mathmatical physics

1. NPTEL – Physics – Mathematical Physics - 1
Module 4 Dirac Delta function
Lecture 21
P. A. M. Dirac (1920-1984) made very significant contribution to quantum theory
of radiation and relativistic quantum mechanics. The quantum statistics for
the fermions is also associated with his name.
The function that we are interested in describing in this chapter is an important one
in the context of quantum mechanics and electrodynamics in particular, but
is generally found in all areas of physics - namely, the Dirac delta function.
For example, the concept of an impulse appears in many physical situations; that is,
a large force acts on a system for a very short interval of time. Such an impulse can
appropriately be denoted by a Dirac delta ()
In one dimension the  - function has the following properties -
(𝑥 − 𝑎) = ∞ 𝑎𝑡 𝑥 = 𝑎
(𝑥 − 𝑎) = ∞ 𝑎𝑡 𝑥 ≠ 𝑎
1. a) (x-a) =0 for xa
And
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function.

b) ∫ (𝑥 − 𝑎)𝑑
𝑥 = 1
−
- function can be visualized as the Gaussian that becomes narrower and narrower,
but at the same time becomes higher and higher, in such a way that the
area enclosed by the curve remains constant. From the above definition, it is
evident that for an arbitrary function f(x),

2. ∫ 𝑓 (𝑥)(𝑥 − 𝑎)𝑑𝑥 = 𝑓(𝑎)
−
The above property can be interpreted as follows.

2. NPTEL – Physics – Mathematical Physics - 1
The convolution of the arbitrary function and the - function when
integrated yields the value of the arbitrary function computed at the point of
singularity. Thus
- function is, even though, sharply peaked, but still is a well behaved function.
Further properties of the function are –

3. ∫ 𝑓(𝑥) ′(𝑥 − 𝑎)𝑑𝑥 =
𝑓′(𝑎)
−
where the prime denotes differentiation with respect to the argument.
If the - function has an argument which is a function f(x) of an
independent variable x, it can be transformed according to the rule,
4. 𝛿[𝑓(𝑥)] = ∑
1
|
𝑖 𝑑 𝑓 (𝑥
𝑖 ) 𝑑
𝑓
|
𝛿(𝑥 − 𝑥 )
𝑖
where f(x) is assumed to have ‘simple' zeros at x = xi
In more than one dimension,
5. 𝛿(𝑟⃑ − 𝑟⃑0) = 𝛿(𝑥 − 𝑥0)𝛿(𝑦 − 𝑦0)𝛿(𝑧 − 𝑧0)
And for the integral,
6. ∫∆𝑣 𝛿(𝑟⃗⃗ − ⃗𝑟
⃗0 )𝑑
3
𝑟 = {
0
1
Other properties of the function are:
7.𝛿(𝑥) = 𝛿(−𝑥) :: Symmetric with respect to change in the argument.
8.𝛿′(𝑥) = -𝛿′(−𝑥): The derivative is antisymmetric 9. x𝛿(𝑥)
= 0
10. 𝑥𝛿′(𝑥) = −𝛿(𝑥)
11. 𝛿(𝑎𝑥) =
1
𝛿(𝑥) ∶ 𝑎 > 0
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𝑎
These latter relations can be verified by multiplying it with a
continuous differentiable function, followed by an integration. Say, we want to
prove property number 9.
If v includes 𝑟⃗⃗ =
⃗𝑟0
If v does not includes 𝑟⃗⃗ = ⃗𝑟0
}
where
v = d3
r

3. NPTEL – Physics – Mathematical Physics - 1
Let 𝛷(𝑥) be a continuous differentiable function,
∫ 𝛷(𝑥)𝑥𝛿 (𝑥)𝑑𝑥 = 0
Since 𝛷(𝑥) is arbitrary,
𝑥(𝑥) = 0
(1)
(2)
To prove property number 10 consider the relation,
𝑓(𝑥)𝛿(𝑥) = 𝑓(0)𝛿(𝑥) (3)
Differentiating (3)
𝑓′(𝑥)(𝑥) + 𝑓(𝑥) ′(𝑥) = 𝑓(0) ′(𝑥)
Thus,
𝑓(𝑥) ′(𝑥) = 𝑓(0) ′(𝑥)– 𝑓 ′(𝑥) ′(𝑥)
In the special case, when 𝑓(𝑥) = 𝑥;
𝑥 ′(𝑥) = − (𝑥)
(4)
we get,
Rest of the relations are also straightforward to prove.
Further properties of the - function are as follows-
12. 𝛿(𝑥2 − 𝑎2) =
1
[𝛿(𝑥 − 𝑎) + 𝛿(𝑥 + 𝑎)]: a > 0
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2𝑎
13. 𝛿(𝑥) = 𝑓(0) 𝛿(𝑥) : already used.
14. 𝛿(𝑥) =  (𝑥)
Where  (𝑥) = {
1 ∶ 𝑥 > 0
0 ∶ 𝑥 < 0
}
 (x) is called Heaviside step function.

15.∫  (𝑥 − 𝑥′) (𝑥 − 𝑥′)𝑑𝑥 =  (𝑥′
− 𝑥′)
−