Wavefunction normalisation for proton beam

[ChrisP]

Homework Statement

Calculate the normalization parameter A in the wavefunction ## \varPsi(x,t) = A e^{i(k\chi - \omega t)} ## for a beam of free protons traveling in the +x direction with kinetic energy 5 keV and a density of ##7.5 * 10^9 ## particles per meter beam length.

Homework Equations

The Attempt at a Solution

How do I normalize the wavefunction of a beam of particles? If it was just a single particle I just multiply the wavefunction with it's conjugate, set the result to 1 and solve the equation for A. But for a beam of particles I have no idea even what physical rules apply there, I mean is the wavefunction of each proton affected in any way by the other particles in the beam, or is it purely independent?

 

[mfb]

In a beam like this, "the wavefunction of a single proton" is a problematic concept.
The normalization with many particles works in the same way as with a single particle, but the result is not 1, but the (average) number of protons within the range of your integral.

 

[ChrisP]

Thank you for your answer! I also don't understand, how does the kinetic energy of the beam fit in with the normalization of the wavefunction? Because to me it seems now that I set ##\int_0^1 \! \varPsi (x,t)^2 \, \mathrm{d}x = \int_0^1 \! A e^{i(k\chi - \omega t)} A e^{-i(k\chi - \omega t)} \, \mathrm{d}x = 7.5*10^9 \Leftrightarrow A^2 = 7.5 *10^9 ## and then I have found the A. Or am I doing something wrong?

 

[mfb]

There are units missing, but apart from that I would do the same.
Are there other parts of the question where the energy is needed? Like finding k or ω?

 

[ChrisP]

Nope that's the whole question, nothing else is asked. The rest of the questions are completely independent. Out of curiosity can you elaborate a bit on the problematic concept of the wave function of a proton? What would be the rigorous quantum mechanical treatment for a beam of particles?

 

[mfb]

Particles with an exact momentum would have to be spread out over all space (at least in the dimension we consider here) - but then you cannot normalize the wave function of a single particle properly. You need a wave packet with some momentum spread.
If you consider systems with multiple fermions of the same type, the wave function has to swap its sign if you exchange fermions. That leads to the Pauli exclusion principle, but it also means your wave functions get more complicated. And I guess you would have to include some "begin" and "end" of the beam.