Developing Wave Equation

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Developing Wave Equation Powered By Docstoc
					             Solving Schrodinger Equation
         • If V(x,t)=v(x) than can separate
           variables
          2  ( x ,t )

                            V ( x)  i 
     2

2m               x 2                       t

assume  ( x, t )   ( x) (t )
                 d 2                                        d

                          V ( x) (t ) ( x)  i
     2

2m                    2                                      dt
                 dx



  2 d 2
    2 mdx    2      V      
                        1 id
                        
                           dt  G

G is separation constant valid any x or t
             Gives 2 ordinary diff. Eqns.
                                 P460 - Sch. wave eqn.   1
          Solutions to Schrod Eqn
• Gives energy eigenvalues and eigenfunctions
  (wave functions). These are quantum states.
• Linear combinations of eigenfunctions are also
  solutions. For discrete solutions


 ( x, t )  c11  c2 2 ...... n n
                               c
each        i   i e      iE i t / 
                                             If H Hermitian
                                
i orthogonal                       j dx   ij
                                             *
                                            i
                               

normalized              c        2
                                  i    1

                    P460 - Sch. wave eqn.                 2
    id
      dt    G   (t )  e                     iGt / 


G=E if 2 energy states, interference/oscillation

 d 
    2 2

     2
           V  E                          1D time
2mdx                                        independent

 ( x, t )   ( x)e iEt /                Scrod. Eqn.

Solve: know U(x) and boundary conditions
want mathematically well-behaved. Do not
                 want:
   ( x)                    No discontinuities. Usually
    
    x                          except if V=0 or  =0

     2                              in certain regions
    x 2
            
                    P460 - Sch. wave eqn.                   3
                    Linear Operators
• Operator converts one function into another
                    Of ( x)  f ( x)  x 2
                              d f ( x)
                    Of ( x) 
                                 dx
• an operator is linear if (to see, substitute in a
  function)

    if O[ f1 ( x)  f 2 ( x)]  Of 1 ( x)  Of 2 ( x)  linear
               d
    ex : O 
               dx

• linear suppositions of eigenfunctions also solution
  if operator is linear……use “Linear algebra”
  concepts. Often use linear algebra to solve non-
  linear functions….



                          P460 - Sch. wave eqn.              4
           Solutions to Schrod Eqn
• Depending on conditions, can have either
  discrete or continuous solutions or a
  combination


    ( x, t )   C n u n ( x ) e              iE n t / 

                  n

                C ( E )u E ( x)e                iE n t / 
                                                                dE


• where Cn and C(E) are determined by taking
  the dot product of an arbitrary function  with
  the eigenfunctions u. Any function in the space
  can be made from linear combinations


                      P460 - Sch. wave eqn.                          5
            Solutions to Schrod Eqn
• Linear combinations of eigenfunctions are also
  solutions. Assume two energies

       ( x, t )  c11  c2 2 
      c1 1e    iE1t / 
                              c2 2 e             iE 2t / 



• assume know wave function at t=0

     ( x,0)                  2
                               7 1                 5
                                                     72
• at later times the state can oscillate between the two
  states - probability to be at any x has a time
  dependence
| ( x, t ) | | c1 1 ( x) |  | c2 2 ( x) | 
            2                           2                        2


c1c2 (  2e
        *
        1
                 i ( E2  E1 ) t / 
                                          e   *
                                                 2 1
                                                     i ( E1  E2 ) t / 
                                                                            )
                         P460 - Sch. wave eqn.                       6
                     Example 3-1
• Boundary conditions (including the functions being
  mathematically well behaved) can cause only
  certain, discrete eigenfunctions
               d f ( )
             i            f ( )
                 d
             with f ( )  f (  2 )
• solve eigenvalue equation
       1 d f ( )                      d f ( )
   i                 eigenvalue or            i d
     f ( ) d                           f ( )
   int egrate  ln f ( )  i  cons tan t
   or     f ( )  f (0)e i


• impose the periodic condition to find the allowed
  eigenvalues

        e i ( 2 )  1    0,1,2, etc
                       P460 - Sch. wave eqn.      7
           Square Well Potential
• Start with the simplest potential

 V ( x)  V0 | x | a V0  finite or 
                    2

 V ( x)  0 | x | a ("in" the well )
                   2

  For  value
  ( x)  0 for | x | a  V  is finite
                       2

Boundary condition is that  is continuous:give:

    out ( a )   in ( a )  0 if V0  
            2             2


                                               V

                                          0
                -a/2               a/2
                       P460 - Sch. wave eqn.       8
    Infinite Square Well Potential
• Solve S.E. where V=0
         2 d 
                    E
              2

        2 m dx 2

         A sin kx, B cos kx, Ce ikx
Boundary condition quanitizes k/E, 2 classes
      Even                     Odd

   =Bcos(knx)                    =Asin(knx)

      kn=n/a                         kn=n/a

     n=1,3,5...                      n=2,4,6...

    (x)=(-x)                     (x)=-(-x)
            p2
   En      2m         2k 2
                        2m             2 2 n 2
                                         2 ma   2      h2n2
                                                        8 ma 2

   as n  0  Em in  E1  0
                       P460 - Sch. wave eqn.                 9
                            Parity
• Parity operator P      x  -x (mirror)
                   P ( x)   ( x)
• determine eigenvalues

Pu( x)  u ( x)
P 2u ( x)  Pu ( x)  2u ( x)
but P[ Pu( x)]  Pu( x)  u ( x)  2  1    1
• even and odd functions are eigenfunctions of P
  Odd : Px   x P sin x   sin x P x   x
  Even : Px2  x 2 P cos x  cos x P x 2  x 2
                                                     2    2




• any function can be split into even and odd

     ( x)  1 [ ( x)   ( x)]  1 [ ( x)   ( x)]
             2                      2

     ( x)    ( x)    ( x)
     P ( x)    ( x)    ( x)
                                                  1 (1  P)
                                                     2

                                                  1 (1  P)
                                                     2
                       P460 - Sch. wave eqn.                  10
                                Parity
• If V(x) is an even function then H is also even then
  H and P commute
                             [ H , P]  HP  PH  0
• and parity is a constant. If the initial state is even it
  stays even, odd stays odd. Semi-prove:
• time development of a wavefunction is given by
                         
                    i        H ( x, t )
                          t

• do the same for P when [H,P]=0
              ( P )
         i            H [ P ( x, t )]  P[ H ( x, t )]
                 t

• and so a state of definite parity (+,-) doesn’t change
  parity over time; parity is conserved (strong and
  EM forces conserve, weak force does not)
                           P460 - Sch. wave eqn.             11
    Infinite Square Well Potential
• Need to normalize the wavefunction. Look up in
  integral tables
                                     a
                                    2


    
   
      |  ( x) |2 dx                
                                     a
                                          A2 sin 2       nx
                                                          a    dx  1
                                      2


    A            2/a

What is the minimum energy of an electron confined
        to a nucleus? Let a = 10-14m = 10 F
                             ( hc ) 2                (1240 MeVF) 2
Em in     2 2
          2 ma 2
                          8 mc 2 a 2
                                                  8.51MeV(10 F ) 2

 4000 MeV  relativistic                                         redo
Em in      m p  2             2
                                               m  (k )
                                                     2             2


    k       hc
               2a       124010 F F  60MeV
                            2
                              MeV


                           P460 - Sch. wave eqn.                   12
             Infinite Square Well
               Density of States
• The density of states is an important item in
  determining the probability that an interaction or
  decay will occur
• it is defined as
                  dn
       (E)             n  number of states
                  dE
• for the infinite well          8ma 2
                               2
                                    n              2
                                                        E  cE
                                                h
                                         dn   c 1 c
                    2ndn  cdE               
                                         dE 2n 2 E

• For electron with a = 1mm, what is the number of
  states within 0.0001 eV about 0.01 eV?
        8  511000eV  (.1cm ) 2
     c             4
                                  2.7  1012 eV 1
          (1.24  10 eVcm) 2


          dn      1     c      1 2.7  1012
     n     E          E               .0001eV  820
          dE      2     E      2  .01eV

                        P460 - Sch. wave eqn.                    13
                       Example 3-5
• Particle in box with width a and a wavefunction of
          ( x )  A( x / a ) 0  x  a / 2       A  12 / a
          ( x )  A(1  x / a ) a / 2  x  a
• Find the probability that a measurement of the
  energy gives the eigenvalue En
           An un ( x )  un  a sin na x
                                 2     

                n

                                                       nx
                a                  a/2
                                          12 x   2
          An   ( x )un dx  2                 sin     dx
                0                    0
                                           a a   a      a
                      24 1
                2           ( 1) n 1
                       n 2

• With only n=odd only from the symmetry
• The probability to be in state n is then
          96                          .986
 | An |  4 4  Pr ob1  .986 Pr ob3  4  .012
     2

          n                           3

                          P460 - Sch. wave eqn.                14
       Free particle wavefunction
• If V=0 everywhere then solutions are

      A cos kx , A sin kx , eikx , e ikx
           p2
    E     2m       2k 2
                     2m


• but the exponentials are also eigenfunctions of the
  momentum operator
                 
    pop  i
                 x
    pop (eikx )  i  ik eikx  eigenvalue  k  p
     pop (e ikx )  i  ik e ikx    k   p

• can use to describe left and right traveling waves
• book describes different normalization factors



                             P460 - Sch. wave eqn.      15

				
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