Find density matrice at zero and infinity temperature

E_1## and ##E_2## evolving towards the same limit as the temperature increases. The density matrix also shows that at this limit, the probability of finding the system in either state is equal, indicating that the system is in a state of thermal equilibrium. The equations used to reach this solution involve the energy and inverse temperature (##\beta##) of the system.
  • #1
romeo31415
5
0

Homework Statement


upload_2015-1-16_9-52-47.png


Homework Equations



https://www.physicsforums.com/attachments/upload_2015-1-15_18-57-56-png.77691/
Density matrix for a 2 state system is:
https://www.physicsforums.com/attachments/upload_2015-1-15_19-1-30-png.77694/

The Attempt at a Solution


For a 2 finite state system, the matrice density at limit temperature is
https://www.physicsforums.com/attachments/upload_2015-1-15_18-59-39-png.77692/
https://www.physicsforums.com/attachments/upload_2015-1-15_18-59-56-png.77693/ [/B]

The Attempt at a Solution

 
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  • #2
You wrote the density matrix in terms of the energy and ##\beta##. What does the finite temperature density matrix in the problems tell you about ##E_1## and ##E_2##?

Could you give some details about how you reached the solutions you wrote?
 
  • #3

Homework Equations



Density matrix for a 2 state system is:
upload_2015-1-16_23-10-22.png


The Attempt at a Solution


For a 2 finite state system, the matrice density at limit temperature is
upload_2015-1-16_23-11-35.png
[/B]
Evolution of the density matrix
upload_2015-1-16_23-12-31.png
 

Related to Find density matrice at zero and infinity temperature

What is the significance of finding the density matrix at zero and infinity temperature?

The density matrix at zero temperature provides information about the ground state of a system, while the density matrix at infinity temperature provides information about the system's highest energy state. This allows scientists to gain a better understanding of the behavior and properties of a system at extreme temperature conditions.

How is the density matrix calculated at zero and infinity temperature?

The density matrix at zero temperature is calculated using the Hamiltonian of the system and the ground state wave function, while the density matrix at infinity temperature is calculated using the partition function and the Boltzmann distribution.

Can the density matrix at zero and infinity temperature be used to calculate other thermodynamic properties?

Yes, the density matrix at zero and infinity temperature can be used to calculate various thermodynamic properties such as energy, entropy, and specific heat. This is done by taking derivatives of the density matrix with respect to temperature.

What insights can be gained from analyzing the density matrix at zero and infinity temperature?

By analyzing the density matrix at zero and infinity temperature, scientists can gain insights into the phase transitions and critical behavior of a system. It can also provide information about the stability and behavior of a system at different temperature regimes.

Are there any limitations to using the density matrix at zero and infinity temperature?

Yes, the density matrix at zero and infinity temperature assumes that the system is in equilibrium, which may not always be the case in real-world systems. Additionally, it may not accurately describe the behavior of systems with strong quantum fluctuations or interactions.

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