Statistical physics. Density matrix

In summary, the system is subject to Hamiltonian ##\hat{H}=-\gamma B_z \hat{S}_z## and the density matrix can be written as ##\hat{\rho}=\frac{1}{Tr(e^{-\beta \hat{H}})}e^{-\beta \hat{H}}## for a system in thermal equilibrium using the canonical ensemble. The exact definition of the density matrix will depend on the state of the system and information may be missing from the problem statement.
  • #1
LagrangeEuler
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Homework Statement


A system is subject to Hamiltonian ##\hat{H}=-\gamma B_z \hat{S}_z##. Write down the density matrix.[/B]

Homework Equations


For canonical ensemble
##\hat{\rho}=\frac{1}{Tr(e^{-\beta \hat{H}})}e^{-\beta \hat{H}}##

In general ##\rho=\sum_m |\psi_m\rangle \langle \psi_m|##

The Attempt at a Solution


How to know which definition of density matrix to use? Thanks.
 
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  • #2
The density matrix (statistical operator) is, of course, not determined by just giving the Hamiltonian but by the state of the system, because the statistical operator describes the state of the system. So there must be something missing from the problem statement or it is implied to consider a system in thermal equilibrium and to use the canonical ensemble. In the latter case it's not that difficult, if (again some information missing in the problem statement!) you know what spin your particle, i.e., ##s \in \{0,1/2,1,\ldots \}##.
 

1. What is statistical physics?

Statistical physics is a branch of physics that uses statistical methods to study and understand the behavior of large ensembles of particles, such as molecules, atoms, and subatomic particles. It aims to explain and predict the macroscopic properties of matter based on the microscopic behavior of its constituent particles.

2. What is a density matrix in statistical physics?

A density matrix, also known as a density operator, is a mathematical tool used in statistical physics to describe the state of a quantum mechanical system. It is a matrix that contains information about the probabilities of finding a system in different possible states, as well as the coherence between those states.

3. How is the density matrix related to the concept of entropy?

The density matrix and entropy are closely related in statistical physics. The entropy of a system is a measure of its disorder or randomness, while the density matrix contains information about the probabilities of different states. As the system evolves, the density matrix changes, and the entropy of the system increases, reflecting the increasing uncertainty about the system's state.

4. What are the applications of statistical physics?

Statistical physics has many applications in various fields, including condensed matter physics, cosmology, astrophysics, and biophysics. It is used to study and understand complex systems, such as materials, fluids, and biological systems, and to make predictions about their behavior and properties.

5. How is statistical physics different from classical mechanics?

Statistical physics and classical mechanics are both branches of physics, but they differ in their approach to studying and understanding physical systems. Classical mechanics deals with the behavior of individual particles, while statistical physics deals with large ensembles of particles. Statistical physics also takes into account the probabilistic nature of quantum mechanics, while classical mechanics assumes deterministic behavior.

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