Physical difference between various wave functions

[jror]

Homework Statement

Is there a physical difference between the following wave functions? If yes, why? If no, why not?

$\Psi(x,0) =5e^{-ax^2}$
$\Psi(x,0) =\frac{1+i}{\sqrt{3}}e^{-ax^2}$
$\Psi(x,0) =e^{i\pi/7}e^{-ax^2}$

Homework Equations

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The Attempt at a Solution

They only differ in the constant factors, and since we are speaking about physical significance we are interested in $|\Psi(x,0)|^2$, which contain no imaginary parts. If I assume that these are realizable states they would have to be normalizable, in which case the a's in the exponents would have set values depending on the constant factor in front? If that is the case then yes these wave functions differ in how quickly they decay as we move away from $x=0$. I am just not completely convinced by my own argument, and if this is what is meant in the exercise. Any input would be very welcome.

 

[andrewkirk]

If I assume that these are realizable states they would have to be normalizable, in which case the a's in the exponents would have set values depending on the constant factor in front?

'Normalisable' only means that they can be converted to a ket of norm (magnitude) 1 by multiplying by a constant. It doesn't mean they already have a norm of 1. That would be 'normalised' not 'normalisable'.

If we normalise those three wave functions we will end up with three identical functions.