Consider two non-interacting Fermions (half-integer spin) confined in a “box”.
Construct the antisymmetric wave functions and compare the corresponding
ground-state 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 wave functions for two non-interacting Fermions (particles with half-integer spin) confined in a "box," we need to consider the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state simultaneously.
Let's denote the single-particle states as |ψ₁⟩ and |ψ₂⟩ for the first and second particles, respectively. These single-particle states form a complete orthogonal set of states within the box.
For two particles with identical spin, we need to consider the symmetric combination of the single-particle states. The general form of the wave function can be written as:
|ψ⟩ = A(|ψ₁⟩⨂|ψ₂⟩ + |ψ₂⟩⨂|ψ₁⟩),
where A is a normalization constant.
However, we must remember that fermions are subject to the Pauli exclusion principle, which requires the wave function to be antisymmetric under the exchange of particle labels. Therefore, the actual wave function for two identical spin fermions must be:
|ψ₁₂, σ⟩ = A(|ψ₁⟩⨂|ψ₂⟩ - |ψ₂⟩⨂|ψ₁⟩),
where σ represents the spin state of the particles.
On the other hand, for two particles with opposite spin, we need to consider the antisymmetric combination of the single-particle states. The wave function becomes:
|ψ₁₂⟩ = A(|ψ₁⟩⨂|ψ₂⟩ - |ψ₂⟩⨂|ψ₁⟩).
Now, let's consider the ground-state energies for these two systems.
The total energy of a two-particle system can be obtained by summing the energies of the individual particles. Since the two particles are non-interacting, their energies are independent of each other.
Let's denote the energy of the single-particle state |ψ₁⟩ as E₁ and the energy of the single-particle state |ψ₂⟩ as E₂.
For the system with particles of identical spin, the ground-state energy is given by:
E₁₂, σ = E₁ + E₂.
For the system with particles of opposite spin, the ground-state energy is also given by:
E₁₂ = E₁ + E₂.
As you can see, the ground-state energies for both systems are the same. This is because the particles are non-interacting, and their energies are independent of each other.
In summary, for two non-interacting Fermions confined in a "box," the wave functions for particles with identical spin are antisymmetric, while the wave functions for particles with opposite spin are symmetric. The ground-state energies for both systems are equal and are given by the sum of the energies of the individual particles.