Two Non Interacting Particles?

It is possible to construct a two-particle wave function for non-interacting particles. The total wave function is simply the product of the individual wave functions in both position and momentum representations. However, for identical particles, symmetry (bosonic or fermionic) must be imposed.

Two-Particle Wave Function

1. Two-Particle Wave Function in Position Space

For two non-interacting particles with wave functions ψ₁(x₁, t) and ψ₂(x₂, t), the total wave function is given by:

Ψ(x₁, x₂, t) = ψ₁(x₁, t) ψ₂(x₂, t).

Since the particles do not interact, the total wave function is just the product of their individual wave functions.

2. Two-Particle Wave Function in Momentum Space

Similarly, in momentum space, the wave function factorizes as:

Ψ̃(p₁, p₂) = ψ̃₁(p₁) ψ̃₂(p₂).

Thus, the position-space wave function expressed in terms of momentum eigenstates is:

Ψ(x₁, x₂, t) = (1/(2πħ)) ∫-∞∞ ∫-∞∞ ψ̃₁(p₁) ψ̃₂(p₂) ei(p₁x₁ + p₂x₂ – Et)/ħ dp₁ dp₂.

where the total energy is:

E = p₁² / 2m₁ + p₂² / 2m₂.

3. Symmetry Considerations

If the particles are identical, we must symmetrize (for bosons) or antisymmetrize (for fermions) the wave function:

  • Bosons: Ψ(x₁, x₂) = Ψ(x₂, x₁).
  • Fermions: Ψ(x₁, x₂) = -Ψ(x₂, x₁).

For distinguishable particles, no such symmetry constraint applies.

Conclusion

It is possible to construct a two-particle wave function for non-interacting particles. The total wave function is simply the product of the individual wave functions in both position and momentum representations. However, for identical particles, symmetry (bosonic or fermionic) must be imposed.