Question

The probability of occupying a given excited state \(p_{i}\) is given by \(p_{i}=n_{i} / N=e^{-\beta \kappa_{i}} / q,\) where \(n_{i}\) is the occupation number for the state of interest, \(N\) is the number of particles, and \(\varepsilon_{i}\) is the energy of the level of interest. Demonstrate that the preceding expression is independent of the definition of energy for the lowest state.

Short answer

To demonstrate that the given expression for the probability of occupying an excited state, \(p_{i}\), is independent of the energy definition for the lowest state, we redefine the energy of the lowest state with a constant, \(\eta\), and determine the new probability expression. The new probability expression becomes, \[p_i ' = C \frac{e^{- \beta\kappa_i}}{q'}\] Comparing this with the original probability expression, we observe that the ratio between probabilities remains unchanged, as the only difference is the presence of the constant factor \(C\) and the new partition function \(q'\). Since \(C\) does not depend on the specific energy level \(\kappa_i\), the overall expression for the probability remains independent of the definition of energy for the lowest state.

Step-by-step solution

Identify the given expression for the probability p_i

The given expression for the probability of occupying a given excited state, \(p_{i}\), is: \[p_{i} = \frac{n_{i}}{N} = \frac{e^{- \beta \kappa_{i}}}{q}\] where, - \(p_{i}\) is the probability of occupying an excited state i - \(n_{i}\) is the occupation number for the state of interest - \(N\) is the number of particles - \(\kappa_{i}\) is the energy of the level of interest - \(\beta = \frac{1}{k_B T}\), where \(k_B\) is Boltzmann's constant and \(T\) is the temperature - \(q\) is the partition function

Define the energy of the lowest state with respect to a new reference

Now, let's redefine the energy of the lowest state (state 0) with respect to a new reference point by adding a constant, \(\eta\), to all energy levels. The new energy levels will be given by: \[\kappa_i ' = \kappa_i + \eta\] where \(\eta\) is a constant and \(\kappa_i ' \) is the new energy level.

Determine the new probability expression

With the new energy levels, the probability expression becomes, \[p_i ' = \frac{e^{- \beta (\kappa_i + \eta)}}{q '}\] where \(q'\) is the new partition function corresponding to the new energy levels.

Analyze the new probability expression

Our goal is to show that \(p_i ' = p_i\). Let's rewrite the probability expression using the exponential property: \[p_i ' = \frac{e^{- \beta\kappa_i}e^{-\beta\eta}}{q'}\] Notice that \(e^{-\beta\eta}\) only depends on the constant \(\eta\) and does not depend on the particular energy level, and so we can regard it as a constant. Thus, \[p_i ' = C \frac{e^{- \beta\kappa_i}}{q'}\] where \(C = e^{-\beta\eta}\).

Show independence of the lowest energy level definition

In order to show that the new probability expression, \(p_i '\), is independent of the lowest energy level definition, we need to compare it with the initial probability expression, \(p_i\), and see if there is any dependence on \(\eta\), which represents the change in energy level definition. We can rewrite the new probability expression, \(p_i '\), as: \[p_i ' = C \frac{e^{- \beta\kappa_i}}{q'}\] By comparing this with the initial probability expression, \[p_i = \frac{e^{- \beta\kappa_i}}{q}\] we can see that the only difference is the presence of the constant factor \(C\) and the new partition function \(q'\). However, although \(C\) depends on \(\eta\), this constant factor multiplies all terms equally, and does not depend on the specific energy level \(\kappa_i\). Thus, the fundamental ratio between the probabilities does not depend on the definition of the lowest energy level, confirming that the given expression for the probability, \(p_i\), is independent of the definition of energy for the lowest state.

Key concepts

Boltzmann Distribution

The Boltzmann Distribution is a fundamental concept in statistical thermodynamics that helps us understand the probability of different energy states being occupied by particles in a system. Essentially, it describes how particles distribute themselves among their available energy levels at a given temperature. The formula for the probability of a system being in a particular energy state is given by:

• \[ p_{i} = \frac{e^{-\beta \kappa_{i}}}{q} \]
• where \( p_{i} \) is the probability of finding a particle in state \( i \)
• \( \beta = \frac{1}{k_B T} \), with \( k_B \) as Boltzmann's constant and \( T \) the temperature
• \( \kappa_{i} \) is the energy of state \( i \)
• \( q \) is the partition function

In simpler terms, the lower the energy of a state, the more likely particles will occupy it. Conversely, higher energy states are less populated, especially as temperature decreases. The Boltzmann Distribution thus offers a clear picture of how energy and temperature play a crucial role in the distribution of particles across various energy levels.

Partition Function

The partition function, denoted as \( q \), is a central component of statistical mechanics. It sums up all the possible states in a system, weighted by their exponential factors, and serves as a key to connecting microscopic states to macroscopic thermodynamic properties. The formula is:

• \[ q = \sum_{i} e^{-\beta \kappa_{i}} \]
• where \( \kappa_{i} \) represents the energy of each possible state \( i \)
• The sum goes over all states \( i \)

This function essentially acts as a normalizing factor, ensuring that all probabilities derived from Boltzmann's equation add up to one. It provides insight into the statistical distribution of a system and is crucial for calculating other thermodynamic quantities, like internal energy and entropy. In essence, the partition function encodes the information of how particles are distributed across the energy levels and facilitates the calculation of average properties of the system.

Energy Levels

Energy levels denote the specific energies that particles can occupy within a system. In the context of statistical thermodynamics, these energy levels dictate how particles distribute themselves among these states. Understanding the energy levels is crucial because they determine:

• How particles populate different states through the Boltzmann Distribution
• The calculation of the partition function, which is a sum over all these energy levels

When we analyze the problem of redefining the energy of the lowest state, adding a constant \( \eta \), it demonstrates that the absolute energy values are not physically meaningful as long as the differences between energy levels remain unchanged. What truly matters in statistical thermodynamics is the relative spacing of these energy levels and how they affect population distribution and macroscopic properties. This principle helps simplify complex systems by allowing flexible definitions of zero potential energy, thus emphasizing the change in energy levels over the absolute values.