Understanding Transition btwn Steps of Dirac Delta Function

In summary, the conversation is discussing the use of a general property of delta functions to understand the transition between two steps in an equation. The formula is used in the equation and the speaker has confirmed that it applies for functions of more than one variable.
  • #1
cyberdeathreaper
46
0
Can someone help me understand the transition between these two steps?
[tex]
<x> = \iint \Phi^* (p',t) \delta (p - p') \left( - \frac{\hbar}{i} \frac{\partial}{\partial p} \Phi (p,t) \right) dp' dp
[/tex]
=
[tex]
<x> = \int \Phi^* (p,t) \left( - \frac{\hbar}{i} \frac{\partial}{\partial p} \Phi (p,t) \right) dp
[/tex]

Assume the integrals go from -infinity to +infinity, and assume the delta function is the Dirac delta function.
 
Last edited:
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  • #2
Dear cyberdeathreaper,

It is a general property of delta functions that:

[tex]\int_{-\infty}^{\infty} f(p') \delta(p-p') dp' = f(p)[/tex]

This formula is used in what you have written.

Carl
 
Last edited:
  • #3
Thanks, I knew it was related to that. I just wasn't sure if it applied for functions of more than one variable or not.
 

1. What is the Dirac delta function?

The Dirac delta function, also known as the impulse function, is a mathematical function that is used to model the concentration of a point mass or impulse at a specific point in space or time. It is commonly represented by the symbol δ(x) and has properties such as being infinitely tall and narrow, with an area under the curve of one.

2. How is the Dirac delta function used in science?

The Dirac delta function is commonly used in various fields of science, such as physics, engineering, and signal processing. It is particularly useful in modeling and analyzing systems with sudden changes or impulses, such as electric circuits, impulse response in mechanical systems, and the behavior of particles in quantum mechanics.

3. What is the relationship between the Dirac delta function and the unit step function?

The unit step function, also known as the Heaviside function, is defined as the integral of the Dirac delta function. This means that the unit step function is equal to zero for all values less than zero and equal to one for all values greater than zero. In other words, the Dirac delta function can be thought of as the derivative of the unit step function.

4. Can the Dirac delta function be graphed?

Technically, the Dirac delta function cannot be graphed as it is an infinitely tall and narrow function. However, it is often graphically represented as a spike or pulse at the origin on a graph, with a height of infinity and a width of zero. This representation helps to visualize the properties of the function, but it is important to remember that it is not an actual graph of the function itself.

5. How do you evaluate the Dirac delta function at a specific point?

The Dirac delta function is evaluated at a specific point by using the sifting property, which states that the integral of the Dirac delta function multiplied by any continuous function f(x) is equal to the value of f(x) at the point where the Dirac delta function is centered. In other words, δ(x-a) = 0 for all x ≠ a and δ(x-a) = ∞ at x = a.

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