# Boltzmann Factor in QM Language

• andrewm
In summary, the definition of the Boltzmann Factor in statistical mechanics, as described by Feynman, is the probability of measuring a system with energy E. This probability is determined by the thermal density matrix, which takes into account the ensemble of systems in different energy eigenstates. At non-zero temperature, the system is assumed to be interacting with a heat bath, causing the Boltzmann factor to not depend on the ket of the system. This results in a probability of 1 for measuring the system in the energy eigenstate it is prepared in.
andrewm
Hi,

I'm stuck on "the summit of statistical mechanics" (as Feynman calls it): the definition of the Boltzmann Factor.

The probability of measuring the system with energy E is P(E) = 1/Z * e^-E/kT.

I've taken courses in QM and can't understand why P(E) does not depend on the ket of the system |psi>.

From QM, I want to write down P(Ei) = <psi|Ei><Ei|psi>. So why doesn't 1/Z * e^-E/kT depend on |psi>? Is it a hidden assumption about the nature of the system that all |psi> in the Hilbert Space have P(Ei) equal?

Andrew

The issue lies in the difference between a pure state, which represents an ensemble of identical systems, and a mixed state, which represents an ensemble of systems in different states. So you might want to consider an ensemble of systems in different energy eigenstates, represented by a thermal density matrix. The diagonal components are then the Boltzmann factors.

What if I consider a system in a single energy eigenstate?

If the system is prepared in an energy eigenstate, say E1, then surely a measurement of E1 has probability 1 and not 1/Z * e^-E1/kT ?

Or can I not prepare a system that has a definite energy?

The "problem" is that you are at non-zero temperature (otrhwise the Boltzmann factor would be 1), meaning you are implicitly assuming that your system is actually interacting with a heat bath of some sort.

## What is the Boltzmann Factor in QM Language?

The Boltzmann Factor is a concept in quantum mechanics that describes the probability of a particle being in a particular energy state at a given temperature. It is based on the Boltzmann distribution, which relates the energy state of a system to its temperature and the number of particles in that state.

## How is the Boltzmann Factor calculated?

The Boltzmann Factor is calculated using the formula e-E/kT, where E is the energy of the particle, k is the Boltzmann constant, and T is the temperature in Kelvin. This formula is derived from statistical mechanics and is used to determine the relative likelihood of a particle being in a certain energy state.

## What is the significance of the Boltzmann Factor in quantum mechanics?

The Boltzmann Factor is an important concept in quantum mechanics as it allows us to understand the behavior of particles at different temperatures and energies. It also helps us to calculate the thermodynamic properties of a system, such as the entropy and free energy.

## How does the Boltzmann Factor affect the distribution of particles in a system?

The Boltzmann Factor is directly related to the probability of a particle being in a certain energy state. As the temperature increases, the Boltzmann Factor decreases, meaning that there is a higher probability of particles being in higher energy states. This leads to a broader distribution of particles in a system at higher temperatures.

## Can the Boltzmann Factor be used to predict the behavior of individual particles?

No, the Boltzmann Factor is a statistical concept and cannot be used to predict the behavior of individual particles. It only describes the overall distribution of particles in a system. The behavior of individual particles is governed by other quantum mechanical principles such as the wave function and the uncertainty principle.

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