Download EP-307 Introduction to Quantum Mechanics

Generalization to Infinite Dimension
The dimensionality(n) of the space has so far been assumed to be finite.
What are the complications that result when we try to generalize our result to
Infinite dimensional space.
To visualize an Infinite dimensional vector, let us consider a string clamped at
x = 0 and at x = l.
We want to describe the shape of the string.
We can divide it into 20 intervals and measure the displacement and quote the
numbers as f(xi).
Denote by fn(x) the discrete approximation to f(x), that coincides with it at n points
and vanishes in between.
We are trying to visualize the ordered n tuple {fn(x1), fn(x2), …, fn(xn)} as
Components of ket |fn in a vector space Vn(R)
The orthonormality &
Completeness relation
are given by
f(xi)
राघव वर्ाा
xi
fn
 f n ( x1 ) 
0 
 f ( x )
1
  n 2  the basis vectiors in this space are xi   
  



 
0 
 f n ( xn ) 
xi xi  1,
n
i
i 1
i I
Infinite Dimensional Space

Imagine a space containing n mutually perpendicular axes one for each
point xi. Along each axis is a unit vector |xi. The function fn(x) is
represented by a vector whose projection along the ith is fn(xi)

To every possible discrete approximation gn(x), hn(x) there corresponds
a ket |gn, |hn
Vector addition as addition of components
Scalar multiplication as multiplying each component with a scalar
String is clamped at two ends. So you have the null vector also.
f g   f (x )g (x )
The inner product is also defined




n
n
n
i
n
i
i






f n gn  0
The orthogonality of two functions is defined as
Now consider the maximal specification of the string displacement
Vector addition and scalar multiplication will still work
n
f n g n   f n ( xi ) g n ( xi )
However, the inner product as it is defined will diverge
i 1
If n goes to infinity so does the sum.
Clearly if we are going to go to infinity the inner product should be
defined in such a way that as n a smooth limit is obtained
f g   f ( x ) g ( x )
n
n
n
i

राघव वर्ाा
l
n 1
i
n
i
Basis Vectors for this Infinite
Dimensional Space
Lets dot this with

x

f on the right and x on the left
x  x  dx  f  x I f
-


x x  x  f dx   f ( x )
-
Orthogonal ity requires x x  0 for x  x
Then as n  
x 
l

f g   f ( x) g ( x)dx
x x  x  f dx   f ( x )
x -
0
x 
If we want to be consistent with our earlier definition s and we
leave the string then we have
l
f g   f  ( x) g ( x)dx

x x f ( x) dx  f ( x)
x-
The interval  can be made as small as possible. One can then
assume that f(x) does not differ from f(x) in this interval.
Therefore we can take out f(x) from the integral as f(x)
0
x 
Next question is what is our basis vectors. Each point gets a basis f ( x)  x x dx  f ( x)
x -
vector x . We will like to have orthogonal ity x x  0.
x 
What about orthonorma lity. Should we require
x x  1. This
will not work out. The easiest wa y to see this through t he
completene ss relation


x  x  dx   I
-
राघव वर्ाा
  ( x  x)dx  1
x 
There is no way that an integral over an infintesim al region
would give you a finite result. The quantity  ( x  x) is infinite
in such a way that the above integral gives you 1. This is called
a Dirac Delta function and it fixes the normalizat ion of the
infinite basis. It arises in many fields of physics wherever you
require finite contributi on from infintesim al region.
It is also called a sampling function as it samples the value of
the function at a point

 δ(x-x)f(x)dx 

f ( x)
A short note on Dirac Delta Function







It occurs everywhere in physics– in quantum mechanics, electrodynamics, and in general in any
branch of advanced theoretical physics.
Here it fixes the normalization of infinite basis.
In electrodynamics, it is the function that describes the charge density of a point charge.
It enables us to treat Green’s functions (whose definition involves the Dirac -function) on a
rigorous footing.
How and why we can use these idealizations in computations?
Lets take the instance of a point charge, which is easier to visualise than the normalization of an
infinite set of basis vectors.
A point charge in a macroscopic problem in electrostatics is in actuality a charge concentrated on a
small region in space as compared to the dimensions of the problem. The point charge is
mathematically convenient limit that allows two things
–
–




It simplifies algebra involved in computations and at the same time
It allows for evaluation of physical quantities to a high level of accuracy.
Need not use the notion of a concentrated point charge as such. We could do all the calculations
with q distributed over a small sphere which then would do away with singular functions.
This would just be complicating calculations for essentially the same physics
The above procedure of distributing q over a sphere then is to regard the charge density (r) as the
function
ρε(r)  0 r  ε
The above function (r) describes the charge density of a small uniformly charged sphere of
q
radius . The point charge is regarded in the limit
ρ (r)  lim  (r )
ρ (r) 
rε
po int
 0


That is not the only way to get the point charge. We could also take

This is infintely differentiable, and hence more tractable in calculations
ρ (r) 

1

3
e r
2 
ρ point(r)  lim ρ (r)
 0
राघव वर्ाा
4 3
ε
2
/ 2 2
 0
Dirac Delta Function
It should be remembered that the delta function has meaning only under the integral
sign. By itself it has no meaning.
Consider a Gaussian representation
  x  x2 
1
g  ( x  x) 
exp 
 1/ 2  2 
centered at x  x  and having a width 
As   0 g    and becomes a better and better
approximat ion to the Delta Function
Its an even function. Also clear from the Gaussian representation
Consider now
 ( x  x) 
d
d
 ( x  x)    ( x  x)
dx
dx
What is its action of this function on an integral-
राघव वर्ाा