Hartree Ansatz Method: Fermion Antisymmetry & Product vs. Linear Combination

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In summary, the conversation discusses the use of Hartree products and linear combinations of basis functions to obtain the total wave function in the context of Fermions and the LCAO method. The question is raised about how the Hartree product does not reflect the antisymmetric nature of Fermions and why a product is used instead of a linear combination. The response suggests that the linear combination may also include products.
  • #1
cotyledon
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How does the Hartree product not reflect the antisymmertic nature of Fermions? Why do you take a product rather than a linear combination of the basis functions to get the total wave function (a la LCAO method, i.e. Linear Combination of Atomic Orbitals)?
 
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cotyledon said:
How does the Hartree product not reflect the antisymmertic nature of Fermions?

This is answered in every textbook on the topic. It's also completely obvious if you know what the antisymmetry requirement is.

Why do you take a product rather than a linear combination of the basis functions to get the total wave function (a la LCAO method, i.e. Linear Combination of Atomic Orbitals)?

What makes you think that the linear combination doesn't include products?
 

1. What is the Hartree Ansatz method?

The Hartree Ansatz method is a mathematical technique used in quantum mechanics to approximate the wave function of a many-particle system. It is based on the idea that the wave function can be expressed as a product of single-particle wave functions, with each particle occupying its own single-particle state.

2. How does the Hartree Ansatz method take into account fermion antisymmetry?

The Hartree Ansatz method takes into account fermion antisymmetry by imposing the Pauli exclusion principle, which states that no two fermions can occupy the same quantum state. This is achieved by requiring that the single-particle wave functions in the Hartree Ansatz are antisymmetric with respect to particle exchange.

3. What is the difference between a product and a linear combination in the Hartree Ansatz method?

In the Hartree Ansatz method, a product refers to expressing the wave function as a simple multiplication of single-particle wave functions, while a linear combination involves adding together multiple single-particle wave functions with different coefficients. The choice between a product or linear combination depends on the specific system being studied and the accuracy desired in the approximation.

4. Can the Hartree Ansatz method be used for any many-particle system?

The Hartree Ansatz method can be used for many-particle systems with a large number of particles, as it provides a computationally efficient way to approximate the wave function. However, it may not be accurate for systems with strong interactions or correlations between particles.

5. How does the Hartree Ansatz method compare to other methods for approximating the wave function of a many-particle system?

The Hartree Ansatz method is one of the simplest methods for approximating the wave function of a many-particle system. It is often used as a starting point for more advanced methods, such as the Hartree-Fock method, which takes into account the effects of particle interactions. The accuracy of the Hartree Ansatz method also depends on the specific system being studied, as well as the number of particles and energy levels involved.

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