# Second moment of occupation number for bosons

• Rayan
Rayan
Homework Statement
Show that for bosons

$$\overline{(\Delta \eta)^2} \quad=\quad \overline{ \eta_{BE}} (1 + \overline{ \eta_{BE}} )$$

where

$$(\Delta \eta) \quad=\quad \eta - \overline{ \eta }$$
Relevant Equations
.
I tried to show this equality by explicitly determining what

$$\overline{(\Delta \eta)^2}$$

is, but I got a totally different answer for some reason, here is my attempt to solve it, what did I miss?

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Last edited:
Note that
$$\Delta n^2 = \overline{n^2}-\overline{n}^2=\langle n^2 \rangle - \langle n \rangle^2.$$
Then indeed you have in the grand-canonical ensemble
$$Z(\beta,\alpha)=\prod_{\vec{p}} \sum_{n=0}^{\infty} \exp[n (-\beta \omega_{\vec{p}}+\alpha)] = \prod_{\vec{p}} \frac{1}{1-\exp(-\beta \omega_{\vec{p}}+\alpha)}.$$
It's easier to work with
$$\Omega=\ln Z=-\sum_{\vec{p}} \ln [ 1-\exp(-\beta \omega_{\vec{p}}+\alpha)].$$
Now
$$\langle n \rangle=\frac{1}{Z} \partial_{\alpha} Z =\partial_{\alpha} \ln Z=\sum_{\vec{p}} f_{\text{B}}(\omega_{\vec{p}})=\sum{\vec{p}} \frac{1}{\exp(\beta \epsilon-\alpha)}.$$
Further you have
$$\partial_\alpha^2 \Omega =\partial_{\alpha} \left ( \frac{1}{Z} \partial_{\alpha Z} \right) =\frac{1}{Z} \partial_{\alpha}^2 Z -\left (\frac{1}{Z} \partial_{\alpha} Z \right)=\langle n^2 \rangle -\langle n \rangle^2=\Delta n^2.$$
Then you get
$$\Delta n^2 = \partial_{\alpha}^2 \ln Z=\sum_{\vec{p}} \partial_{\alpha} f_{\text{B}}(\omega_{\vec{p}}).$$
It's easy to check, that this gives what you try to prove.

## What is the second moment of occupation number for bosons?

The second moment of occupation number for bosons refers to the expectation value of the square of the number of particles occupying a particular quantum state. It provides information about the fluctuations in the number of particles in that state.

## How is the second moment of occupation number calculated for bosons?

For bosons, the second moment of the occupation number n_i for a given state i can be calculated using the formula: ⟨n_i^2⟩ = ⟨n_i⟩ + ⟨n_i⟩^2, where ⟨n_i⟩ is the average occupation number of the state. This takes into account the Bose-Einstein statistics that bosons obey.

## Why is the second moment of occupation number important in quantum statistics?

The second moment of occupation number is important because it provides insight into the quantum statistical properties of a system, especially regarding the fluctuations and correlations between particles in different states. These fluctuations are critical in understanding phenomena like Bose-Einstein condensation.

## What does a high second moment of occupation number indicate in a bosonic system?

A high second moment of occupation number indicates significant fluctuations in the number of particles occupying a particular state. This can be a sign of collective behaviors such as Bose-Einstein condensation, where many bosons occupy the same quantum state.

## How does the second moment of occupation number differ between bosons and fermions?

For fermions, the second moment of occupation number is given by ⟨n_i^2⟩ = ⟨n_i⟩, due to the Pauli exclusion principle which restricts fermions from occupying the same state more than once. In contrast, bosons can occupy the same state multiple times, leading to ⟨n_i^2⟩ = ⟨n_i⟩ + ⟨n_i⟩^2, which reflects the enhanced fluctuations due to Bose-Einstein statistics.

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