Statistical Physics - counting states

In summary, the conversation discusses the number of states and density of states for a system of N 3-dimensional quantum harmonic oscillators. The total number of states from energy E_0 to E is given by a formula shown in class, and the density of states for energy E is determined by counting the number of solutions of a particular equation. This can be visualized by imagining painting a certain number of objects with a certain number of colors.
  • #1
1. Homework Statement [/b]
There are N 3-dimensional quantum harmonic oscillators, so the energy for each one is:
[tex]E_i = \hbar \omega (\frac{1}{2} + n_x^i + n_y^i + n_z^i)[/tex]. What is the total number of states from energy E_0 to E, and what is the density of states for E?

The Attempt at a Solution

In class they've shown that for a 1-D harmonic oscillator:

[tex] \Omega(E) = \frac{(\frac{E-E_0}{\hbar \omega} + N-1)!}{(\frac{E-E_0}{\hbar \omega})!(N-1)!} \delta E[/tex].

How did they get that? I Don't understand how it becomes (roughly) [tex] (\sum_i n_i + N) choose N [/tex]. And so I really have no idea how to generalize it to 3D (even though I guess it should be just the same).
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  • #2
In the one dimensional case:

E_0 = N hbar omega/2


n_1 + n_2 + ...n_N = R (1)

where R = (E - E_0 )/(h omega)

So, you need to count the number of solutions of eq. (1). Then because we always specify a small energy interval delta E and count the number of states in that inteval, you need to multiply the answer by delta E/(h omega), because this is the number of possible values of R when E is specified with an uncertainty delta E.

Suppose I want to paint R objects with N colors. If n_j is the number of objects with color j, then all the n_j sum up to R, so any coloring is a solution of (1). Also, any solution of (1) defines a coloring. We can count the number of colorings by imagening a bog box with N compartments filled with the N types of paint. There are then N - 1 separation walls. We can then schematically denote any coloroing as a string:


where the "o" are the objects and the "|" are the separations between the compartments. The number of possible strings is thus given by:

Binomial[R + N-1,R] = (R+N-1)!/[R! (N-1)!]

1. What is statistical physics and why is it important?

Statistical physics is a branch of physics that uses statistical methods to study the behavior of large systems of particles. It is important because it allows us to understand and predict the behavior of complex systems, such as gases, liquids, and solids, by considering the statistical properties of their individual particles.

2. What is the concept of counting states in statistical physics?

In statistical physics, counting states refers to the process of determining the number of different ways a system can be arranged or configured. This is important because it allows us to calculate the probability of a particular state occurring and make predictions about the behavior of the system.

3. How is entropy related to counting states in statistical physics?

Entropy is a measure of the disorder or randomness of a system. In statistical physics, it is related to counting states because the higher the number of possible states a system can have, the higher its entropy will be. This means that a more disordered or random system will have a higher number of possible states.

4. What is the difference between microstates and macrostates in statistical physics?

Microstates refer to the individual configurations or arrangements of particles in a system, while macrostates refer to the overall properties of the system, such as temperature, pressure, and volume. Counting microstates allows us to determine the probability of a system being in a particular macrostate.

5. How does statistical physics explain the behavior of gases?

Statistical physics explains the behavior of gases by considering the interactions of individual gas particles, such as their velocities and positions. By counting the possible states of these particles, we can predict macroscopic properties of the gas, such as pressure and temperature, and explain phenomena such as diffusion and thermal expansion.

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