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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
                                  2            n x n y  n z
           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
            E           2
                            ( n1  n2  n3 )
                                    2        2
                                                                         (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 ( ,  )

  where              L2Ym   2 (   1)Y m
  and                LzYm  mYm
 The radial wavefunction RE (r ) satisfies

    d2 2 d              2         l(l  1) 2             2E
    2
    dr       R nl (r )  2 V(r ) 
                                                  R nl (r )  2 R nl (r )  0
       r dr                          2r 
                                             2
                                                                

 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 
                               2  E  V( r )               u nl (r )  0
                     dr 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        2  R (r )  k 2 R (r )  0
           dr  r dr   r 
 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
                h ( )  j (  )  i n (  )
                  (1, 2 )


 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 e ik r  e ikr 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 Ylm 2 .
 4-2. Calculate Ylm1 L2x Ylm 2                  and Ylm1 L2y Ylm 2 .
 4-3. The Hamiltonian for an axially symmetric rotator is given by
                                              L2x  L2y       L2z
                                        H                
                                                2I1           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
                                    
                                j       * ( r ) ( r )   * ( r ) ( r )
                                    2i
                                                                             
                                                                        ˆ
                                                                         
Calculate the radial flux integrated over all angles, that is, dir  j for wave functions
                               ikr
                                      Ylm ,  . The unit vector in the radial direction is iˆr .
                          e
of the form   (r )  C
                              r
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).

				
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