Normalization of a quantum particle

[DODGEVIPER13]

Homework Statement

Why is it important for a wave function to be normalized? Is an unnormalized wave function a solution to the Schrödinger equation?

Homework Equations

∫ ψ^2 dx=1 (from neg infinity to infinity)

The Attempt at a Solution

So I know normalization simply means that the sum of all dx is equal to 1 and the squared function is know as the probabily density so it gives that you can find a particle with 100% certainty and this is why it is important. Is this correct? I am not sure on the second part because when a wave is not normalized we can't know with 100% probability where a particle is appeasing the uncertainty principle which, I would guess the normalized version would not. I rememeber my instuctor saying something about it being a solution if it satisfys the conservation of energy and de broglies hypothesis or something to that effect so yes I would assume an unormalized wave would pass the test. Is this correct?

 

[dextercioby]

There's a mere simplification of all formulae derived from the probabilistic interpretation. All vectors are by convention set to modulus 1, the ones which can't be 'normalized' are said to be 'generalized eigenstates', like the ones for the free particle in n-dimensions.

 

[DODGEVIPER13]

Thanks man I appreciate the help.