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 two non-interacting Fermions with half-integer spin, we need to consider the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state simultaneously.
Let's consider the two particles with spin 1/2 as A and B. We can represent their possible spin states as |up> and |down> respectively. Since we have a total of two particles and each particle has two possible spin states, we have a total of four possible states: |up>⨂|up>, |up>⨂|down>, |down>⨂|up>, and |down>⨂|down>.
Now, according to the Pauli exclusion principle, the antisymmetric wave function for the two particles should change sign if we swap the particles. We can express the antisymmetric wave function as:
ψ_antisymmetric = (1/√2) [(|up>⨂|down>) - (|down>⨂|up>)]
Similarly, for the case of particles with opposite spin, we can express the antisymmetric wave function as:
ψ_antisymmetric_opposite = (1/√2) [(|up>⨂|down>) + (|down>⨂|up>)]
Now, let's calculate the groundstate energies for the two different systems.
The energy of each particle in a one-dimensional box of length L is given by the expression:
E_n = (n^2 * h^2)/(8mL^2)
where n is the quantum number (n = 1, 2, 3, ...) representing different energy levels, h is the reduced Planck's constant, and m is the mass of the particle.
In the case of identical spin particles, the two particles will occupy different energy levels since they have different spins. So, the total groundstate energy of the system will be the sum of the groundstate energies of each particle:
E_identical = 2 * E_1
In the case of particles with opposite spin, the two particles will occupy the same energy level since they have opposite spins. So, the total groundstate energy of the system will be twice the groundstate energy of a single particle:
E_opposite = 2 * E_1
Therefore, we can see that the groundstate energies for the two systems are the same: E_identical = E_opposite = 2 * E_1.