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Integration question

According to a textbook, this intergral

U=12∫ℏω3NWδ(ω−ωE)  dωU=\dfrac{1}{2}\displaystyle\int \hbar \omega \dfrac{3N}{W} \delta (\omega - \omega_E ) \; d \omega

is evaluated as

U=3N2WℏωEU=\dfrac{3N}{2W} \hbar \omega_E

(where N and W are constants)

I don't get how this is done, or how the delta is treated. What does a delta even mean in this context? And why is there no omega squared term?

Anyone any ideas?
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davros
16
Original post by Plato's Trousers
According to a textbook, this intergral

U=12∫ℏω3NWδ(ω−ωE)  dωU=\dfrac{1}{2}\displaystyle\int \hbar \omega \dfrac{3N}{W} \delta (\omega - \omega_E ) \; d \omega

is evaluated as

U=3N2WℏωEU=\dfrac{3N}{2W} \hbar \omega_E

(where N and W are constants)

I don't get how this is done, or how the delta is treated. What does a delta even mean in this context? And why is there no omega squared term?

Anyone any ideas?


Is this an infinite integral, or an integral over all space or time, by any chance?

Looks like that delta is the Dirac delta function which is technically a distribution - it's a very weird object which is zero everywhere except at the value within the brackets at which point it becomes infinite!

The basic useful property you need to know is that

∫f(x)δ(x−a)dx=f(a)\int f(x) \delta(x-a) dx = f(a)
Original post by davros
Is this an infinite integral, or an integral over all space or time, by any chance?

Looks like that delta is the Dirac delta function which is technically a distribution - it's a very weird object which is zero everywhere except at the value within the brackets at which point it becomes infinite!

The basic useful property you need to know is that

∫f(x)δ(x−a)dx=f(a)\int f(x) \delta(x-a) dx = f(a)


aha! That'll be it.

Thanks :smile: +rep
(edited 11 years ago)

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