What is the physical significance of having a wavefunction with a discontinuous derivative? Sometimes we impose continuity of the derivative in order to find values of unknown parameters and it's helpful. But sometimes it's not required and so it seems arbitrary.
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The relation of the first derivative of the wave function is meant to keep the flux of propability constant across the interface. Therefore, if the effectice masses are different across a heterojunction ($m_1$ at left and $m_2$ at right), the contiuity of flux leads to the following relation across a heterojunction at $x = x_0$: $$ \frac{1}{m_1}\frac{\partial \Psi}{\partial x}\Big\vert_{x_0^-} = \frac{1}{m_2} \frac{\partial \Psi}{\partial x}\Big\vert_{x_0^+} $$
If there is a $\delta(x-x_0)$ potential at the boundary, it will involve in the continuity of flux throught the Shr$\ddot{\text{o}}$dinger equation.
ytlu
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Continuity of wave function is essential for finding momentum, momentum is the first derivative of wave function.
hat
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