How BEC being described by the single-particle density matrix?

In summary, the conversation is about a beginner seeking help in understanding the one-body density matrix in Bose-Einstein Condensation. The equation in question, <\psi|\mathbf{\Psi(r)^\dagger\Psi(r')}|\psi>, is discussed and the individual steps in deriving it are outlined. The conversation ends with a suggestion to use knowledge of the commutation properties of Psi operators to work out the matrix element.
  • #1
csky
1
0
Hello everybody,

this is my first time being here. I am a beginner learning some introductions on Bose-Einstein Condensation (BEC) on my own. Often times in the literature (say, [1], [2] (p.409) ) it comes the one-body(single-particle) density matrix, as

[tex]<\psi|\mathbf{\Psi(r)^\dagger\Psi(r')}|\psi>=N\int dx_2...dx_N~\psi^*(r,x_2,...,x_N)\psi(r',x_2',...,x_N')
[/tex]

I am not sure how to derive the above equation... My first step is to write [itex]<\psi|\mathbf{\Psi(r)^\dagger\Psi(r')}|\psi>[/itex] as

[tex]
<\psi|\mathbf{\Psi(r)^\dagger\Psi(r')}|\psi>=\int dx_1...dx_N \int dx_1'...dx_N' \psi_t^*(x_1,...,x_N)<x_1,...,x_N|\mathbf{\Psi(r)^\dagger\Psi(r')}|x_1',...,x_N'>\psi_t(x_1,...,x_N)
[/tex]

then I am not sure how to handle [itex]<x_1,...,x_N|\mathbf{\Psi(r)^\dagger\Psi(r')}|x_1',...,x_N'>[/itex]. Any ideas?

thanks in advance for help and comments,
C.H.
 
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  • #2
You probably know that e.g. ##\mathbf{\Psi(x_1)\Psi(x_2)}|0>=|x_1,x_2>## and so on for the position eigenstates of n particles in genera. Furthermore, you know the commutation properties of the Psi operators, ##\{\mathbf{\Psi^+(x_1),\Psi(x_2)}\}=\delta(x_1-x_2)##. This should be sufficient to work out the matrix element.
 

1. What is the single-particle density matrix?

The single-particle density matrix is a mathematical representation used to describe the quantum state of a many-particle system, such as a Bose-Einstein condensate (BEC). It contains information about the probability amplitudes for each possible configuration of the particles in the system.

2. How is the single-particle density matrix related to BEC?

In the case of BEC, the single-particle density matrix describes the collective state of all the particles in the condensate. It provides a complete description of the system's quantum state, including information about its coherence, superfluidity, and other properties.

3. What does the single-particle density matrix tell us about BEC?

The single-particle density matrix allows us to calculate important physical quantities related to the BEC, such as the condensate fraction, the correlation functions, and the momentum distribution. It also provides insights into the spatial and temporal behavior of the condensate.

4. How is the single-particle density matrix used in experiments?

In experiments involving BEC, the single-particle density matrix is often measured indirectly through techniques such as time-of-flight imaging or Bragg spectroscopy. These methods allow researchers to extract information about the quantum state of the BEC and validate theoretical predictions.

5. Can the single-particle density matrix be used to describe other systems besides BEC?

Yes, the single-particle density matrix can be used to describe a wide range of quantum systems, including fermionic systems, superfluids, and superconductors. It is a powerful tool for understanding the collective behavior of many-particle systems and has applications in fields such as condensed matter physics, quantum information, and atomic physics.

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