Dimensionality of Phase Space

[JordanGo]

Homework Statement

A classical gas consists of N molecules; each molecule is composed of two atoms
connected by a spring. Identify the dimensionality of the phase space that can be used to
describe a microstate of the system.

The Attempt at a Solution

I believe the answer is 12, but I'm not sure why. Since there are three cartesian coordinates (say x,y,z) and 3 corresponding conjugate momenta (px,py,pz). Thus the dimensionality is 6. But why would you multiply it by 2 to get 12?

* Should I say 3N coordinates and 3N conjugate momenta?

 

[kevinferreira]

Your phase space should represent the degrees of freedom of your system, don't you think? Then, it would be dependent on N, as each particle has its own degrees of freedom.

 

[JordanGo]

so each molecule has two atoms, each atom has 6 degrees of freedom, thus the system has 12 degrees of freedom and since there is N molecules, the dimensionality is 12N?

 

[Fredrik]

You seem to be ignoring the information that each molecule consists of two atoms connected by a "spring".

Instead of counting the degrees of freedom of each atom, I suggest that you first pretend that each molecule is a point particle and count the coordinates you need in this case. Then you start thinking about the coordinates you need to deal with a single molecule at a known location.

 

[JordanGo]

Ok, so if I look at the molecule as two points (since there is two atoms per molecule) connected by a spring, each point has 6 coordinates that describe it. The coordinates being, say, x, y, z, θ, ∅ and d (the length of the spring). Now, since there is 2 points in our system, we have x1, x2, y1, y2, z1, z2, θ1, θ2, ∅1, ∅2, d1 and d2 to describe our system, which is 12 coordinates. Now, since there is N number of molecules, the dimensionality is then 12N.

Is this correct?

 

[kevinferreira]

How many coordinates do you need to describe one single molecule (for example its center of mass)?
Then, in the center of mass frame of one molecule, how many coordinates do you need to describe the two atoms?
Then sum everything.

 

[Fredrik]

I think you will need to use the approach suggested by me in #4 and explained further by kevinferreira in #6. It's a good way to see e.g. that d1 and d2 aren't good choices.

Spherical coordinates are not a good enough way to represent a direction in space. Do you see why?

I recommend that you don't use ∅ instead of φ. That symbol only represents the empty set.