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Reading my text book on quantum physics, I found the following statement:

Let's suppose we have a short-scale force, so we have a potential energy such that $$ V(x) = V_- \quad \text{if} \quad x \ll 0 \\ V(x) = V_+ \quad\text{if}\quad x \gg 0 $$ with $V_+ \ge V_-$, the wavefunction on these regions is $$ \Phi_E = A e^{ik_-x} + B e^{-ik_-x} \quad\text{if}\quad x \ll 0 \qquad \text{with}\quad k_- = \sqrt{2m(E-V_-)}/\hbar \\ \Phi_E = C e^{ik_+x} + D e^{-ik_+x} \quad\text{if}\quad x \gg 0 \qquad \text{with}\quad k_+ = \sqrt{2m(E-V_+)}/\hbar $$ Since there exist a linear relationship between $\Phi_E(x\ll 0)$ and $\Phi_E(x\gg 0)$, the coefficients also have a linear relationship $$ \left(\begin{array}{c}A\\B\end{array}\right) = \left(\begin{array}{cc}t_{11}&t_{12}\\t_{21}&t_{22}\end{array}\right) \left(\begin{array}{c}C\\D\end{array}\right) $$ and only two coefficients are indepent. Thus, all the energies $E>V_+$ have a multiplicity of two.

I understand that the degeneracy must be two because of the two independent coefficients, but I don't see where the linear relationship between $\Phi_E(x\ll 0)$ and $\Phi_E(x\gg 0)$ comes from to explain this result. Is there any easy proof of this?

Qmechanic
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SrJaimito
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1 Answers1

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  1. The point is that the 1D TISE for fixed energy $E>V_{\pm}$ is a 2nd-order linear homogeneous ODE. We assume that the solutions are globally well-defined on the whole real line $\mathbb{R}$. So the general solution $\psi$ is a complex linear combination of 2 linearly independent solutions $\psi_{1/2}$.

  2. We next choose $\psi_{1/2}$ such that they satisfy the boundary condition $$ \lim_{x\to \infty}e^{(-1)^a ik_+ x}\psi_a(x)~=~1, \qquad a~\in~\{1,2\}.$$ Note that $\psi_1$ corresponds to $(C,D)=(1,0)$ while $\psi_2$ corresponds to $(C,D)=(0,1)$.

  3. We finally define the transfer matrix elements as $$\psi_a(x)~\sim~t_{1a}e^{ ik_- x}+t_{2a}e^{ -ik_- x} \quad{\rm for}\quad x\to -\infty.$$ Note that $\psi_1$ corresponds to $(A,B)=(t_{11},t_{21})$ while $\psi_2$ corresponds to $(A,B)=(t_{12},t_{22})$.

  4. This explains the linear relationship between $(C,D)$ and $(A,B)$. See also e.g. my related Phys.SE answer here.

Qmechanic
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