Dear PF:(adsbygoogle = window.adsbygoogle || []).push({});

I'm currently working in a problem that has had me stranded for several weeks now. The problem reads as follows:

(See attachment)

Consider a beam of quantum particles (that is, the particles are small enough to exhibit non-negligible quantum effects) that propagates through a two-dimensional waveguide of width [itex] H [/itex] from [itex]x=-\infty [/itex] to [itex]x=+\infty [/itex]. At [itex] x=0 [/itex] the particles encounter a step of height [itex]0<H_0<H[/itex]. All walls are impenetrable. Calculate the reflection and transmission coefficients.

Approach

The potential within the waveguide can be described as:

[tex]

V(x,y)=

\begin{cases}

\infty & \text{at} \ AO, OH_0, H_0D, CE, OF\\

0 & \text{elsewhere}

\end{cases}

[/tex]

A particular solution I worked out was:

[tex]

\psi(x,y)=

\begin{cases}

A_1^+\sin k_1x\sin\frac{\pi y}{H}, & x<0, 0<y<H_0 \\

A_1^+e^{ik_1x}\sin \frac{\pi y}{H}+ \sum\limits_mA_m^-e^{-ik_mx}\sin\frac{m\pi y}{H},& x<0, H_0<y<H\\

\sum\limits_mB_m^+e^{ik'_mx}\sin\frac{m\pi(y-H_0)}{H-H_0}, & x>0, H_0<y<H

\end{cases}

[/tex]

where [itex] k_1=\sqrt{K^2-(\pi/H)^2}, k_m = \sqrt{K^2-(m\pi/H)^2}, k'_m=\sqrt{K^2-(m\pi/(H-H_0))^2}[/itex], [itex]E=\hbar^2K^2/2m[/itex].

The wavefunction above must satisfy the following boundary conditions:

[tex] \psi(AO, OH_0, H_0D, CE, OF)=0 ,

[/tex]

[tex]

\psi(HH_0^-)=\psi(HH_0^+),

[/tex]

[tex]

\psi'(HH_0^-)=\psi'(HH_0^+)

[/tex]

From the above, I can calculate the constants [itex]A[/itex] and [itex]B[/itex], but all I get is nonsense. In particular, the above solution isnotcontinuous along the boundary [itex]BH_0[/itex], but as hard as I try, I cannot make a satisfactory modification such that this discontinuity is healed.

What am I doing wrong? Could someone direct me to a similar problem? I'm sure there has to be a treatise for presence of steps in rectangular waveguides, but I can't seem to find any.

Thanks in advance.

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# Scattering Within Rectangular Waveguide With A Step Obstacle

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