How Do You Convert d3p to a Function of Frequency Nu in Spherical Coordinates?

[erok81]

Homework Statement

Write the volume element of d³p as a function of "nu". Assume spherical symmetry in doing this change of variables so write d³p = 4$$\pi$$p²dp.

Homework Equations

$$n(\nu)=\frac{1}{e^{\frac{h \nu}{kT}} -1}$$

$$\epsilon=\frac{2}{h^3}\int h \nu \cdot n(\nu)d^3p$$

The Attempt at a Solution

I have zero idea of where to even start with this. As stupid as this is, I don't even understand "d³p = 4$$\pi$$p²dp" or what the d³p even is. I don't think I've ever come across an integral that has used this type of notation before.

Any help to even get me started would be appreciated.

 

[shreepy]

The p in the equation above is simply the momentum of photons. There is a relationship between momentum and frequency for photons:

p = $$\frac{h\nu}{c}$$

With this you can figure out p^2 ( simply square it) and a relationship between dp and dv. With both of these pieces of information you should be able to replace d3p with an expression that is entirely in terms of dv , v and constants. Your new integration range will simply be over all of the frequencies so from 0 to infinity.

 

[erok81]

The p in the equation above is simply the momentum of photons. There is a relationship between momentum and frequency for photons:

p = $$\frac{h\nu}{c}$$

With this you can figure out p^2 ( simply square it) and a relationship between dp and dv. With both of these pieces of information you should be able to replace d3p with an expression that is entirely in terms of dv , v and constants. Your new integration range will simply be over all of the frequencies so from 0 to infinity.

Ah got it. That helps out a ton. Thank you.

I couldn't figure out what that d^3p term meant.

 

[erok81]

So I am pretty sure I stumbled through this and got it mostly right, but I would I really like to understand this for future problems. Here is what I did...at least from what I remember. I already turned in my assignment a like I said...I stumbled through it.

$$n(\nu)=\frac{1}{e^{\frac{h \nu}{kT}} -1}$$

$$\epsilon=\frac{2}{h^3}\int h \nu \cdot n(\nu)d^3p$$

Combining the two.

$$\epsilon=\frac{2}{h^3}\int h \nu \cdot \frac{1}{e^{\frac{h \nu}{kT}} -1} d^3p$$

Part 1 I don't get.

d³p = 4Π²dp Where does this change occur? i.e. how would one know to do this?

Using that and subbing into my integral so far.

$$\epsilon=\frac{2}{h^3}\int h \nu \cdot \frac{1}{e^{\frac{h \nu}{kT}} -1} 4\pi p^2 dp$$

Here is where I probably messed up as I don't think I ever actually learned how to do this. My answer did match another integral in book. So that's a plus.

$$p=\frac{h\nu}{c}$$

Using my masterful, and most likely wrong subbing skills I come up with this.

$$p=\frac{h\nu}{c}$$

$$dp=\frac{h}{c}d\nu$$

$$\frac{c}{h}dp=d\nu$$

Then substitute again...

$$\epsilon=\frac{2}{h^3}\int h \nu \cdot \frac{1}{e^{\frac{h \nu}{kT}} -1} 4\pi \left(\frac{h \nu}{c}\right)^2 \frac{c}{h} d\nu$$

This eventually simplifies to...

$$\frac{8\pi}{hc}\int \frac{\nu^3}{e^{\frac{h \nu}{kT}} -1} d\nu$$


Now...I am quite certain this isn't right. I really had no idea what to do here so I just tried to make it match the final integral I found in my text using u-sub type rules. I don't think I've ever had to find relationships between dp and dv or any other sort of integral. So you can see my confusion.

Any pointers or tips would be appreciated. I tried searching google for videos/lectures on this subject but couldn't find anything.

 

[rwisz]

Don't worry, it's completely normal to feel overwhelmed when encountering new notation or concepts in science. Let's break down the problem together.

First, let's define some terms:

- d3p: This is the volume element, which represents an infinitesimal volume in three-dimensional space. In other words, it is a very small volume that is being considered in the problem.
- nu (\nu): This is the frequency variable that is being used in the problem.
- dp: This is the momentum variable, which represents the change in momentum in the infinitesimal volume d3p.

Now, let's look at the equation d3p = 4\pip2dp. This is a change of variables that is assuming spherical symmetry. This means that the volume element d3p can be written in terms of the frequency variable nu and the momentum variable dp. The notation 4\pip2 is just a way of writing the constant 4π^2.

So, to write the volume element d3p as a function of nu, we can use the following equation:

d3p = 4\pip2dp = 4\pip2 \cdot \frac{dp}{d\nu} \cdot d\nu

This equation is essentially saying that the infinitesimal volume d3p can be written as the product of the infinitesimal change in momentum dp, the derivative of momentum with respect to frequency (dp/d\nu), and the infinitesimal change in frequency d\nu.

Now, let's look at the second part of the problem, which involves the black body energy spectrum. The equation n(\nu)=\frac{1}{e^{\frac{h \nu}{kT}} -1} represents the number of particles with a given frequency nu in a black body at a certain temperature T. This equation is known as the Planck distribution.

The equation for energy density \epsilon=\frac{2}{h^3}\int h \nu \cdot n(\nu)d^3p is a way of calculating the amount of energy per unit volume in a black body at a certain temperature. It involves integrating the product of the frequency nu, the Planck distribution n(\nu), and the volume element d3p over all possible momentum values.

So, to summarize, the problem is asking you to write the volume element d3p in terms of the frequency variable nu, assuming spherical symmetry. This volume element will then