# Average Energy for n-State Systems?

• danyull
In summary, to find the average energy for a given state with discrete energy values, one can use the weighted sum of probabilities, where the probabilities are given by the Boltzmann distribution. In the case of a harmonic oscillator, the sum goes to infinity and can be evaluated using a common trick of taking the partial derivative and using the fact that it commutes with summation. The final result can be obtained by normalizing the expectation value.
danyull

## Homework Statement

Find the average energy ##\langle E \rangle## for
(a) an n-state system in which a given state can have energy 0, ε, 2ε, 3ε... nε.
(b) a harmonic oscillator, in which a state can have energy 0, ε, 2ε, 3ε... (i.e. with no upper limit).

## Homework Equations

Definition of temperature: ##β = \frac 1 {K_BT} = \frac {d lnΩ(E)} {dE}##
Boltzmann distribution: ##P(ε) ∝ e^{-εβ}##

## The Attempt at a Solution

Since the energy here takes on discrete values, the average is found by taking the weighted sum of the probabilities, $$\langle E \rangle = \sum_{n=0}^n nε~e^{-nεβ}$$
and in the case of part (b), the sum goes to infinity. My problem is I don't know how to evaluate these sums. Any help would be appreciated, thanks!

One commonly used trick to evaluate such sums is the following observation that
$$n\varepsilon\,e^{-n\varepsilon \beta} = - \frac{\partial}{\partial \beta } (e^{-n\varepsilon \beta})$$
and use the fact that the partial derivative commutes with the summation to get
$$\sum_{n=0}^{N} n\varepsilon\,e^{-n\varepsilon \beta} = - \frac{\partial}{\partial \beta } \sum_{n=0}^{N} e^{-n\varepsilon \beta}$$
The summation is now simply just a geometric series.
(and don't forget to normalize your expectation value)

danyull, TSny and DrClaude

## What is the definition of "average energy for n-state system"?

The average energy for n-state system refers to the average amount of energy possessed by a system with n possible states. It takes into account the probability of each state and calculates the overall average energy of the system.

## How is the average energy for n-state system calculated?

The average energy for n-state system is calculated by taking the sum of the product of each state's energy and its corresponding probability. This is then divided by the total number of states in the system.

## What is the significance of the average energy for n-state system in thermodynamics?

In thermodynamics, the average energy for n-state system is used to determine the equilibrium state of a system. It helps in understanding the behavior and properties of a system at a macroscopic level.

## Can the average energy for n-state system change?

Yes, the average energy for n-state system can change if there is a change in the energy levels of the individual states or if the probabilities of the states change. This can happen due to external factors such as temperature or pressure.

## How does the concept of entropy relate to the average energy for n-state system?

Entropy is a measure of the disorder or randomness in a system. The average energy for n-state system is directly related to the entropy as it is a measure of the average energy of the system's states. An increase in the number of possible states or an increase in the energy of the states leads to an increase in entropy.

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