Why isn't momentum quantized in quantum mechanics?

[Darren93]

I'm going through a textbook, on what is pretty much my first course in Quantum mechanics. I've got to a section were it says that in order for a sinusoidal wave function to correspond to a non infinite probability distribution, there must be a range of momentum, or at least more than 1 momentum term on x.. Which I understand, that makes complete sense. However the next bit they utilize an integral term to sum momentum over a unit step like distribution. That is they consider the wave function to be built up from an integral w.r.t momentum, composed of various sinusoidal terms. This integral considers momentum to be a continuous range.

Equation of concern (probably makes little sense): Ψ(x,t)=(2πħ)^-½ ∫ e^{i[px-Et]/ħ} *Φ(p) dp

However why isn't momentum quantized? Surely energy exists in steps related to h, which will extend to quantum's of momentum. Hence in a momentum range, a sum would be needed.

Am I wrong in thinking that momentum is quatized, or is it that the integral is a close match to the sum due to the size of h? The reason I'm annoyed at this is that they use this to relate wave function and momentum together in a Fourier transform manner. Which I presume is the basis for Heisenberg's uncertainty principle. I feel uneasy about leaving such a hole in my knowledge with the significance this derivation poses.

 

[The_Duck]

However why isn't momentum quantized? Surely energy exists in steps related to h, which will extend to quantum's of momentum.

You are getting ahead of yourself. Energy is only quantized in certain cases. Your textbook will discuss some examples soon if it is any good. For a free particle, energy and momentum are *not* quantized.

The reason I'm annoyed at this is that they use this to relate wave function and momentum together in a Fourier transform manner. Which I presume is the basis for Heisenberg's uncertainty principle.

Yes. You are basically working through a special case of the uncertainty principle here.

 

[Darren93]

Thanks for the reply, so momentum doesn't exist in quanta's then? Makes sense as when I googled it I found nothing.