lec21.ppt
Download as PPT, PDF0 likes48 views
1. The Dirac delta function is an important concept in quantum mechanics and electrodynamics that describes an impulse or large force acting over a very short time interval. 2. The key properties of the Dirac delta function are that it is equal to infinity at a single point and zero everywhere else, and that the integral of the function over its entire range is equal to one. 3. The Dirac delta function can be used to find the value of an arbitrary function f(x) at a specific point a, as the integral of f(x) multiplied by the Dirac delta function over all x is equal to f(a).
![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
Page 2 of 15
Joint initiative of IITs and IISc – Funded by MHRD
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](https://image.slidesharecdn.com/lec21-231023100922-01ca4bc0/85/lec21-ppt-2-320.jpg)
![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
Page 3 of 15
Joint initiative of IITs and IISc – Funded by MHRD
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′)
−](https://image.slidesharecdn.com/lec21-231023100922-01ca4bc0/85/lec21-ppt-3-320.jpg)
lec21.ppt
- 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 xa And Page 1 of 15 Joint initiative of IITs and IISc – Funded by MHRD 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 Page 2 of 15 Joint initiative of IITs and IISc – Funded by MHRD 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 Page 3 of 15 Joint initiative of IITs and IISc – Funded by MHRD 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′) −