Momentum Operator - Derivation From Infinitesimal Translations

Derivation From Infinitesimal Translations

See also: Noether's theorem

The translation operator T(ϵ), where ϵ represents the length of the translation, satisfies the following identity:

that becomes

Assuming the function ψ to be analytic (or simply differentiable, for simplicity), one may write:

so for infinitesimal values of ϵ:

As it is known from classical mechanics, the momentum is the generator of translation, so the relation between translation and momentum operators is:

thus

Read more about this topic:  Momentum Operator

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