Question

In quantum theory, a system of oscillators, each of fundamental frequency \(v\), interacting at temperature \(T\) has an average energy \(\bar{E}\) given by $$ \bar{E}=\frac{\sum_{n=0}^{\infty} n h v e^{-n x}}{\sum_{n=0}^{\infty} e^{-n x}} $$ where \(x=h v / k T, h\) and \(k\) being the Planck and Boltzmann constants respectively. Prove that both series converge, evaluate their sums, and show that at high temperatures \(\bar{E} \approx k T\) whilst at low temperatures \(\bar{E} \approx h v \exp (-h v / k T)\)

Short answer

\( € \bar{E} = \frac{h v }{e^{x} - 1} \) converges. At high temperatures: \( \bar{E} \approx k T \), and low temperatures: \( \bar{E} \approx h v e^{-\frac{h v}{k T}} \).

Step-by-step solution

Understanding the Series

The average energy \(\bar{E}\) is given by \(\bar{E} = \frac{\beta}{\theta}\), where \(\beta = \frac{\text{sum of series with n}}{\text{sum of series without n}}\). We will first prove the convergence of both series.

Prove Convergence of Denominator Series

Consider the denominator series \(\theta = \sum_{n=0}^{\text{∞}} e^{-n x} \). This is a geometric series of the form \(\theta = 1 + e^{-x} + (e^{-x})^2 + \text{...}\), which converges if \(|e^{-x}| < 1\). For any real \(x > 0\), the series converges and sums to \( \theta = \frac{1}{1 - e^{-x}}\).

Prove Convergence of Numerator Series

Now consider the numerator series \(\beta = \sum_{n=0}^{\text{∞}} n h v e^{-n x} \). This series is equivalent to \(hv \sum_{n=0}^{\text{∞}} n e^{-n x} \). Using the result of the geometric series sum, its derivative w.r.t. \(e^{-x}\) is \( \frac{e^{-x}}{(1 - e^{-x})^2} \). Thus, \( \beta = h v \frac{e^{-x}}{(1 - e^{-x})^2} \) and the series converges.

Sum of the Series

Thus, \(\bar{E} = \frac{h v \frac{e^{-x}}{(1 - e^{-x})^2}}{\frac{1}{1 - e^{-x}}} = \frac{h v e^{-x}}{1 - e^{-x}}\) remains to be simplified.

Simplify the Expression

Simplify the result: \( \bar{E} = \frac{h v }{e^{x} - 1}\).

Evaluate at High Temperatures

At high temperatures, \( x = \frac{h v}{k T} \) is very small. Use \(e^{x} \approx 1 + x\), and \(\bar{E} \approx \frac{h v }{x} = k T\).

Evaluate at Low Temperatures

At low temperatures, \( x = \frac{h v}{k T} \) is very large, \(e^{x} \gg 1\), then \(\bar{E} \approx \frac{h v e^{-x}}{1} = h v e^{-\frac{h v}{k T}}\).

Key concepts

Quantum Theory

Quantum theory describes the physics of subatomic particles. Unlike classical mechanics, quantum mechanics uses probabilities to predict how particles move and interact. One critical aspect is quantization, where particles exist in discrete energy levels rather than a continuous spectrum.
In terms of oscillators, each has a fundamental frequency denoted as \(v\). When these oscillators interact with their environment at a temperature \(T\), the average energy they possess is quantized and can be calculated using the given formula.

Oscillator

An oscillator is a system that undergoes periodic motion, such as a pendulum. In quantum mechanics, oscillators are modeled mathematically to explain phenomena at the atomic scale.
The average energy of a quantum oscillator involves both its fundamental frequency \(v\) and temperature \(T\), accounting for factors like Planck's constant \(h\) and Boltzmann's constant \(k\).
The concept is essential for interpreting how energy is distributed among particles at different temperatures.

High and Low Temperature Limits

Temperature affects the behavior of quantum oscillators significantly. At high temperatures, where \(T\) is large, the term \(x = \frac{hv}{kT}\) becomes very small.
This makes the exponential function \(e^{x} \approx 1 + x\), simplifying the average energy to approximately \(kT\).
At low temperatures, where \(T\) is small, \(x\) becomes large, making \(e^{x} \gg 1\). In this case, the average energy can be approximated as \(hv e^{-\frac{hv}{kT}}\). This demonstrates the relationship between temperature and quantum energy levels.

Series Convergence

Both the numerator and the denominator of the average energy formula are infinite series. To work with these series, we must prove they converge.
For the denominator, which is a geometric series \(\sum_{n=0}^{\infty} e^{-nx}\), it converges if \(|e^{-x}| < 1\).
Similarly, for the numerator \(\sum_{n=0}^{\infty} n h v e^{-nx}\), represented as \(hv \sum_{n=0}^{\infty} n e^{-nx}\), the convergence follows if the series can be summed and has a finite value.
Both series meet these criteria under standard conditions for \(x > 0\).

Geometric Series

A geometric series is a series with a constant ratio between successive terms. It takes the form \(1 + r + r^2 + r^3 + \text{...}\). This series converges if the absolute value of the common ratio \(|r|<1\).
In the context of the quantum oscillator energy equation, the denominator is modeled as a geometric series with a ratio \(e^{-x}\).
This series sums to \(\frac{1}{1 - e^{-x}}\), showcasing the series' convergence and how it simplifies complex expressions in quantum calculations.