Free Particle Wave Function

The free-particle wave function can be expressed as an integral over momentum space:

ψ(x,t) = (1/√(2πħ)) ∫-∞∞ ψ̃(p) ei(px – Et)/ħ dp

where ψ̃(p) is the momentum-space wave function, and the energy E is given by the classical dispersion relation:

E = p² / 2m.

Limits of Integration

  • The momentum p is integrated from -∞ to ∞ because, in quantum mechanics, momentum can take both positive and negative values.
  • Unlike classical mechanics, where momentum might be restricted to nonnegative values in certain contexts (e.g., motion confined to one direction), quantum mechanics allows for superpositions of positive and negative momenta.
  • If the wave function represents a wave packet, it typically includes both positive and negative momentum components.

Thus, the integral is taken over all p values, from -∞ to ∞, not just from 0 to ∞.