Question about origin of wave number K in wave equations

[daselocution]

Homework Statement

Hey All,

I'm hoping that you might be able to help me understand where the wave number "k" referred to in equations 6.62a and b below comes from. It makes sense to me that IF k=(2mE/hbar²) normally, that 6.62b would have E-V instead of E. That this change is made makes me think that it must have something to do with the kinetic energy, as I am assuming that E-V signifies kinetic energy, but I just don't know how to get any value of K to start with or how to derive either of the values shown above. I would really like to know exactly what K means/represents and how to derive its value for myself

Any help would be appreciated.

 

[daselocution]

I actually may have just asked a very dumb question.

I just realized that by the definition given in my book, I can at least come up with the values of k given, though I still don't really understand what they mean.

I saw that:

k=2π/lambda = p/hbar = (p^2/hbar^2)^.5 = (2mE)^.5 / hbar.

If anyone could help me understand what this means, or what the PHYSICAL significance of this value (other than its being 1/m in units) is, I would be super grateful.

 

[Basic_Physics]

It tells you the amount of waves that will fit in one meter. The inverse of the wavelength that is. Say it is 1/4 meter wavelength. Then we get 4 wavelengths per meter. The 2π changes it into a phase difference with units rad/m. So the wavelength increases in the region of the potential barrier. The particle becomes more wavelike or tenuous in nature. k is more correctly called the angular wave number of the wave.