1. Chap 3. Formalism Hilbert Space Observables
Eigenfunctions of a Hermitian Operator
Generalized Statistical Interpretation
The Uncertainty Principle
Dirac Notation

2. Vector Space & Inner Product
Vector space : Linear space closed under vector addition & scalar multiplication. 
Inner product :
   : V  V  
satisfying
Conjugate symmetric.
with 
Positive.
Linear.
which implies

3. Dual Space Dual space V* of a vector space V  Set of all linear maps
V* can be associated with an inner product on V by setting
V* is itself a vector space and isomorphic to V.
Thus, the dual   |  V* of a vector |    V is defined as the linear mapping such that

4. Quantum state space is a Hilbert space.
State of system : Wave function
Observables : Operators
( -D ) Vectors
Linear transformations
Linearity 
Hilbert space  Complete inner product space.
( Cauchy sequence always converges )
E.g., Set of all square-integrable functions over a domain  :
exists
with inner product :
L2  L2   :
Quantum state space is a Hilbert space.

5. Schwarz inequality :
( cos   1 )
[ Guarantees inner product is finite in Hilbert space. ]
conjugate symmetric
orthogonal
 r component of vector f 
completeness
Read Prob 3.1, Do Prob 3.2

6. Hermitian Operators Determinate States
3.2. Observables
Hermitian Operators
Determinate States

7. 3.2.1. Hermitian Operators Expectation value of Q :
Outcomes of experiments are real : 
  
Q is hermitian (self-adjoint)
Observable are represented by hermitian operators.
E.g. :
Read Prob 3.3, 3.5
Do Prob 3.4

8. Determinate States
Determinate state : A state  on which every measurement of Q gives the same value q.
i.e., 
Determinate states are eigenstates.
E.g., solutions to the Schrodinger eq. H  E 
are determinate states of the total energy,
as well as eigenfunctions of the hamiltonian.
Spectrum of an operator  Set of all of its eigenvalues
If two or more independent eigenfunctions share the same eigenvalue, the spectrum is degenerate.

9. Example 3.1. Let , where  is the polar coordinate in 2-D.
Is Q hermitian ?
Find its eigenfunctions and eigenvalues.
Ans.
Consider the Hilbert space of all functions
Q is hermitian.
Eigenequation
has eigenfunctions  
Spectrum of Q is the set of integers, & it’s non-degenerate.

10. 3.3. Eigenfunctions of a Hermitian Operator
Discrete Spectra
Continuous Spectra

11. Phys : Determinate states of observables.
Math : Eigenfunctions of hermitian operators.
Discrete spectrum : n  L2 normalizable & physically realizable.
Continuous spectrum : k not normalizable & not physically realizable.
Can be used to form wave packets.
Examples:
Purely discrete spectrum : Harmonic oscillator.
Purely continuous spectrum : Free particle.
Mixed spectrum : Finite square well.

12. Discrete Spectra
Theorem 1 : Eigenvalues of hermitian operators are real.
Proof :
Let
( f is eigenfunction of Q with eigenvalue q )
( Q is hermitian )  
QED

13. Theorem 2 : Eigenfunctions belonging to distinct eigenvalues are orthogonal.
Proof :
( f , g are eigenfunctions of Q with eigenvalue q and r )
Let
( Q is hermitian ) 
( Theorem 1 )
r  q 
QED
Using the Gram-Schmidt orthogonalization scheme on the degenerate subspaces, all eigenfuctions of a hermitian operator can be made orthonormal.

14. Axiom (Dirac) :
Eigenfunctions of an observable operator are complete.
( Required to guarantee every measurement has a result. )
Note:
Eigenfunctions of a hermitian operator on a finite dimensional space are complete.
Not necessarily so if the space is infinite dimensional.

15. 3.3.2. Continuous Spectra Example 3.2. Momentum Operator
Eigenfunctions not normalizable.
Example Momentum Operator
Example Position Operator

16. Example 3.2. Momentum Operator
Find the eigenfunctions & eigenvalues of the momentum operator.
Ans.
Set 
Placement of the 2 is a matter of taste.
( Dirac orthogonality )

17. is complete ( Fourier transform )
For any real function f : 
i.e.

18. Example 3.3. Position Operator
Find the eigenfunctions & eigenvalues of the position operator.
Ans.
Let gy be the eigenfunction with eigenvalue y. 
Dirac orthonormality : 
Completeness:
For any real function f
with

19. Preferred Derivation    

20. 3.4. Generalized Statistical Interpretation
Measurement of an observable Q(x, p) on a state (x, t) always gets one of the eigenvalues of the hermitian operator Q(x, i d / dx).
2.a) Discrete eigenvalues, orthonormalized eigenfunctions :
Probability of getting the eigenvalue qn is
2.b) Continuous eigenvalues, Dirac orthonormalized eigenfunctions :
Probability of getting an eigenvalue q(z) with z between z and z + dz is
3. Upon measurement,  collapses to fn or fz .

21. Proof for Discrete Eigenvalues
If  is normalized. 
| cn |2 could be a probability. 
| cn |2 = Probability of getting the eigenvalue qn .

22. Position Eigenfunctions
Eigenfunction of position operator : 
= probability of finding particle within ( y, y+dy ).

23. Momentum Eigenfunctions
Eigenfunction of momentum operator :
Momentum space wave function.
Position space wave function.
= probability of finding particle with momentum within ( p, p+dp ).
Note :

24. Example 3.4.
A particle of mass m is bound in the delta function well V(x) =   ( x).
What is the probability that a measurement of its momentum would yield a value greater than p0 = m  / .
Ans.

25. 
Do Prob 3.11
Read Prob 3.12

26. 3.5. The Uncertainty Principle
Proof of the Generalized Uncertainty Principle
The Minimum Uncertainty Wave Packet
The Energy-Time Uncertainty Principle

27. 3.5.1. Proof of the Generalized Uncertainty Principle
System in state .
A hermitian,  A  real. 
where
where
Schwarz inequality

28. 
A, B hermitian,
 A ,  B  real. 
f  g , A  B 
where
Generalized Uncertainty Principle  or

29. 
Observables A and B are incompatible if
Measuring A collapses state to an eigenstate of A, and similarly for B.
If A and B are incompatible, repeated measurements of A, B, A, B, ..., will never, except by accident, get the same values.
If A and B are compatible, repeated measurements of A, B, A, B, ..., will get the same values provided A and B are also compatible with H.
Stationary states of a system can be specified by the eigenvalues of a maximal (complete) set of observables compatible with H.
Do Prob 3.15
Read Prob 3.13

30. 3.5.2. The Minimum Uncertainty Wave Packet
E.g., ground state of a harmonic oscillator ,
Gaussian wave packets of a free particle.
Minimum Uncertainty
Setting the Schwarz inequality to equality 
c = constant
Uncertainty principle keeps only Im < f | g > to give
Minimal uncertainty thus implies
i.e.  
a = real  or
minimal uncertainty state

31. For position-momentum uncertainty :
Prob 3.16

32. 3.5.3. The Energy-Time Uncertainty Principle
Position-momentum uncertainty :
4-vectors:
( c t, x ), (E / c, p )
Special relativity suggests energy-time uncertainty :
Non-relativistic theory :
t is a parameter, not a dynamic variable.
t  t .
t = time for system to change appreciably.
 Energy-time uncertainty is NOT like the other uncertainty pairs.

33. Let Q( x, p, t ) be some characteristic observable of the system.
H hermitian  
Q = Q( x, p )
Define 

34. Example 3.5. For a stationary state, for all observable Q.
 E  0, t  
Time dependence occurs only for linear combinations of stationary states.
E.g.,
a, b, 1 , 2 real.
 E  E2  E1 

35. Example 3.6.
How long does it take for a free-particle wave packet to pass by a particular point ?
Roughly,
x  width of wave packet.
Note :  
c.f. Prob 3.19

36. Example 3.7.
The  particle lasts about 1023 s before spontaneously disintegrate.
A histogram of all measurements of its mass gives a bell-shaped curve centered at 1232 MeV/c2, with a width about 120 MeV/c2.
Why does the rest energy ( mc2 ) sometimes come out larger than 1232, and sometimes lower? Is this experimental error?
Ans.
Measured data :
while 
Spread in measured mass is close to minimum uncertainty allowed.
 It’s not experimental error.

37. Caution :
does NOT mean you can violate energy conservation by borrowing energy E and paying it back within t   / (2E).

38. 3.6. Dirac Notation
The set of all eigenfunctions of any observable is complete.
It can be used as a basis for the system’s Hilbert space.
State of system ( vector in Hilbert space ) :
t is a parameter
r-representation : Basis = Eigenstates of the position operator.
completeness
orthogonality, Dirac normalization

39. p - Representation
p-representation : Basis = Eigenstates of the momentum operator.
completeness  
orthogonality, Dirac normalization

40. Let
| n  is an energy eigenstate.
completeness
orthonormality    
where 

41. Operators: Discrete Basis
Operators are linear transformations :
Orthonormal basis   Or
where
matrix elements

42. Operators: Continuous Basis
Operators are linear transformations :
Dirac-orthonormal basis   Or
where
matrix elements
If Q is diagonal, i.e.,
then

43. Example: x-representation
Matrix elements :
Since ( prove it ! )
we have

44.  

45. Example 3.8 Consider a system with only 2 independent states
General normalized state :
with
Most general Hamiltonian :
with h1, h2 real.
Assume
with h, g real.
If the system starts out ( at t = 0 ) in state | 1 , what is its state at time t ?
Ans :
Time-dependent Schrodinger eq.
Time-independent Schrodinger eq.

46. 
Characteristic eq. 
 Eigenenergies are :
Corresponding eigenvectors are given by: 
 Eigenenergies are :
Normalized :

47. System starts out ( at t = 0 ) in state | 1  :
neutrino oscillation: e   .

48. Dirac notation :
ket : | f  is a vector in L2 .
bra :  f | is in its dual L2*.
For L2 infinite dimensional,
| f  is a function,
 f | a linear functional:
so that
For L2 finite dimensional,
|   is a column vector,
  | is a row vector :
so that

49. Let |   be normalized, then
is the projection operator onto the 1-D space spanned by |  .
E.g.
is a vector in the direction |   with magnitude  |  .
Let { | en  ; n = 1, ..., N } be an orthonormal basis, i.e.,
then
= unit operator in N-dimensional space spanned by { | en  } .
E.g.
magnitude of |  along | en 
If { | ez  } is a Dirac orthonormalized continous basis, i.e.,
then
= unit operator in function space spanned by { | ez  }.
Read Prob 3.21, 3.24
Do Prob 3.22, 3.23