Express arbitrary state in second quantization

In summary, expressing an arbitrary 2-particle state in second quantization involves using the sum of creation operators and a constant to represent all possible pairs. However, when summing over all pairs, it is important to consider the statistics of the particles, in this case being fermions, which requires only considering pairs with a lower index being less than the higher index.
  • #1
daudaudaudau
302
0
How do I express an arbitrary 2-particle state in second quantization? I could write this
[tex]
|\psi\rangle=\sum_{mn}c_{mn} a_m^\dagger a_n^\dagger |0\rangle
[/tex]
where [itex]c_{mn}[/itex] is a constant, [itex]a_n^\dagger[/itex] is the creation operator and [itex]|0\rangle[/itex] is the vacuum state. The only problem is that I want to sum over all PAIRS, and when I write the sum like this, all pairs are included twice, which is a mess.
 
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  • #2
Well, you did not specify the statistics (Bose, Fermi, distinguishable particles?).
 
  • #3
Fermi.
 
  • #4
Then it is enough to take m<n (because of the anticommutation).
 

Related to Express arbitrary state in second quantization

1. What is second quantization?

Second quantization is a mathematical framework used to describe quantum systems with many particles. It involves expressing the state of a system in terms of creation and annihilation operators, which act on states to create or destroy particles.

2. How is second quantization used in physics?

Second quantization is used in many areas of physics, including quantum mechanics, condensed matter physics, and nuclear physics. It allows for a more efficient and concise way of describing systems with many particles, making it a valuable tool for studying complex physical phenomena.

3. What is an "arbitrary state" in second quantization?

An arbitrary state in second quantization refers to a general state of a system with multiple particles, where the specific number and arrangement of particles is not specified. This allows for a more general and flexible approach to describing quantum systems.

4. How is the state of a system expressed in second quantization?

The state of a system is expressed in second quantization using creation and annihilation operators, which act on a vacuum state to create or destroy particles. The state is then represented as a linear combination of all possible configurations of particles.

5. What are the advantages of using second quantization?

Second quantization offers several advantages, including a more concise and efficient representation of complex systems, the ability to easily incorporate interactions between particles, and the ability to describe systems with variable numbers of particles. It also allows for the use of advanced mathematical techniques, making it a powerful tool for studying quantum systems.

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