Download quantum and stat approach

New chapter – quantum and stat approach
We will start with a quick tour over QM basics in order to
refresh our memory.

Wave function  x, t , as we all know, contains all information
about a physical system. I can be obtained by solving the

Schrodinger Equation:

 ( x , t )
H op ( x , t )  i
Questions:
Can the wave function
be measured?
does the wave function
exist in the real world,
or is it only an abstract
mathematicla object?
with H op 
2
pop
t

 Vop ( x )
2m
  
where pop  i , , 
 x y z 
so : H op

2 2

  Vop ( x )
2m
In QM, observables are represented by operators
Values of observables allowed by nature are called eigenvalues
The results of a single physical
measurement is always an eigenvalue
The set of all possible eigenvalues is called spectrum
Eigenvalues are real numbers
The state functions corresponding to
eigenvalues (“pure states”) are called eigenstates
~

In Dirac notation, eigenstates are denoted as
n
(in Dr. Wasserman’s text, I mean – the notation
used in different textbooks may differ considerably).
Eigenvalues and eigenstates corresponding to an Ωop
operator can be obtained by solving the equation:
~  
~
op 
n
n
n
which is called the eigenvalue problem
eigenvalue
The subscript n represents the fact that the equation
usually has many solutions – n may be finite or infinite
depending on the given physical situation.
~

The eigenfunctions
n
are called kets in Dirac notation.
The conjugate of a “ket” is called bra:
 ~
n


~
 
n
“≤” is used as a symbol of Hermitian conjugation in Dr. Wasserman’s text
~
Hermitian conjugatio n rule :  op 
n


~  
 
n
op
Note: operators operate on “kets” from the left, and
on “bras” from the right.
If op  op , i.e., the operator is equal to its conjugate,
~
then  op 
n



~
 n  op
The eigenvalues of Hermitian operators are always real.
Operators correspoding to observables are always
Hermitian operators.
~
The full set of eigenfunctions 
constitute a basis
n
that spans the entire space of the states that the physical
system may assume.
Those states are linear combinations of the eigenstates:
   cn ~n
n

in Dirac notation is a “scalar product”
(a.k.a. “inner product”; it is the equivalent
of a “dot product” of ordinary vectors).
The eigenstates – as any basis in
any vector space – must satisfy
the orthonormality relation:
~
~
m n   mn
The orthonormality relation enables us to find a simple recipe
For finding the expansion coefficients cn in a linear
Combination representing an arbitrary state function:
~
Arbitrary state function :    cn n
n
Take an inner product wi th ~m :
~m   ~m  cn ~n   cn ~m ~n
  cn mn  cm
n
n
n
So : cn  ~n 
cn

Note: it means that
is the length of
the projection of
on
~n
QUICK QUIZ:
So, a b is an inner product.
But what does it mean if I write a b ?
Answer : a b is an ....
OPERATOR!
How to show that |a><b| is indeed an operator?
It’s not a big challenge: an operator converts one
function (vector) into another, right? Let’s then try,
and operate on an arbitrary ket vector, call it |c> :
a b c
 a bc

number,
right?
Inner product of two vectors is a scalar, right? It is just a
number. But specifically, a number of what kind?
COMPLEX number! Call it z :
Yes, U R right!!! It’s a..................
So:
 a b c
 a zza
In particular, an interesting operator of such kind is this one:
~
~
n  n
, made of the ket and bra of the same eigenvector.
OPERATOR
It is called the PROJECTION
............................

~n
~ 
~ ~
the redprojection
vector length


n
n n
~
~
Conclusion: the projection n n operator acting on a vector ψ
returns the component of the ψ vector that is parallel to the eigenvector
~n
Question: and if we take the sum of projection
operators constructed from ALL eigenvectors
comprising the basis? In other words, what we get
if we take:
~ ~

n
all n
Hint:
y
      
or  jn  r    jn   r  r
all n
 all n 

r
 
k r
n  ?
    
j  r  k  r  r , right?

k

j
 
j r
x


~
~
Then,   n n  
 all n

In other wor ds :
~ 
~  I (" unit operator" )

 n n
all n
This is the so-called “completeness relation”, sometimes
also called “closure relation”. We will need it soon!
Probabilities
Any state function, as we said, can be written as a linear
combination of the eigenvectors comprising the basis:
~
   cn n
~
with cn  n 
n

must be normalized, i.e.,    1
Since
~m ~n   mn , by carrying out  
the combination sum,
one readily obtains:
c
n
n
2
1
taking
Now, suppose that you are making a measurement of
the observable associated with a Ωop operator, on a
system that is in a quantum state ψ.
As the result, you may obtain only one of the allowed
eigenvalues. In general*, it is not possible to predict
which one. But you can find the probability of obtaining
a particular value.
One of the GREAT POSTULATES of QM states that
this probability is:
2
~
pn   cn
Therefore, the cn coefficients are often called the
“probability amplitudes”
*There are some special situations in which it is possible to predict the outcome
of a QM measurement – can you think of an example?
Average values (a.k.a. “expectation values”)
Suppose that you perform measurements of a
quantity associated with a Ωop operator, on a quantum
system that at the time of each measurement is in the
same state ψ . Each measurement yields an eigenvalue,
but each time it may be a different one from the allowed
ωn set.
After collecting a sufficient number of results, you may
calculate the average. Conventionally, it’s denoted
as <Ωop> .
It is possible to predict the value of this average, by
using the well-known formula:
op
~
  pn   n   n n 
n
n
2
The sum from the preceding slide can be converted into
an elegant compact expression. For doing that, we will
need to use at one moment the eigenvalue equation:
~
~
op n  n n
OK, let’s go ahead:
 op

n
  n ~n 
n
2
  n  ~n ~n 
n
~
~
~
~
 n n n      op n n 




n
use eigenvalue equation!
   op 
 I (recall "com pleteness relat.")
(familiar expression - right?)
More about the expectation value:
~
If the system is in a pure quantum state
n
with a corresponding eigenvalue ωn , then
the expectation value of Ωop is, of course,  op

 n
This is obvious – but it is instructive to show it
by calculations:
 
op
pure state
~
~
 n  op n
use the eigenequat ion :  op ~n  n ~n
 ~n n ~n  n ~n ~n  n QED.



1
~  
~ - equivalent , of course, to   
Sometimes we use :  op   p(n ) 
n
op
n
op
n