Question

Consider a quantum state of energy $E$, which can be occupied by any number $n$ of some bosonic particles, including $n=0$. At absolute temperature $T$, the probability of finding $n$ particles in the state is given by $P_n=N\exp (-nE/k_{\mathrm{B}}T)$, where $k_{\mathrm{B}}$ is Boltzmann's constant and the normalization factor $N$ is determined by the requirement that all the probabilities sum to unity. Calculate the mean or expected value of $n$, that is, the occupancy, of this state, given this probability distribution.

Short answer

Answer: The expected occupancy (mean) of the quantum state with energy E can be expressed as: ⟨n⟩ = e^(-E / k_B * T) / (1 - e^(-E / k_B * T)).

Step-by-step solution

Calculate the normalization factor N

To calculate the normalization factor N, we need to make sure that the sum of all probabilities is equal to 1: $$\sum _{n=0}^{\mathrm{\infty }}{P}_n=1$$ Plug the equation for P_n: $$\sum _{n=0}^{\mathrm{\infty }}N{e}^{-nE/k_{B}T}=1$$ N can be factored out since it's a constant: $$N\sum _{n=0}^{\mathrm{\infty }}{e}^{-nE/k_{B}T}=1$$ Now compute N: $$N=\frac{1}{\sum _{n=0}^{\mathrm{\infty }}{e}^{-nE/k_{B}T}}$$

Calculate the expected value of n

The expected value or mean of n is given by: $$⟨n⟩=\sum _{n=0}^{\mathrm{\infty }}n{P}_n$$ Using the equation for P_n: $$⟨n⟩=\sum _{n=0}^{\mathrm{\infty }}nN{e}^{-nE/k_{B}T}$$ Plug in the expression for the normalization factor N: $$⟨n⟩=\sum _{n=0}^{\mathrm{\infty }}n\frac{e^{-nE/k_{B}T}}{\sum _{n=0}^{\mathrm{\infty }}{e}^{-nE/k_{B}T}}$$ Now, use the fact that the derivative of a geometric series with respect to x is: $$\frac{d}{dx}\sum _{n=0}^{\mathrm{\infty }}{x}^n=\sum _{n=1}^{\mathrm{\infty }}n{x}^{n-1}$$ We can rewrite the expected value of n as: $$⟨n⟩=\frac{\frac{d}{dx}\sum _{n=0}^{\mathrm{\infty }}{e}^{-nx}}{\sum _{n=0}^{\mathrm{\infty }}{e}^{-nx}}$$ Since we need to do this at x=E/(k_B T): $$⟨n⟩=\frac{\frac{d}{dx}\sum _{n=0}^{\mathrm{\infty }}{e}^{-nx}{|}_{x=E/(k_{B}T)}}{\sum _{n=0}^{\mathrm{\infty }}{e}^{-nE/k_{B}T}}$$ Let S = the series ∑e^(-nx). Using the definition of the derivative: $$⟨n⟩=\frac{\underset{h\to 0}{lim}\frac{S(e^{-Eh/k_{B}T})-S}{h}}{S}$$ Once you compute this limit, you end up with: $$⟨n⟩=\frac{e^{-E/k_{B}T}}{1-e^{-E/k_{B}T}}$$ This is the expected value or mean occupancy of the quantum state with energy E given the probability distribution P_n.

Key concepts

Bose-Einstein statistics

Understanding the behavior of particles at quantum levels requires a shift from classical statistical mechanics to quantum statistics. One of the fundamental pillars in quantum statistics is Bose-Einstein statistics, which deals specifically with indistinguishable particles known as bosons. These particles, unlike fermions, do not adhere to the Pauli Exclusion Principle and can occupy the same quantum state without any restriction.

The significance of Bose-Einstein statistics lies in its ability to describe phenomena such as superfluidity and Bose-Einstein condensates where particles coalesce into the lowest energy state, especially near absolute zero temperatures. This statistical approach is crucial when dealing with particles like photons, which are bosons, and helps us understand how laser light behaves, as all photons can be in the same quantum state, leading to a coherent light beam.

In the context of this exercise, Bose-Einstein statistics govern the probability distribution over the number of bosonic particles that can occupy a quantum state of energy E. The application of these statistics is evident when considering the occupancy of energy levels by particles such as photons in a blackbody radiation context or helium atoms in a superfluid state.

Boltzmann constant

The Boltzmann constant ($k_{\text{B}}$ ) is a fundamental physical constant that plays a key role in the realm of thermodynamics and statistical mechanics. It essentially provides a bridge between the macroscopic and microscopic worlds by connecting temperature, an observable macroscopic property, with the energy of states on a microscopic scale.

The constant is named after Ludwig Boltzmann, a significant contributor to the field of statistical mechanics, which explains and predicts how the properties of atoms and molecules result in the macroscopic behaviors we observe. The Boltzmann constant has a value of approximately 1.38×10⁻²³ joules per kelvin, establishing a proportional relationship between temperature and energy.

In our textbook exercise, the Boltzmann constant is instrumental in determining the probability of bosonic particles occupying a quantum state at a given energy level, signifying how thermal energy, quantified through temperature ($T$ ), influences quantum systems.

Geometric series

A geometric series is a series of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This series can be represented as $a,ar,a{r}^2,a{r}^3,\dots$ where $a$ is the first term and $r$ is the common ratio.

Geometric series have significant importance in mathematics, particularly when it comes to convergent series where $|r|<1$. The sum of an infinite convergent geometric series is given by $S=\frac{a}{1-r}$, an elegant expression that deeply ties into various aspects of financial mathematics, physics, and theoretical frameworks such as probability theory.

Within the scope of this exercise, we use the properties of the geometric series to sum the probabilities and calculate the normalization factor necessary for determining the expected value. This showcases the geometric series as a powerful tool in simplifying complex infinite sums arising in quantum statistics.

Expected value

The concept of expected value, also known as mean or average, is a fundamental idea in probability and statistics that provides a measure of the center of a probability distribution. It signifies the weighted average of all possible outcomes of a random variable, where each outcome is weighted by its probability of occurrence.

In practical terms, the expected value helps to predict the long-term average result of a random event when the process is repeated multiple times. For example, the expected value in gambling games like roulette or dice games can inform players about their long-term wins or losses.

As applied in our exercise, the expected value is used to calculate the mean occupancy of a quantum state by bosonic particles. Using Bose-Einstein statistics and the properties of the geometric series, students can determine the average number of particles in an energy state given a specific temperature, vital for understanding the behavior of systems such as lasers or superfluids.