Physics 451
Quantum mechanics I
Fall 2012
Nov 9, 2012
Karine Chesnel
Phys 451
Announcements
• HW #18 today Nov 9 by 7pm
Homework next week:
• HW #19 Tuesday Nov 13 by 7pm
• HW #20 Thursday Nov 15 by 7pm
Phys 451
The hydrogen atom
How to find the stationary states?
nlm r ,  ,    Rnl (r )Yl m ( ,  )
Step1: determine the principal quantum number n
1
kn 
na
Step 2: set the azimuthal quantum number l (0, 1, …n-1)
Step 3: Calculate the coefficients cj in terms of c0
(from the recursion formula, at a given l and n)
Step 4: Build the radial function Rnl(r) and normalize it (value of c0)
Step 5: Multiply by the spherical harmonics Yl ( ,  ) (tables)
and obtain 2l +1 functions nlm for given (n,l)
m
(Step 6): Eventually, include the time factor:

 (r , t )  nlm (r ,  ,  )e  iEnt / 
Phys 451
The hydrogen atom
Representation of
nlm r , ,  
Bohr radius
4 0 2
10
a

0.529

10
m
2
me
Quantum mechanics
The hydrogen atom
 nlm  r, ,   Rnl  r  Yml  , 
Expectation values
r   r R r 2dr
2
r
2
  r R r dr
2
2
2
Pb 4.13
x   d  d sin   r cos  sin  R r 2dr
2
Most probable values
Pb 4.14
 r 
2
d  r2
2
2
max
dr
0
Quantum mechanics
The hydrogen atom
 nlm  r, ,   Rnl  r  Yml  , 
Expectation values for potential
 e 2  2 2
V  
 R r dr
 4 0 r 
Pb 4.15
Phys 451
The angular momentum
Lrp
 Lx , Ly   i Lz
 Ly , Lz   i Lx
 Lz , Lx   i Ly
 L2 , Lx    L2 , Ly    L2 , Lz   0
Pb 4.19
Phys 451
The hydrogen atom
Representation of
nlm r , ,  
Anisotropy
along Z axis
Phys 451
The angular momentum
L  Lx  iLy
Ladder operator
 Lz , L   L
 Lz , L    L
 L2 , L   0
• If
f
eigenvector of L2, then
• If
f
eigenvector of Lz with eigen value m
then
L f eigenvector of L2, same eigenvalue
L f eigenvector of Lz, new eigenvalue
m
Phys 451
The angular momentum
L  Lx  iLy
Ladder operator
L2  L L  L2z
Top
Value
=+l
Lz
m
m
Eigenstates
f l m  Yl m
m
L
Lz f l m  mf l m
L2 fl m 
2
l (l  1) f l m
L fl m   lm fl m 1
L
Bottom
Value = -l
Pb 4.18
Phys 451
Quiz 25
When measuring the vertical component
of the angular momentum (Lz )
3 2
of the state L Y5 , what will we get?
A. 0
B.
C. 2
D.
3
E.
5
Phys 451
The angular momentum
in spherical coordinates
z


r
Lrp
r  

i
y
x



1
 
L  r  r  r   r 

r 
i
r
 sin 
 

 
1  
L  


i  
sin   




Lz 
i 
Phys 451
z


The angular momentum
In spherical coordinates
r
y
L  Lx  iLy
x
L   e
 i
 
 
 i cot 


 
 
2


1


1



2
2
L  
 sin 
 2
2 
  sin   
 sin   
Pb 4.21, 4.22
Phys 451
z

The angular momentum
eigenvectors
r
y

Lz fl m 
x
and
L fl  
2
m
2
 m
fl  m fl m
i 
 1  

sin



sin






1 2  m

f 
 2
2  l
 sin   
2
l (l  1) fl m
were the two angular equations for the spherical harmonics!
Spherical harmonics
are the
eigenfunctions
H nml  En nml
L2 nml 
2
l (l  1) nml
Lz nml  m  nml
Phys 451
z


The angular momentum
and Schrödinger equation
r
y
x
1
2mr 2

  2   2

 r  r r   L   V  E




3 quantum numbers (n,l,m)
• Principal quantum number n: integer
• Azimutal and magnetic quantum numbers (l,m)
can also be half-integers.