1.26: Matching piecewise solutions
- Page ID
- 50130
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The Schrödinger Equation is a second order differential equation. From Equation (1.25.2) we observe the second derivative of the wavefunction is finite unless either E or V is infinite.
Infinite energies are not physical, hence if the potential is finite we can conclude that \(\frac{d\psi}{dx}\) and \(\psi(x)\) are continuous everywhere.
That is, at the boundary (\(x = x_{0}\)) between piecewise solutions, we require that
\[ \psi_{-}(x_{0})=\psi_{+}(x_{0}) \nonumber \]
and
\[ \frac{d}{dx}\psi_{-}(x_{0}) = \frac{d}{dx}\psi_{+}(x_{0}) \nonumber \]