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Lesson 7
I.
Dirac Delta Function - (x)
The Dirac delta function is not actually a function, but a distribution
(see either Arfkin or Butkov) which means that it is defined in terms
of its integral properties.
The two integral properties that define the Dirac delta function are
1
  x  x dx  0
b
1)
o
a
if x o in the region between a and b
otherwise
 f x  if x in the region between a and b
 f x  x  x dx   0 otherwise
b
2)
o
o
o
a

EXAMPLE: Evaluate the integral
e

x 2
 nπ x 
sin 
  x  a  dx
 L 
5
EXAMPLE: Evaluate
4
8
  3x
2
y z 3 δx  1 δy  2 δz  3 dx dy dz
x  0 y  0 z  5
II.
Kroniker Delta Function - m,n
1 if m  n
0 if m  n
 m, n  
Example: Orthogonality and Normalization

2
 mππ   nππ 
sin 
 sin 
 dx  δ m, n

L   L   L 
III.
Expansion Postulate
For an arbitrary observable Ô with eigenvectors Φ n (x) and real
eigenvalues on, the eigenfunctions form an orthogonal set that can be
normalized so that



m
 n dx   m, n

Furthermore, the set of eigenfunctions span the space so the wave
function describing any quantum state can be expressed as a linear
combination of these eigenstates as
 x    c n  n x 
n
This fact leads us to consider the similarity between our work with
eigenfunctions and our previous work in physics with unit vectors.
IV.
Dirac Notation (Bra-ket Notation)
We will consider Dirac notation as a sort of vector shorthand for
doing algebra in quantum mechanics but it is actually much more
powerful and profound.
A.
B.
ket
1.
Symbol - Ψ
2.
Ψ is interpreted as a vector representing  .
Bra
1.
2.
Symbol - 
 is interpreted as a vector representing   .
C.
Inner Product
1.
Symbol -  

2.
  is defined as
   dx


3.
  is interpreted as giving the projection of the vector
 onto the vector  .
Special Cases
1.
Orthogonality
Two vectors are said to be orthogonal if their inner product is
zero.
  0
2.
Normalization
A vector is normalized if the inner product of the vector with
itself is equal to 1.
  1
D.
Expansion Postulate
We can now write the expansion postulate in terms of Dirac notation
as
   cn n
n
where  is the state vector representing the quantum system and
 n is the nth eigenvector for the operator Ô . The coefficients may
be real or complex.
In Dirac notation, eigenvectors act like unit vectors. Each operator has
a set of eigenvectors which one can use to represent the state vector of
the quantum system. Thus, you may represent the state vector of a
quantum system as a combination of many different sets of
eigenvectors in the same way that we can represent the position of a
classical particle in different coordinate systems (cylindrical,
Cartesian, spherical, etc.). However, in quantum mechanics the
measurement process and its connection to the eigenvectors has a
deeper importance as we shall see shortly.
E.
Normalization of a State Vector
We know that the probabilistic interpretation of quantum mechanics
requires that



 dx    1

If we now write the expansion of the bra and the ket in terms of a set
of eigenvectors of some operator, we have
   cn n
n
   c m  m
m
Thus, our normalization requirement becomes
    c m c n  m  n
m
n
    c m c n  m, n
m
n
    c n c n   c n 1
2
n
F.
n
Operators and Eigenvalues
When an operator is applied to a ket that is one its eigenvectors, we
obtain the ket times the associated eigenvalue.
  n  Â n  a n  n
G.
Expectation Values
The expectation value is the average value that one would obtain if
one made a specific measurement of a large number of identical
quantum systems (ensemble). Each measurement would produce an
eigenvalue for the physical quantity which may or may not be equal
tothe expectation value and would leave the system in the eigenstate
that corresponds to the measured eigenvalue of the operator associated
with the physical quantity being measured.
In Dirac notation, we can write the expectation value as

A     Â  dx   Â

In our expansion representation, we find the following useful result
for a discrete set of eigenvectors.
A   Â    cm c n a n  m  n
m
n
A 
 c
m

m
c n a n  m, n
n
A   a n cn
2
n
We see that expectation value is found by multiplying the eigenvalue
2
an by the probability of obtaining that eigenvalue, c n .
H.
Additional Computation Properties
All of the properties of Dirac algebra are defined either by our
previous definition of the inner product or due to physical
requirements.
1.
Constants can be pulled outside of a bra or ket using the following
relationships:
i)
a  a 
ii)
a  a 
Both of these statements indicate that the constant increases the length
of the vector (multiplication of a vector by a scalar).
Proof of (i) Let x  a x 

      a  dx


   a     dx

  a  
     a 

by comparison of the two sides of the equation, we have that
a a 
Q.E.D.

    
2.
Proof:

 

   





dx       dx

       dx 





dx


    
Q.E.D.
3.
Addition Property
The ket of the sum of two wave functions is the sum of the ket
vectors of the individual wave functions.
    
The bra of the sum of two wave functions is the sum of the bra
vectors of the individual wave functions.
    
Dirac Calculation Example:
The chief quantum mechanic of the quantum artillery has told you that they
have discovered a new operator Ĝ (gun operator). The eigenvectors and
eigenvalues for the gun operator are
 n

2
g n  2n
nth eigenvecto r
nth eigenvalue
You are also told that an atomic artillery unit at Ft. Small has a state vector
given by
  K 2 1  3  2   3

A.
Normalize the state vector
B.
What are the possible values that can be obtained for the guns in the
artillery unit?
C.
What is the probability that we measure 8 guns?
D.
What would be the artillery state vector immediately following a
measurement in which we found 18 guns?
E.
If we made a second gun measurement on the artillery system in D,
what would we obtain for the number of guns?
F.
For the artillery unit in part B, what is the expectation value for the
number of guns?