Correction in Phase Space Calculation for Identical Particles

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Questions and Answers

Why must the phase space equation Ω be corrected by dividing by N! when dealing with N identical particles?

To account for the difficulty with an earlier presented formula in statistical mechanics

To account for the fact that swapping identical particles does not lead to a new microstate (correct)

To account for the indistinguishability of the particles

To account for the units of measurement for position (q) and momentum (p)

How does the entropy formula S change when considering the indistinguishability of particles?

The entropy formula is modified to incorporate the correction by dividing Ω by N! (correct)

The entropy formula is modified to account for the difficulty with an earlier presented formula in statistical mechanics

The entropy formula is modified to account for the units of measurement for position (q) and momentum (p)

The entropy formula remains the same, as the indistinguishability of particles is not relevant

How does the text explain the need to consider the indistinguishability of particles when calculating entropy?

The text explains that classical mechanics already accounts for the indistinguishability of particles

The text does not explain the need to consider the indistinguishability of particles when calculating entropy

The text explains that the indistinguishability of particles is only relevant in quantum mechanics, not in classical statistical mechanics

The text explains that the indistinguishability of particles needs to be considered to correctly calculate the entropy of a system of identical particles (correct)

What is the relationship between the indistinguishability of particles and the formula for entropy change when mixing gases?

The text explains how the entropy change for mixing gases can be recalculated when considering volume ratios, which accounts for the indistinguishability of particles Signup and view all the answers

What is the role of quantum mechanics in addressing the issue of indistinguishability of particles?

Quantum mechanics resolves the issue of indistinguishability of particles by symmetrization of the wave function Signup and view all the answers

What is the specific difficulty mentioned in the text regarding an earlier presented formula in statistical mechanics?

The difficulty is that the formula contains an arbitrary constant that changes with the units of measurement for position (q) and momentum (p) Signup and view all the answers

What does the correction factor N! represent in the modified entropy formula?

The number of possible permutations of N particles Signup and view all the answers

Why is the indistinguishability of particles important in calculating entropy?

It affects the statistical weight of microstates Signup and view all the answers

What is the limitation of classical mechanics mentioned in the text?

It cannot account for the indistinguishability of particles Signup and view all the answers

What is the role of quantum mechanics in addressing the indistinguishability of particles?

It symmetrizes the wave function to account for indistinguishability Signup and view all the answers

What is the specific difficulty mentioned regarding an earlier presented formula in statistical mechanics?

The formula contained an arbitrary constant that changed with units Signup and view all the answers

How does the text explain the need to recalculate entropy change when mixing gases?

It considers the volume ratios of the gases being mixed Signup and view all the answers

Study Notes

Correction for Identical Particles in Phase Space Calculation

The phase space Ω needs to be corrected by a division by N! (number of possible permutations of N particles) to account for the indistinguishability of identical particles.
Swapping identical particles does not lead to a new microstate, meaning we have overcounted the phase space.

Modified Entropy Formula

The corrected formula for entropy (S) incorporates the division by N! to account for the indistinguishability of particles.
This correction is important for accurately calculating the entropy of a system of identical particles.

Entropy Change for Mixing Gases

The entropy change for mixing gases can be recalculated by considering volume ratios.
Location permutations of identical particles need to be considered to obtain an accurate calculation.

Limitation of Classical Mechanics

Classical mechanics does not account for the indistinguishability of particles.
Quantum mechanics resolves this issue through symmetrization of the wave function.

Difficulty in Statistical Mechanics Formula

An earlier presented formula in statistical mechanics has a difficulty due to an arbitrary constant that changes with the units of measurement for position (q) and momentum (p).

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Description

This quiz covers the correction needed in the phase space calculation for N identical particles due to the indistinguishability of the particles. It explains how to account for overcounting in phase space by correcting the formula with a division by N! to consider the non-uniqueness of particle permutations.