Question

For a two-level system, the weight of a given energy distribution can be expressed in terms of the number of systems \(N\) and the number of systems occupying the excited state \(n_{1}\). What is the expression for weight in terms of these quantities?

Short answer

For a two-level system with \(N\) total systems and \(n_{1}\) systems in the excited state, the expression for the weight of a given energy distribution is: \( W = \frac{N!}{n_{1}!(N - n_{1})!} \)

Step-by-step solution

Define the problem

In a two-level system, the total number of systems \(N\) can be partitioned into two groups: systems in the excited state \(n_{1}\) and systems in the ground state \(N - n_{1}\). We are to find the total number of different ways of distributing the systems into these two groups (i.e., the number of unique configurations), which represents the weight.

Apply the combinatorial formula

To find the weight, we use the combinatorial formula for choosing \(k\) elements from a set of \(n\) elements, also known as the binomial coefficient, which is expressed as: \( \binom{n}{k} = \frac{n!}{k!(n - k)!} \) In our case, the total number of systems is \(N\), and we need to choose \(n_{1}\) systems to be in the excited state. Therefore, we can write the expression for the weight as: \( W = \binom{N}{n_{1}} = \frac{N!}{n_{1}!(N - n_{1})!} \)

Final answer

So, for a two-level system with \(N\) total systems and \(n_{1}\) systems in the excited state, the expression for the weight of a given energy distribution is: \( W = \frac{N!}{n_{1}!(N - n_{1})!} \)

Key concepts

Combinatorial Formula

Understanding the combinatorial formula is key to grasping the concept of statistical weight in two-level systems. Put simply, this formula is a way to calculate the number of possible combinations or arrangements of objects without regard to the order in which they are selected. In mathematical terms, if you have a set of ‘n’ items and want to select ‘k’ of them, the number of combinations is given by the binomial coefficient. This is crucial for determining how many ways you can distribute energy levels among particles in a system.

For example, consider you have a bowl of fruit with apples and oranges. If you wish to know how many ways you can pick two pieces of fruit from a selection of five, regardless of the order, you would use the combinatorial formula. This mathematical principle lays the foundation for calculating statistical weights in more complex systems like the two-level quantum systems described in the exercise.

Binomial Coefficient

The binomial coefficient, often represented as \( \binom{n}{k} \), plays a pivotal role in combinatorics and appears frequently in probability, statistics, and many areas of mathematics. It is a numerical value that describes the number of ways to choose ‘k’ elements out of a set of ‘n’ distinct elements. The formula for the binomial coefficient is \( \binom{n}{k} = \frac{n!}{k!(n - k)!} \), where ‘!’ denotes the factorial operation, meaning the product of all positive integers up to that number.

Real-World Applications

In everyday life, the binomial coefficient is used to solve problems involving combinations, from simple choices like how many different teams of players can be formed, to complex problems in engineering and sciences. In quantum systems, the binomial coefficient helps in determining the degeneracy of energy levels, which is crucial for understanding the distribution of particles in various quantum states.

Excited State in Quantum Systems

In quantum systems, particles such as electrons can exist in different energy levels. The 'excited state' refers to a level where a particle has more energy than it does at its most stable level, known as the 'ground state'. When a particle absorbs energy, it may jump from the ground state to an excited state. This concept is vital in fields such as quantum mechanics and quantum chemistry, influencing the behavior of atoms and molecules.

Significance in Two-Level Systems

In the context of a two-level system, you would consider only two energy states: the ground state and one excited state. The number of particles in the excited state (\(n_{1}\)) and the ground state (\(N - n_{1}\)) are crucial for calculating the statistical weight or the number of microstates compatible with a particular energy distribution. This helps scientists predict how the system will behave and is particularly important when dealing with lasers, semiconductors, and other quantum phenomena.

Partition Function

The partition function is a central concept in statistical mechanics and thermodynamics. It provides a link between the microstates of a system (the quantum states of its individual particles) and the macroscopic properties that we can measure, like temperature, pressure, and volume. Mathematically, the partition function is a sum that goes over all possible states of a system, each term weighted by the Boltzmann factor, which is an exponential factor depending on the energy of the state and the temperature of the system.

Role in Statistical Weight

The importance of the partition function becomes evident when you're calculating the probability of a system being in a certain state. In the two-level system, the partition function would consider both the ground and excited states and their respective statistical weights. Understanding the partition function enables one to discern not just the most likely energy distribution, but also to derive other thermodynamic properties that describe systems at the macroscopic scale.