind of angular momentum
•Think about the Earth going around the Sun
•Orbital angular momentum
•Intrinsic angular momentum = Spin S

Particles (like electrons) also have spin
S 2  2 s2  s
•The rules are very similar to rules for orbital angular momentum
•Difference #1: Half-integers are allowed for s
S z  ms
•Difference #2: Total spin s is an intrinsic property of the particle
•Electron has spin s = ½.
ms  s, , s
S 2  34 2
To describe one electron, we
ms   12
also need to give s and ms
n  1, 2,3,
But always s = ½
S m
z
To completely describe one electron,
you have to give n, l, m, and ms
s
l  0,1, 2, , n  1
m  l , l  1, , l
ms   12

Nomenclature
To describe one electron, we give the following information:
•n – this is called the shell Shells are labeled by numbers 1, 2, 3, . . .
•l – this is called the subshell  Subshells are denoted by letter s, p, d, f, g
•m – this is called the orbital  orbital depends on choice of axes
•ms – this is called the spin  spin also depends on choice of axes
Normally, just give n and l, like 4p
l
0
If we have one electron in a 4p state:
n  1, 2,3,
1
•What are possible quantum values?
2
•How many states are there?
l  0,1, 2, , n  1
3
m  l , , l
m  l , l  1, , l
n4
4
1
5
m


1,
0,1
m


l 1
s
2
6
ms   12
total  2  3  6
Can easily be generalized to any subshell
•2(2l+1) total states in any subshell
?
s
p
d
f
g
h
i
#
2
6
10
14
18
22
26
Quantum Mechanics of Many Particles
 


2
 r , t   r1 , r2 ,
•The wave functions get very complicated
•There are additional rules about the “symmetry” of the wave function
•Interactions get confusing
 r , r , , r , t   r , r ,
1
2
n
•We will think about electrons one at a time:
•Symmetry condition leads to the Pauli Exclusion Principle
•Can’t have two electrons in the same quantum state
•Interactions between electrons make things complicated
1
Many-electron atoms
•The nuclear charge is large
•Electron charges are relatively small
•Maybe treat each electron as orbiting in a
Hydrogen-like atom
•Goal: Figure out which electrons will be there,
list which states they are in
, rn , t 
-e
-e
+Ze
-e
, rn , t 
-e
The ultra-naïve picture of atoms
•All atoms are (sort of) hydrogen-like
•Normally, they will go into their lowest energy states
•Each electron goes into the lowest possible energy
•List all possibly energy states
•n determines the energy
•For each n, we have l = 0, 1, 2, …, n -1 subshells
•There are 2l + 1 orbitals in each subshell
•You can fit two electrons in each orbital
n=5
n=4
n=3
Fill them up in order,
from bottom to top
g
f
n=2
n=1
d
p
s
Example: Chlorine, Z = 17
How do you decide between 3s, 3p, 3d?
A less naïve view
•If you ignore the interaction of electrons, then 3s = 3p = 3d
•In fact, the inner electrons are so close to the nucleus, the nuclear
charge looks like it is reduced
•Unless the “outer” electron penetrates inside a bit
•Behavior near the nucleus is governed by l
•Small l values can go near nucleus
•Small l values have slightly lower energy
n=6
n=5
n=4
n=3
n=2
n=1
d
p
s
-e
+Ze
-2e
f
Energies in atoms depend on both n and l
•The level with the smaller n + l has lower energy
•In case of tie, level with smaller n wins
1s
2s 2p
List all the possible states in a table, like this
3s 3p 3d
•Draw in diagonal lines, like this
4s 4p 4d 4f
•Make a list in order
5s 5p 5d 5f
•Put a superscript to remember how many electrons 6s 6p 6d 6f
you can fit in each level
7s 7p 7d 7f
1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 6s2 4f14 5d10 6p6 7s2 5f14 6d6 7p6
2 2s2 2p6 3s2 3p6 4s2
1s
Which electron states exist for Tin (Sn, Z = 50)?
3d10 4p6 5s2 4d10 5p2
Filling Up an Atom
n=5
n=4
n=3
d
n=2
n=1
p
s
Does this rule always work?
•104 elements
•83 exactly right (80%)
•18 off by one electron (17%)
•3 off by two electrons (3%)
•Overall: 24/5640 electrons
wrong (0.4%)
The Periodic Table
Chemical properties depend on which type of electrons you are adding
•Example: If outer electron is an isolated s electron, it is farthest out and most easily
shared – alkali metals
•Example: If outermost electrons are 1s2 or np6, we won’t want to add another electron,
since it would go in a higher level – noble gases
•Makes sense to organize atoms by what type of subshell is being filled
s electrons
End of material
for Test 3
d electrons
f electrons
Noble gases
p electrons