# [Statistical Physics] Probability of finding # photons in the mode

• Flucky
In summary: It should be fixed now.In summary, the conversation discusses calculating the probability of finding 0 or 1 photons in an optical mode in a cavity containing black body radiation at a temperature of 500 K. The mean number of photons in the mode is found to be 1.855. The relevant equation for finding the probability of a certain number of photons is P(n) = (1 - e^(-ħω/kT))/e^(-nħω/kT), where n is the number of photons in the mode. However, this equation may yield probabilities greater than 1, indicating that it may not be entirely accurate.
Flucky

## Homework Statement

A cavity contains black body radiation at temperature T = 500 K. Consider an optical mode in the cavity with frequency ω=2.5x10$^{13}$ Hz. Calculate

a) the probability of finding 0 photons in the mode
b) the probability of finding 1 photon in the mode
c) the mean number of photons in the mode.

## Homework Equations

Possibly <n> = $\frac{1}{exp(\frac{\hbar \omega}{k_{b} T})}$

## The Attempt at a Solution

Plugging the numbers into the equation above gives the answer to c (I think), which comes out to 1.855. However I thought that you could only have 0 or 1 photons in a given mode.

Not sure how to go about a) and b).

Photons are bosons. Bosons do not follow Pauli's exclusion principle. Any number (from zero to infinity) of photons may occupy any given mode.

Ah ok, so the mean number of photons might still be ok.

Just out of curiosity: how can you get <n> > 1 if all the factors in the exponential are > 0 ?

You've just made me realize the equation should have a -1 on the bottom. Can't edit my original post for some reason.

Ok think I've got the relevant equation for a) and b) now.

P(n) = $\frac{1 - exp(-\frac{ħw}{kT})}{exp(\frac{nħw}{kT})}$

where n is the number of photons in the mode.

Last edited:
Flucky said:
Ok think I've got the relevant equation for a) and b) now.

P(n) = $\frac{1 - exp(-\frac{ħw}{kT})}{exp(-\frac{nħw}{kT})}$

where n is the number of photons in the mode.

Note that in your expression P(n) → ∞ as n→∞. So, it can't be correct since a probability can't be greater than 1.

Maybe you need to switch the numerator and denominator.

Ooh thanks for pointing that out, I accidentally put a minus in there.

## 1. What is the significance of the probability of finding photons in a mode in statistical physics?

The probability of finding photons in a mode is an important concept in statistical physics because it helps us understand the behavior of large systems of particles. In statistical physics, we use probability distributions to describe the likelihood of different states of a system, and the probability of finding photons in a mode is one such distribution that can tell us how likely it is to find a certain number of photons in a given mode.

## 2. How is the probability of finding photons in a mode related to the energy of the system?

In statistical physics, the probability of finding photons in a mode is directly related to the energy of the system. This relationship is described by the Boltzmann distribution, which states that the probability of finding a system in a given state is proportional to the energy of that state. This means that the higher the energy of a system, the more likely it is to find more photons in a particular mode.

## 3. Can the probability of finding photons in a mode change over time?

Yes, the probability of finding photons in a mode can change over time. In statistical physics, we use equations such as the Boltzmann distribution and the Maxwell-Boltzmann distribution to describe the evolution of a system over time. These equations take into account factors such as temperature and energy to determine how the probability of finding photons in a mode will change over time.

## 4. How does the number of photons in a mode affect the overall probability of finding photons in a mode?

The number of photons in a mode has a direct impact on the overall probability of finding photons in that mode. As the number of photons increases, the overall probability also increases. This is because the probability distribution is skewed towards higher energy states, and as the number of photons increases, the system is more likely to be in a higher energy state where more photons are present in a particular mode.

## 5. How does the probability of finding photons in a mode relate to the concept of entropy?

In statistical physics, entropy is a measure of the disorder or randomness of a system. The probability of finding photons in a mode is related to entropy through the Boltzmann distribution, which states that the entropy of a system is directly proportional to the natural logarithm of the number of microstates (possible arrangements of particles) corresponding to a given macrostate (observable properties of the system, such as the number of photons in a mode). This means that as the probability of finding photons in a mode increases, the entropy of the system also increases.

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