Number of States in a 1D Simple Harmonic Oscillator

In summary, the total number of states with energy E in a system of N 1D simple harmonic oscillators is given by \Omega(E) = \frac{(M+N-1)!}{(M!)(N-1)!}, where M is the number of excited particles and N is the total number of particles in the system. This is due to the fact that the excited oscillators do not have to be in the n=1 state, and thus the total energy can be distributed among different states.
  • #1
Ang Han Wei
9
0

Homework Statement


A system is made of N 1D simple harmonic oscillators. Show that the number of states with total energy E is given by [itex]\Omega[/itex](E) = [itex]\frac{(M+N-1)!}{(M!)(N-1)!}[/itex]


Homework Equations


Each particle has energy ε = [itex]\overline{h}[/itex][itex]\omega[/itex](n + [itex]\frac{1}{2}[/itex]), n = 0, 1

Total energy is given by E = [[itex]\frac{N}{2}[/itex] + M][itex]\overline{h}[/itex][itex]\omega[/itex], M is an integer

The Attempt at a Solution



If the entire system is in the ground state, n = 0 for all particles and the energy will be [itex]\frac{N}{2}[/itex][itex]\overline{h}[/itex][itex]\omega[/itex]

So M must be the number of excited particles having the value of n = 1

Hence, the total number of states should be the number of ways that I can choose M particles to be "excited" out of N particles.
[itex]\Omega[/itex] = [itex]\frac{N!}{(M!)(N-M)!}[/itex] but that is not the case.

I am not sure where I have gone wrong.
 
Physics news on Phys.org
  • #2
Your mistake is in assuming the excited oscillators have to be in the n=1 state. For example, say you had three oscillators. If two are in the n=1 state and one is in the n=0 state, the system has the same energy as one with one oscillator in the n=2 state and two oscillators in the n=0 state.
 
  • Like
Likes 1 person

1. How many states does a 1D simple harmonic oscillator have?

A 1D simple harmonic oscillator has an infinite number of states.

2. What is the energy of the ground state in a 1D simple harmonic oscillator?

The energy of the ground state in a 1D simple harmonic oscillator is equal to 1/2 of the oscillator's quantum number, which is equal to 1/2 of Planck's constant times the oscillator's frequency.

3. How do you calculate the number of states in a 1D simple harmonic oscillator?

The number of states in a 1D simple harmonic oscillator can be calculated using the formula N = n + 1, where n is the oscillator's quantum number.

4. Can a 1D simple harmonic oscillator have a negative number of states?

No, a 1D simple harmonic oscillator cannot have a negative number of states. The number of states must always be a positive integer or zero.

5. How does the number of states in a 1D simple harmonic oscillator relate to its energy levels?

The number of states in a 1D simple harmonic oscillator is directly proportional to its energy levels. As the energy levels increase, so does the number of states.

Similar threads

  • Advanced Physics Homework Help
Replies
24
Views
632
  • Advanced Physics Homework Help
Replies
1
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
1K
Replies
11
Views
1K
  • Advanced Physics Homework Help
Replies
5
Views
1K
  • Advanced Physics Homework Help
Replies
2
Views
1K
  • Advanced Physics Homework Help
Replies
21
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
991
Back
Top