Slater Determinant & Permanent

In summary, the conversation discusses how to use Slater determinants and permanents to determine if a state is symmetrical or anti-symmetrical. It is mentioned that determinants are used for fermionic systems because they result in an antisymmetric wave function, while permanents are used for bosonic systems because they result in a symmetric wave function. It is clarified that this is not something that is determined, but rather something that is chosen for the basis set.
  • #1
M. next
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How to use Slater determinant and permanent to find out whether the state is symmetrical or anti-symmetrical?
How to use them? I got the concept but I didn't get for example when to know if there was a plus or a minus and thus whether what we're talking about is a permanent or determinant (respectively).
 
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  • #2
If you have minus signs, its a determinant and totally antisymmetric, if you don't, it's a permanent and totally symmetric.

It's not something you determine. It is something you put into your ansatz: You choose determinants as a basis set for fermionic systems precisely because you know that you need an antisymmetric wave function, and determinants are the simplest kinds of wave functions which are totally antisymmetric. And for the same reason you choose permanents as basis for bosonic systems because they are the simplest wave functions which are totally symmetric.
 
  • #3
Thank you a lot for clearing things up. I thought it is something to be determined.
 

FAQ: Slater Determinant & Permanent

1. What is a Slater Determinant and how does it differ from a Permanent?

A Slater Determinant is a mathematical concept used in quantum mechanics to describe the state of a system of identical particles. It is a combination of single-particle wave functions that represents the wave function of the entire system. A Permanent, on the other hand, is a similar concept but is used in combinatorics to describe the number of ways to choose a subset of objects from a larger set without regard to their order. They differ in their applications and the types of systems they describe.

2. What is the significance of Slater Determinants and Permanents in quantum chemistry?

Slater Determinants and Permanents are important concepts in quantum chemistry because they allow us to describe the electronic structure of molecules and atoms. A Slater Determinant is used to describe the wave function of a system of electrons, while a Permanent is used to calculate the number of possible electronic configurations for a given number of electrons in a molecule. These concepts are essential for understanding chemical bonding and molecular properties.

3. Can Slater Determinants and Permanents be calculated analytically or do they require numerical methods?

Both Slater Determinants and Permanents can be calculated analytically for simple systems. However, for more complex systems, numerical methods are often necessary. This is because the number of terms in these calculations grows exponentially with the number of particles, making analytical solutions infeasible for large systems.

4. How are Slater Determinants and Permanents related to each other?

Slater Determinants and Permanents are closely related, with the Permanent being a generalization of the Slater Determinant. In fact, the Permanent can be thought of as a special case of the Slater Determinant, with all the single-particle wave functions being equal. Additionally, the coefficients in a Slater Determinant can be obtained by taking the square root of the corresponding Permanent.

5. Are there any real-world applications of Slater Determinants and Permanents?

Yes, Slater Determinants and Permanents have numerous real-world applications, particularly in quantum chemistry and physics. They are used to describe the electronic structure of molecules, atoms, and solids, and to calculate molecular energies and properties. They are also used in statistical mechanics to describe the behavior of many-particle systems. Additionally, Permanents have applications in computer science, such as in the design of error-correcting codes and in the analysis of algorithms.

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