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If you try to integrate the solution of the potential step,

http://en.wikipedia.org/wiki/Solution_of_Schr%C3%B6dinger_equation_for_a_step_potential#Solution

you will notice that it diverges! Doesn't that mean that the solution is physically incorrect?

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In quantum physics there are often "large-size" (also called infrared) divergences. They're not very problematic, since generic real life systems are typically finite slabs of matter, so no solution can extend to infinity. –  Vibert Apr 1 '13 at 10:26

1 Answer 1

This pathology is not peculiar to the potential step problem, or 1D problems even. It is a general feature of expanding solutions of Schrodinger equation in plane waves (or their equivalent for some other potential). These solutions are not normalisable, so technically they are not member of the Hilbert space of states. You are right. These states are not physical. But they can be linearly combined into wavepacket states which are normalisable.

There is a mathematical formalism making all of this rigorous, but for most physical purposes it is unnecessary since at the end of the day all you care about are wavepackets. Look up "rigged Hilbert space" for more information.

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Good answer. However, there is something that confused me about this. In Zettili's Quantum Mechanics, the author mentioned mentioned while trying to find the constants A, B, C (the constants of the incident, reflected, and transmitted waves, respectively): "As for the constant A, it can be determined from the normalization condition of the wave function, but we don't need it here...." So it seems that the author is mistaken about this? –  Promather Apr 1 '13 at 10:24
    
They are possibly using a delta function normalisation (this is the practical version of the rigged Hilbert space construction). You text should discuss their conventions somewhere under plane wave or momentum eigenstates. –  Michael Brown Apr 1 '13 at 11:50

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