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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 2

The <Schrödinger equation> and its complex conjugate are
$$
i\hbar\frac{\partial\psi}{\partial t}
=-\frac{\hbar^2}{2m}\nabla^2\psi+U\psi,
\qquad
-i\hbar\frac{\partial\psi^*}{\partial t}
=-\frac{\hbar^2}{2m}\nabla^2\psi^*+U\psi^*.
$$
Because the potential $U$ is real, its two contributions cancel when differentiating the <probability density> $\rho=\psi^*\psi$. Therefore
$$
\begin{aligned}
\frac{\partial\rho}{\partial t}
&=\psi^*\frac{\partial\psi}{\partial t}
+\psi\frac{\partial\psi^*}{\partial t}\\
&=\frac{i\hbar}{2m}
\left(\psi^*\nabla^2\psi-\psi\nabla^2\psi^*\right)\\
&=-\nabla\cdot
\left[
-\frac{i\hbar}{2m}
\left(\psi^*\nabla\psi-\psi\nabla\psi^*\right)
\right].
\end{aligned}
$$
The expression in square brackets is the <probability current> $J$, so
$$
\boxed{\frac{\partial\rho}{\partial t}+\nabla\cdot J=0}.
$$
This <probability continuity equation> says that probability can leave a region only through the current across its boundary.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 15D / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 15D / a / Solution

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Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 15D / a / Solution

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