Telling whether a wave function might have physical graph

[Von Neumann]

Problem:

Which of the wave functions shown might conceivably have physical significance?

Solution:

I have attached a drawing of the two wave functions. According to my book, the one on the right could have physical significance, while the one on the left does not. Can anyone explain why not? They are both single valued, differentiable functions, so I don't see why the left one is apparently disqualified. Thanks for any help.

 

[SammyS]

Problem:

Which of the wave functions shown might conceivably have physical significance?

Solution:

I have attached a drawing of the two wave functions. According to my book, the one on the right could have physical significance, while the one on the left does not. Can anyone explain why not? They are both single valued, differentiable functions, so I don't see why the left one is apparently disqualified. Thanks for any help.

I assume the graph on the left continues upward toward ∞ without a bound.

 

[Von Neumann]

Hmmm, that never occurred to me. How would that take away from its physical significance?

 

[Von Neumann]

Well, since the wave function is proportional to finding the object at a specific place at a specific time, and unbounded wave function doesn't make sense. Right?

 

[SammyS]

Well, since the wave function is proportional to finding the object at a specific place at a specific time, and unbounded wave function doesn't make sense. Right?

Yes.

Why doesn't it make sense?

 

[Von Neumann]

Yes.

Why doesn't it make sense?

It doesn't make sense logically because, according to the unbounded function, the object described by the wave function is located at an infinite number of places at the same time.

 

[SammyS]

It doesn't make sense logically because, according to the unbounded function, the object described by the wave function is located at an infinite number of places at the same time.

In order for the square of the wave function to represent a probability distribution, what has to be true of the integral of the square of the wave function ?

 

[Von Neumann]

In order for the square of the wave function to represent a probability distribution, what has to be true of the integral of the square of the wave function ?

The integral of the square of the wave function must be normalized, correct? Are you hinting that it is impossible to normalize the diverging function?

 

[SammyS]

The integral of the square of the wave function must be normalized, correct? Are you hinting that it is impossible to normalize the diverging function?

It is impossible if the integral diverges.