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Chapter 8   Lecture 8 - Overlap Integrals, Expectation Values and Superpositions

8.1   Superpositions

The Completeness Theorem tells us an arbitrary wavefunction Y (r,t) can be described as a superposition of eigenfunctions of any Hermitian operator A .
Y (r ,t)=
 
å
n
cnUn(r,t)
As in the double slit experiment (lecture 3) we can think of Y (r,t) as a state of a particle that is simultaneously in several states ( Un(r,t) ). A measurement of A will select one of the potential eigenstates with proability |cn|2 . If we consider some Y (r,t) as a superposition of energy eigenstates we can write
Y (r ,t)=
 
å
n
cnUn(r )e
En
 
t
 
    (8.1)
A measurement of the energy (using H ) will determine this to be say Em with a probability of finding Em is
P(Em)=|cm|2

8.2   Overlap Integral

How do we determine these values of cn ?
cn= ó
õ
 


all space
Un*(r)Y (r)dV     (8.2)
this is called the overlap integral. The proof is
ó
õ
 


all space
Un(r)Y (r )dV= ó
õ
 


all space
Un*(r) æ
ç
ç
è
 
å
m
cmUm(r) ö
÷
÷
ø		 dV
Þ ó
õ
 


all space
Un(r)Y (r )dV=cn ó
õ
 


all space
Un*(r)Un(r )dV+
 
å
m¹ n
cm ó
õ
 


all space
Un*(r)Um(r)dV
ó
õ
 


all space
Un(r)Y (r)dV=cn

8.3   Expectation Values

If a system is in an eigenstate Un of operator A then
AUn=anUn
a unique eigenvalue of A is always determinded with one hundred percent certainity. If Y is in a superposition each measurement will yield one of the potential eigenvalues with probability of each one given by |cn|2 .

Often we are considering experiments where many identically compared quantum systems are interrogated (aka measured) simultaneously or consecutively. Then we are interested in the weighted mean (aka expectation value) of all the measurements. The weighted mean of A is
<A >=
 
å
n
P(an)an     (8.3)
so
<A >=
 
å
n
|cn|2an     (8.4)
we can also show that the following is true
<A >= ó
õ
 


all space
Y*AY dV     (8.5)
the proof is
ó
õ
 


all space
Y*A dV= ó
õ
 


all space
æ
ç
ç
è
 
å
m
cmUm ö
÷
÷
ø		 A æ
ç
ç
è
 
å
n
cnUn ö
÷
÷
ø		 dV
ó
õ
 


all space
Y*A dV= ó
õ
 


all space
é
ê
ê
ë		 æ
ç
ç
è
 
å
m
cm*Um* ö
÷
÷
ø		 × æ
ç
ç
è
 
å
n
cn*Un* ö
÷
÷
ø		 ù
ú
ú
û		 dV
ó
õ
 


all space
Y*A dV=
 
å
m
 
å
n
an ó
õ
 


all space
(cm*Um*)(CnUn)dV
by using orthogonality of states this reduces to
ó
õ
 


all space
Y*A dV=
 
å
n
an|cn|2

8.3.1   Example - One Dimensional System with Energy Eigenstates

Y (x,t)=
1
2
u1(x)e
-
E1
 
t
 
+
3
2
u2(x)e
-
E2
 
t
 
what is the expectation value of energy
<E >=
 
å
n
|cn|2En=
1
4
E1+
3
4
E2
to determine the expectation value of x as a function of t , here we use equation II.45
<X >= ó
õ
¥


-¥
Y*(x,t)xY (x,t)dx
when the spatial integration is performed
<X>=Acos
æ
ç
ç
è
E2-E1t
 
ö
÷
÷
ø
+B
therefore we can show the expectation oscillates in time
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