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Note-4. Schrodinger equation in three dimensions
We will consider problems where the partial differential equations are separable.
4-1. Free particles in a box--separable in Cartesin coordinates
If the particle is confined in a box L3, clearly the wavefunction is given by
n z
n x
n y
2
un1n2n3 ( x, y , z )  ( ) 3 / 2 sin 1 sin 2 sin 3
(1)
L
L
L
L
and the energies are given by
 2 2 2
(2)
E
(n1  n22  n32 )
2
2mL
Thus there are three quantum numbers, n1,n2, n3 to denote a give state and since the
energy depends is given by (2), there are degeneracy in that different eigenstates can have
the same energy.
4-2. Separable in Spherical coordinates, ( r,  ,  )
If the potential seen by the particle depends only on the distance r, then the
Schrodinger equation is separable in Spherical coordinates.
Some key equations:
2 2
H 
  V (r)
2m

[ H , L]  0
[ H , L2 ]  0
[ Lx , L y ]  iLz
Choose eigenstates of H, L2 and Lz gives quantum numbers n, , m . ( This is just a
convention.)
The eigenstates are given by
Em ( r,  ,  )  RE ( r )Ym ( ,  )
L2Ym   2 (  1)Y m
where
LzYm  mYm
and
The radial wavefunction RE (r ) satisfies
 d2 2 d 
2 
l(l  1) 2 
2E
 2 
 R nl (r )  2 V(r ) 
 R nl (r )  2 R nl (r )  0
2
r dr 
 
2r 

 dr
Note that at large r, the centrifugal potential goes like 1/r2. Thus it is important to
distinguish potential V(r) which drops faster than 1/r2 or not. In general, if the potential
decreases exponentially with r at large r, we called that it is a short-range potential. For
Coulomb interaction it goes like 1/r, it is a long-range potential.
By writing
un ( r )  rRn ( r )
The equation for un (r ) is
d 2 u nl (r ) 2 
l(l  1) 2 

E

V
(
r
)


 u nl (r )  0
dr 2
2 
2r 2 
Here n is a quantum number for the radial equation. In general, the energy E depends on
n and  .
4-3. Spherical harmonics
A good place to find the summary of spherical harmonics is
http://mathworld.wolfram.com/SphericalHarmonic.html
Spherical harmonics are used when there are spherical symmetry.
For diatomic molecules, for example, there is no spherical symmetry, then one also
uses, for example (for m>0 only),
1
i
*
*
Y,cos 
[Ym  Ym ]
Y,sin  
[Ym  Ym ]
2
2
For   1 , they are called px and py, respectively. In this case, Y10 is proportional to pz.
The parity of Ym is (1)  . For small m for a given  , the function peaks more along
the z-direction (quantization axis). For the maximum m, it peaks perpendicular to it.
Terminology in spectroscopy:
  0, 1, 2, 3, 4, …. are called s, p, d f, and g, ….
4-4. Free-particle solutions
For V(r)=0,
 d 2 2 d l(l  1) 
 2  R (r )  k 2 R (r )  0
 2
r dr
r 
 dr
Using   kr,
d 2 R 2 dR l(l  1)


RR 0
d 2  d
2
There are two independent solutions, j (  ) and n (  ) , the spherical Bessel functions
and spherical Neumann functions, respectively. To first order, at large  , they are like
sine functions and cosine functions, respectively. At small  , j (  ) is finite but
n (  ) diverges.
One can also uses a separate set of two independent solutions, called spherical Hankel
functions
(1, 2 )
h (  )  j (  )  i n (  )
These two functions represents a spherical wave going outward (outgoing wave) or a
wave coming toward the center (ingoing wave) at large distances.
Take a look at these functions if they are unfamiliar to you on the web or in your
textbooks.
4-5. The 3D infinite potential well
If the particle is confined to a sphere of radius a, clearly the radial wavefunction which
if finite at r=0 is given by j (kr) . The condition that it vanishes at r=a requires that
j (ka)  0
Thus the allowed energies are related to the zero's of the spherical Bessel functions.
4-6. The expansion theorem for a plane wave
Recall that Ym ( ,  ) are eigenstates of L2 and Lz in the Hilbert space of the two
spherical angles. Thus any function in  ,  can be expanded in terms of the complete set
of functions of Ym ( ,  ) .
 
A plane wave giving by eik r  eikr cos can be expanded as

e ikr cos    2l  1 i l jl kr  P1 cos 
l0
This equation will be useful for discussion scattering where the incident wave is a plane
wave and the scattered wave is a spherical wave.
===========================================
Homework #4
4-1. Check the definition of L+ and L- and their operations on Ym ( ,  ) . Calculate
Ylm1 L x Ylm2 and Ylm1 L y Ylm2 .
4-2. Calculate Ylm1 L2x Ylm2 and Ylm1 L2y Ylm2 .
4-3. The Hamiltonian for an axially symmetric rotator is given by
H
L2x  L2y
2 I1

L2z
2l 2
What are the eigenvalues of H? Sketch the spectrum, assuming that I1 I 2 .
4-4. The three-dimensional flux is given by


 * (r ) (r )   * ( r ) (r )
j
2i

Calculate the radial flux integrated over all angles, that is, diˆr  j for wave functions

of the form
(r )  C
e
 ikr
r
Ylm ,  . The unit vector in the radial direction is iˆr .
4-5. Learn how to count.
(You can use the results from any books to start this problem.)
(a) For a cubic box of dimension L on each side, what is the energy of the highest
occupied level in the ground state for a systems of 20 noninteracting electrons?
(b) Answer the same question if the cube is replaced by a spherical potential well where
the potential is zero inside r=L and infinite outside.
4-6. Use computer to graph (or sketch by hand) in polar plots | Ym ( ,  ) |2 for
( , m )  (1,0), (1,1), (2,0), (2,1) and (2,2).