Consider two non-interacting Fermions (half-integer spin) confined in a “box” of length L.

Construct the antisymmetric wave functions and compare the corresponding groundstate energies of two systems; one with particles of identical spin and the other with
particles of opposite spiN

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To construct the antisymmetric wave functions for the two non-interacting fermions, we need to consider the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state.

Let's label the two fermions as Fermion 1 and Fermion 2. The wave function for Fermion 1 can be denoted as ψ1(x) and for Fermion 2 as ψ2(x), where x represents the position coordinate in the box.

For the case where the fermions have identical spin, the total wave function of the two-particle system, Ψ(x1, x2), must be antisymmetric with respect to the exchange of the particles:

Ψ(x1, x2) = - Ψ(x2, x1)

The general wave function can be written as a product of the single-particle wave functions ψ1(x1) and ψ2(x2):

Ψ(x1, x2) = ψ1(x1) * ψ2(x2) - ψ1(x2) * ψ2(x1)

On the other hand, for the case where the fermions have opposite spin, the total wave function of the two-particle system, Ψ(x1, x2), must be symmetric with respect to the exchange of the particles:

Ψ(x1, x2) = Ψ(x2, x1)

The general wave function can be written as a product of the single-particle wave functions ψ1(x1) and ψ2(x2):

Ψ(x1, x2) = ψ1(x1) * ψ2(x2) + ψ1(x2) * ψ2(x1)

Now let's compare the groundstate energies of the two systems.

To find the groundstate energy, we need to solve the Schrödinger equation for both systems. The Hamiltonian operator for a particle in a one-dimensional box is given by:

H = - (hbar^2 / (2m)) * (d^2 / dx^2)

where hbar is the reduced Planck constant and m is the mass of the particle.

For a system with two non-interacting particles, the total Hamiltonian can be written as the sum of the individual particle Hamiltonians:

H_total = H1 + H2

where H1 and H2 are the Hamiltonian operators for Fermion 1 and Fermion 2, respectively.

Solving the Schrödinger equation for each Hamiltonian, we can find the eigenenergies and corresponding eigenstates for the single-particle systems. The groundstate energy for each system will be the sum of the two lowest eigenenergies.

By comparing the groundstate energies of the two systems, we can determine which one has a lower energy.