1. 2. Time Independent Schrodinger Equation
Stationary States
The Infinite Square Well
The Harmonic Oscillator
The Free Particle
The Delta Function Potential
The Finite Square Well

2. 1.1. Stationary States Schrodinger Equation : Assume
(  is separable ) 
= const
= E
if V = V(r) 
Time-independent Schrodinger eq.

3. Properties of Separable 
1. Stationary States: Expectation values are time-independent.
In particular :

4. 2. Definite E :
Hamiltonian : 
 Time-independent Schrodinger eq. becomes

5. 3. Expands Any General Solution :
Any general solution of the time dependent Schrodinger eq. can be written as
where
The cn s are determined from
{ n } is complete.
Proof : 

6. Example 2.1 A particle starts out as What is ( x, t ) ?
Find the probability, and describe its motion.
Ans.
if c &  are real.
Read Prob 2.1, 2.2

7. 2. The Infinite Square Well
  0 for x  [ 0, a ]
for x  [ 0, a ]
General solution.
Allowable boundary conditions ( for 2nd order differential eqs ) :
 and   can both be continuous at a regular point (i.e. where V is finite).
Only  can be continuous at a singular point (i.e. where V is infinite).

8. Boundary Conditions B.C.:  continuous at x = 0 and a , i.e.,   
n = 0    0 (not physically meaningful). 
 k give the same independent solution. 
E is quantized.
Normalization:

9. Eigenstates with B.C. has a set of normalized eigenfunctions
with eigenvalues
n = 1 is the ground (lowest energy) state.
All other states with n > 1 are excited states.
even, 0 node.
odd, 1 node.
even, 2 nodes.

10. Properties of the Eigenstates
Parity = () n  1 . [ True for any symmetric V ]
Number of nodes  n  1. [ Universal ]
Orthogonality: [ Universal ]
if m  n.
Orthonormality:
4. Completeness: [ Universal ]
Any function f on the same domain and with the same boundary conditions as the n s can be written as

11. Proof of Orthonormality

12. Condition for Completeness
where { n } is orthonormal.
Then  
if { n } is complete.

13. Conclusion Stationary state of energy is General solution: where
so that

14. Example 2.2 A particle in an infinite well has initial wave function
Find ( x, t ).
Ans.
Using
we have

15. Normalization: 
for n odd
B = Bernoulli numbers 

16. In general : 
{ n } orthonormal
 normalized If
then | cn |2 = probability of particle in state n.
  H   average of En weighted by the occupation probability.

17. Example 2.3.
In Example 2.2, ( x, t ) closely resembles 1 (x) , which suggests | c1 |2 should dominate. Indeed,   
where 
Do Prob 2.8, 2.9

18. 3. The Harmonic Oscillator
Classical Mechanics:
Hooke’s law:
Newton’s 2nd law.
Solution:
Potential energy:
parabolic
Any potential is parabolic near a local minimum.
where

19. Quantum Mechanics:
Schrodinger eq.:
Methods for solution:
Power series ( analytic )  Hermite functions.
Number space (algebraic)  a , a+ operators.

20. 3.1. Algebraic Method Schrodinger eq. in operator form:
Commutator of operators A and B :
Eg.
 f 
canonical commutation relation
Let
,  real, positive constants  

21. Set   
Let 
i.e., H , a+a and a a+ all share the same eigenstates.
Their eigenvalues are related by
where

22. Adjoint or Hermitian Conjugate of an Operator
Given an operator A , its adjoint (Hermitian conjugate) A+ is defined by
 ,  
Proof :
,  real & positive
Integration by part gives:
( ,   0 at boundaries )  
Note:

23. n-Representation Consider an operator a with the property
Define the operator
with its eigenstates and eigenvalues
Meaning of a+ n :
 a+ n is an eigenstate with eigenvalue larger than that of  n by 1.
a+  raising op
Meaning of a n :
 a n is an eigenstate with eigenvalue smaller than that of  n by 1.
a  lowering op
Let N be bounded below ( there is a ground state 0 with eigenvalue 0 ).  
i.e.
Hence
with

24. Normalization Let ( ,   normalization const. ) with  n 
Assuming ,  real :

25. Orthonormality 
for m  n

26. Harmonic Oscillator : n-Representation
 a ~ a , a+ ~ a+ , a+ a ~ a+ a  
Equation for 0 : 

27. 0 
Set A real : 
Normalized

28. Example 2.4 Find the first excited state of the harmonic oscillator.
Ans: 

29. Example 2.5 Find  V  for the nth state of the harmonic oscillator.
Ans:  
Do Prob 2.13 (drudgery)

30. 3.2. Analytic Method
is solved analytically.
Set 
Set
where 

31. Asymptotic Form For x,     as    not normalizable Set
h solved by Frobenius method
(power series expansion)

32. h() 
Asume   
recursion formula

33. Termination Even & odd series starting with a0 & a1 , resp.
For large j :  
explodes as  
 Power series must terminate.
Set 
 j  n+2 

34.  with E  En

35. Hermite Polynomials Hermite polynomials : n even : set a0  1 , a1  0
n odd : set a0  0 , a1  1
an  2n
Normalized:

36. n
n  3
n  2
n  1
n  0

37. | 100 |2 ,  (x) Classical distribution : A = amplitude
Do Prob 2.15, 2.16
Read Prob 2.17

38. 4. The Free Particle  Eigenstate: Stationary state: to right to left
phase velocity
k > 0 : to right
k < 0 : to left
Reset:

39. Oddities about k (x,t)  Classical mechanics :  not normalizable
k is not physical (cannot be truly realized physically ).
k is a mathematical solution
( can be used to expand a physical state or as an idealization ).

40. Wave Packets General free particle state : wave packet
Fourier transform

41. Example 2.6
A free particle, initially localized within [ a, a ], is released at t  0 :
A, a real & positive
Find ( x, t ).
Ans:
Normalize ( x, 0 ) :  

42. FT of Gaussian is a Gaussian.

43. Group Velocity In general : (k) = dispersion
For a well defined wave packet,  is narrowly peaked at some k = k0 .
To calculate ( x, t ), one need only 
(  moves with velocity 0 . )
Define group velocity
3-D
Free particle wave packet : 
Reminder: phase velocity
Do Prob 2.19
Read Prob 2.20

44. Example 2.6 A Consider a free particle wave packet with  (k) given by
A, a real & positive
Find ( x, t ).
Ans:
Normalize ( x, 0 ) :  

45. t = { 0, 1, 2 }ma2/

46. 5. The Delta Function Potential
Bound States & Scattering States
The Delta Function Well

47. 5.1. Bound States & Scattering States
Classical turning points x0 :
If V(r)  E for r  D
and V(r) > E for r  D,
then the system is in a bound state for r  D.
If V(r)  E everywhere,
then the system is in a scattering state.
Quantum systems (w / tunneling) :
If V() > E,
then the system is in a bound state.
If V() < E,
then the system is in a scattering state.
If V() = 0, then
E < 0  bound state.
E > 0  scattering state.

48. The Delta Function Dirac delta function : such that
 f , a, and c > b
 is a generalized function ( a distribution).
[ To be used ONLY inside integrals. ]
Setting f (x) = 1 gives
Rule :
( meaningful only inside integrals.)
Proof :

49. 2. The Delta Function Well
For x  0 :
Bound states : E < 0
Scattering states : E > 0
Boundary conditions :
1.  continuous everywhere.
2.   continuous wherever V is finite.

50. Bound States For x  0 : Bound states ( E < 0 ) : Set 
 real & positive
(  ) = 0 
 continuous at x  0  B  C

51. Discontinuity in   (x0) is discontinuous at V(x0)  . Let 
 continuous & finite   

52.   
Only one bound state
Normalization :  

53. Scattering States Scattering states ( E > 0 ) : Set  for x  0
k real & positive
 continuous at x  0 
Let 

54. 2 eqs., 4 unknown, no normalization.
Scattering from left :
A : incident wave C : transmitted wave
B : reflected wave D = 0 x 
Reflection coefficient
Transmission coefficient

55. Delta Function Barrier
 No bound states
For the scattering states, results can be obtained from those of the delta function well by setting   . 
Note: Since E < V inside the barrier,
T  0 is called quantum tunneling. ( T = 0 in CM)
Also: In QM, R  0 even if E > Vmax .
( R = 0 in CM)
Do Prob 2.24
Read Prob 2.26

56. 6. The Finite Square Well

57. Bound States ( E < 0 )  l &  are real & positive  finite 
Symmetry : 
 If (x) is a solution, (x) is also a solution.

58. Even Solutions  continuous at a :  or   continuous at a :   
where  or

59. Graphic Solutions
E < 0  z < z0

60. Limiting Cases 1. Wide, deep well ( z0 large ) : Lower solutions : 
c.f. infinite well
2. Shallow, narrow well ( z0 <  / 2 ) :
Always 1 bound state.

61. Scattering States ( E > 0 )
l & k are real & positive

62. Scattering from the Left
 continuous at a :
  continuous at  a :
 continuous at a :
  continuous at a :
Eliminate C, D and express B , F in terms of A ( Prob 2.32 ) :

64. T  when
( well transparent )
i.e.
( at energies of infinite well )
( Ramsauer-Townsend Effect )
Do Prob 2.34