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

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

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 ∞.

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