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.
To construct the antisymmetric wave functions for the two systems, one with particles of identical spin and the other with particles of opposite spin, we need to consider the Pauli exclusion principle. According to the Pauli exclusion principle, no two Fermions can occupy the same quantum state simultaneously.
Let's consider the system with particles of identical spin first. Since they have an identical spin, we need to make sure their wave function is antisymmetric under exchange. For two particles, the wave function can be written as:
ψ_identical (x1, x2) = A * (ψ1 (x1) * ψ2 (x2) - ψ2 (x1) * ψ1 (x2))
Where ψ1 (x1) and ψ2 (x2) are the wave functions for the individual particles, and A is a normalization constant.
Now, let's consider the system with particles of opposite spin. In this case, the wave function needs to be antisymmetric under exchange, taking into account the opposite spin. The wave function for this system can be written as:
ψ_opposite (x1, x2) = A * (ψ1 (x1) * ψ2 (x2) - ψ2 (x1) * ψ1 (x2)) * (α * β - β * α)
Where α and β are the spin wave functions for the particles. The factor (α * β - β * α) ensures that the overall wave function is antisymmetric.
Now, to compare the groundstate energies of the two systems, we need to solve the Schrödinger equation for each system separately. The Schrödinger equation will give us the energy eigenvalues for each system.
For the system with particles of identical spin, we can solve the Schrödinger equation using the Hamiltonian operator for two non-interacting particles in a box:
H_identical = -ħ^2/2m (∂^2/∂x1^2 + ∂^2/∂x2^2)
Where m is the mass of the particles and ħ is the reduced Planck constant.
Similarly, for the system with particles of opposite spin, the Hamiltonian operator will be the same, but we need to consider the spin wave functions as well.
By solving the Schrödinger equation for each system and finding the eigenvalues, we can compare the groundstate energies of the two systems.
Please note that the above explanation assumes non-interacting particles and a simple box potential. In more realistic scenarios, other factors need to be taken into account like the specific confinement potential, spin-orbit coupling, and interaction among the particles. The detailed calculations and analysis depend on the specific system and its characteristics.