Show that the Slater Determinant states are a complete basis

[kayan]

Homework Statement

"Show that the Slater Determinant states are a complete basis" is the entire statement.

Homework Equations

The Attempt at a Solution

I guess I'm trying to prove that the rank of the states is equal to the basis? I'm not sure where to start on this one.

 

[mark726]

To prove that the Slater Determinant states are a complete basis, we first need to understand what a basis is. A basis is a set of vectors that can be used to represent any other vector in a vector space. In other words, any vector in the vector space can be written as a linear combination of the basis vectors.

In quantum mechanics, the vector space is the space of all possible wavefunctions, and the basis vectors are the eigenstates of the Hamiltonian operator. The Slater Determinant states are a set of anti-symmetric wavefunctions that are constructed from single-particle wavefunctions. These single-particle wavefunctions are the basis vectors in the Slater Determinant states.

Now, to prove that the Slater Determinant states are a complete basis, we need to show that any wavefunction can be written as a linear combination of these anti-symmetric states. This can be done by using the concept of superposition. Any wavefunction can be expressed as a linear combination of basis vectors, i.e. it can be written as a sum of Slater Determinant states.

Therefore, the Slater Determinant states form a basis in the vector space of all possible wavefunctions. This means that any wavefunction can be represented as a linear combination of these states, making them a complete basis.

In conclusion, the Slater Determinant states are a complete basis because they can be used to represent any wavefunction in the vector space of all possible wavefunctions. This is why they are an important tool in quantum mechanics and are used extensively in many applications.