Wavelength of Particle Passing Through Potential Barriers

[xWaffle]

Homework Statement

Compare the wavelengths of a particle when it passes a barrier of height (a) +V₀ and (b) -V₀ where E > |V₀|. Calculate the momentum and kinetic energy for both cases.

Homework Equations

(see below)

The Attempt at a Solution

I know the wavelength changes in the area of potential inside the barriers, but my textbook doesn't really give me much else information. It just leads on to explain how when passing the negative potential region, reflection and transmission may occur, which isn't what I'm interested in. With regards to the positive potential region, it gives what we 'thought' happened classically, and says this is not the case. Not very helpful either.

My book refers to the regions outside of the barriers as regions I and III, and the region within the barriers as region II. So what I'm looking for is how the wavelength changes in region II.

Equation given for 'wave number' in region II is as follows:

$\kappa_{II} = \frac{\sqrt{2m(E - V_{0})}}{\hbar}$

And the 'wave function' in region II is:

$\psi_{II} = Ce^{\kappa x} + De^{-\kappa x}$

How am I supposed to tie these together with wavelength, momentum, and kinetic energy? Am I even supposed to use these equations? I'm so confused as to how I'm supposed to approach this.

 

[mark726]

Thank you for your question. I understand your confusion as the equations given in your textbook may not directly relate to the wavelength, momentum, and kinetic energy. However, I can provide some guidance on how to approach this problem.

Firstly, let's start with the wave number equation given for region II. As you mentioned, this equation is used to calculate the wave number in region II, which is related to the wavelength through the following equation:

\lambda = \frac{2\pi}{k}

where \lambda is the wavelength and k is the wave number. So, by plugging in the equation for wave number in region II, you can calculate the wavelength in region II for both cases (a) and (b).

Next, to calculate the momentum and kinetic energy, you can use the following equations:

p = \frac{h}{\lambda}

and

K = \frac{p^2}{2m}

where p is the momentum, h is Planck's constant, K is the kinetic energy, and m is the mass of the particle.

By plugging in the values for wavelength calculated in the previous step, you can calculate the momentum and kinetic energy for both cases (a) and (b).

I hope this helps you in solving the problem. If you have any further questions, please don't hesitate to ask. Good luck!