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in my text while solving one dimension problems like barriers, step potential etc i noticed that they use a free particle eigenfunction coming from left ,. now why that is so did? i mean , if i need to describe a moving particle , will i not require time dependent scroedinger equation?

yet, as far as i now, these are eigenfunctions of a time independent equation ie. momentum eigenfunctions.

also, such eigenfunctions have actually infinite spread in posision space. so we always have a definite probability beyond the barrier?

any help will be highly appreciated.