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When we solve Schrodinger equation with potential goes to 0 at large distance, if $E>0 $, the wave function dies away to zero (as this shows).

My idea to prove this fact is using curvature, since normalizable it must vanish as the curvatual show.But there is a possibility that curve of wave function can look like a bowl,since normalizable we may claim , it can't happen.

The question here is for free particle,we allow unnormalizable solution,why here unnormalizable solution is not allowed.(the intuition is quite natural,big potential barrier the wave function,but what's the difference compared with free particle case?)

The similar question for harmonic oscillator. Why unnormalizable solution is allowed for free particle but not harmonic oscillator solution?

Qmechanic
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yi li
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  • I find these two unnormalizable is not the same, one is divergence another one is oscillation – yi li Jun 06 '20 at 14:11
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    A perfect homogeneous EM (or acoustic) plane wave is equally unphysical for it has infinite energy, nevertheless, it is still useful for both exact and approximate calculations. – hyportnex Jun 06 '20 at 16:22

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You are correct that a plane wave does not represent a normalizable wave function. Therefore, it is not a valid wave function.

However, from a mathematical point of view, a plane wave can still be used to construct properly normalizable wave functions, and this is why they are discussed in many books, although you are correct in pointing out that many books do not always make it clear why they discuss them.

To give you an example, in the case of a free particle or of a simple potential like a step or a barrier, the most common wave function you will be looking at will be a wave packet: $$ \psi(x,t)=\int_{-\infty}^{\infty}g(k)e^{i(kx-\omega t)}dk, $$
which is made of a superposition of plane waves and, for a good choice of $g(k)$, will be properly normalizable.

ProfM
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  • Why solving an equation that yields unrealizable things , then construct a realizable solution based on it,that's very strange. – yi li Jun 06 '20 at 14:42
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    Plane waves are mathematically correct solutions of the equation. Because the equation is linear, any superposition of plane waves (like the integral in my answer) is also a valid mathematical solution. This is what the maths tells us. When we do physics we have to ask an extra question: which one of all these mathematically allowed solutions is physically allowed? This is where the normalizability of the wave function comes in. – ProfM Jun 06 '20 at 14:46
  • But why physics allows superposition of physicical unrealizable thing?I know math allows it. – yi li Jun 06 '20 at 14:48
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    What physics tells us is that we need square-integrable wave functions (a simple argument is that the absolute value squared of the wave function represents the probability density and therefore it must integrate to 1). Square-integrable wave functions like the one I wrote in my answer are also solutions to the mathematical equation, and are square integrable, so these are the physically allowed solutions. The fact that they are a superposition of plane waves, and these plane waves are themselves not normalizable does not matter, because individual plane waves are not in your solution. – ProfM Jun 06 '20 at 14:53
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    @yi_li . You have to have clear in your mind that physics theories use mathematics and its axioms as a tool, impsoing extra axioms on the solutions that connect physical data with the formulas. These are called "principles" "laws" "postulates", example :http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/qm.html . also axiomatic is the table of elementary particles of the standard model. – anna v Jun 06 '20 at 15:09
  • @ProfM thanks for your comment, do you mean solving time-independent Shrodinger equation is just a mathematical intermediate process to get a final physical solution for the Shrodinger equation?So it doesn't matter intermediate is physical or not,but final output from the "box" must be math solution + physically allowed? So superposition is a intermediate mathematical process,not a real physical process to combine those plane wave ? – yi li Jun 06 '20 at 15:13
  • @anna v Do you mean Step1 solve equation ,get's final wave function .Step 2 throw away some solution based on extra axioms? – yi li Jun 06 '20 at 15:22
  • @yi_li yes, it is generally true in physics (and not only in quantum mechanics), that after you have solved for the mathematics, you still need to consider the physical situation to decide which of the mathematically allowed solutions are physically allowed. – ProfM Jun 06 '20 at 15:44