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Old Oct 16th 2016, 12:08 PM   #1
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Free particle hits potential step. Wavelength of particle and amplitude of wave?

A free particle with energy E hits a potential step with $U_0 > 0$ at $x \geq 0$. Assume that $E > U_0$.

(1) Determine the wavelengths associated with the particle in the area $x > 0$ and in the area $x < 0$.

(2) What is the relationship between the amplitude of the transmitted wave in the area $x > 0$ and the amplitude of the incident wave in the area $x < 0$.

Use the two boundary conditions that at $x=0$
$$
\Psi_{x<0} = \Psi_{x>0}
$$
$$
({\dfrac{\partial\Psi_{x<0}}{\partial_x}})_{x=0} = ({\dfrac{\partial\Psi_{x>0}}{\partial_x}})_{x=0}
$$

Thanks in advance
llawerif is offline   Reply With Quote
Old Oct 16th 2016, 01:34 PM   #2
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Anyone out there, correct me if I'm wrong here.

I would start with wave equations for the Potential Step problem for x<0 and x>0, for the case when the step is constant and E>StepPotential. Do you have those equations? Hint: the x<0 region should have a wave equation which is the sum of two exponential functions, while the x>0 region is just an exponential function with single piece.

Once you have those, can you determine which portion of your exponents determine wavelength vs. phase shift (the latter which we don't care about)? And then, what about amplitude for functions of that form?

Probably should have considered this first, but are you expected to derive these wave functions, and do you need help with that?
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