What does not normalizable ential

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What does "not normalizable" ential

the free particle's wave function is not normalizable...what does that mean??
I understand there are mathematical tricks to help, but i still don't understand why it is not normalizable? does that cast the whole statistical interpretation into doubt when it comes to free particles?
 
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  • #2
jtbell
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When we talk about the "free particle wave function" we usually mean this (in the case of one-dimensional motion):

[tex]\Psi(x,t) = A e^{i(k x - \omega t)}[/tex]

By "not normalizable" we mean that there is no value of A that makes the following integral true:

[tex]\int^{+\infty}_{-\infty}{\Psi^* \Psi dx} = 1[/tex]

which makes the total probability of finding the particle somewhere equal to 1.

It simply means that the wave function given above is not actually a valid wave function for a free particle, strictly speaking. Physically, it means that it is not possible for a particle to have a completely definite, exact value of momentum [itex]p = \hbar k[/itex].

To get a valid wave function for a free particle, that is localized in space and is normalizable, you have to superpose wave functions that span a continuous range of momentum values:

[tex]\Psi(x,t) = \int^{+\infty}_{-\infty} {A(k) e^{i(k x - \omega t) } dk}[/tex]

where a function A(k) gives the amplitude of the wave that has momentum [itex]p = \hbar k[/itex]. This is called a wave packet. One common example (because it's relatively easy to analyze) is the Gaussian wave packet:

http://musr.physics.ubc.ca/~jess/hr/skept/GWP/
 
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  • #3
Matterwave
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The last integral in jtbell's post should have a "dk" at the end.
 
  • #4
Fredrik
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I just want to add that all members of a Hilbert space like [itex]L^2(\mathbb R^3)[/itex] have finite norm. So if v is an arbitrary non-zero vector in the Hilbert space, then [itex]v/\|v\|[/itex] is a unit vector in the direction of v, i.e. it's a "normalized" version of v.

So the "not normalizable" wavefunctions aren't really wavefunctions. They also don't correspond to physically realizable states.
 
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Thanks a lot jtbell, matterwave and fredrik.
 
  • #6
jtbell
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The last integral in jtbell's post should have a "dk" at the end.
Oops! I fixed it. Thanks!
 

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