Justification of a common calculation

  • Thread starter saunderson
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  • #1
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Hi,

isn't it a bit dangerous to claim that

[tex]\left[ x \cdot \left( \psi(x,t) \, \frac{\partial \psi^\ast (x,t)}{\partial x} + \psi^\ast(x,t) \, \frac{\partial \psi(x,t)}{\partial x} \right) \right]_{x=-\infty}^{x=\infty} = 0[/tex]​


for example?

Expressions like this one are often found in popular quantum mechanics textbooks. How do you justify such expressions? I would prefer a mathematical- instead of a physical explanation...

With best regards
 

Answers and Replies

  • #2
1,789
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Very good question. I spent several days trying to figure it out when I first began learning QM (and haven't stopped since!). Here is my take on it.

It is implicitly assumed here that the wavefunction is square integrable meaning

[tex]\int_{-\infty}^{\infty} |\psi(x)|^2\,dx[/tex]

exists and is finite. Books usually do not go further than this, but state conditions of "well behaviour" (but see * below). More rigorous books on QM will tell you that if the long-range behavior of the wavefunction is of the form [itex]1/|x|^{\alpha}[/itex] for [itex]\alpha > 1[/itex], then this condition is (usually) satisfied. [c.f. Schwabl, for instance.]

So in your expression, as [itex]|x| \rightarrow \infty[/itex], the terms involving the product of the wavefunction and its derivative will go to zero faster than [itex]x[/itex] tends to infinity, and hence 'overall' the expression will go to zero at [itex]|x| = \infty[/itex].

If you want to be more rigorous, you can define a subspace of the Hilbert space and regard only square integrable functions or functions satisfying some regularity conditions, as valid wavefunctions. That way, all such boundary terms automatically vanish.

* - Square integrability does not guarantee that the wavefunction goes to zero at infinity. For an explanation, see D.V. Widder, "Advanced Calculus" 2nd ed., Dover, New York, 1998, p. 325.
 
  • #3
62
1
Thank you for your fast and detailed answer! That supports my assumption that several authors leave the reader deliberatly (or not) behind some serious issues :uhh:

Now my thoughts about this are about to come to maturity...

Any other suggestions are appreciated as well =)
 

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