Understanding Transition btwn Steps of Dirac Delta Function

AI Thread Summary
The discussion revolves around understanding the transition between two mathematical expressions involving the Dirac delta function. The key point is that the integral of a function multiplied by the delta function simplifies to the function evaluated at the point where the delta function is centered. This property holds true even for functions of multiple variables, as demonstrated in the provided equations. The participants confirm that the relationship applies universally, reinforcing the foundational aspect of delta functions in integral calculus. The clarification enhances comprehension of the delta function's role in simplifying complex integrals.
cyberdeathreaper
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Can someone help me understand the transition between these two steps?
<br /> &lt;x&gt; = \iint \Phi^* (p&#039;,t) \delta (p - p&#039;) \left( - \frac{\hbar}{i} \frac{\partial}{\partial p} \Phi (p,t) \right) dp&#039; dp<br />
=
<br /> &lt;x&gt; = \int \Phi^* (p,t) \left( - \frac{\hbar}{i} \frac{\partial}{\partial p} \Phi (p,t) \right) dp<br />

Assume the integrals go from -infinity to +infinity, and assume the delta function is the Dirac delta function.
 
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Dear cyberdeathreaper,

It is a general property of delta functions that:

\int_{-\infty}^{\infty} f(p&#039;) \delta(p-p&#039;) dp&#039; = f(p)

This formula is used in what you have written.

Carl
 
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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.
 
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