Derivation From Infinitesimal Translations
See also: Noether's theoremThe 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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