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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2016

Paper 3

For a particle of mass $m$, the <probability density> and one-dimensional <probability current> are
$$
\boxed{\rho=\psi^*\psi,\qquad
j=\frac{\hbar}{2mi}(\psi^*\psi_x-\psi\psi_x^*)
=\frac{\hbar}{m}\operatorname{Im}(\psi^*\psi_x).}
$$
The <Schrödinger equation> and its <complex conjugate>, for real $V$, give
$$
\rho_t=\psi_t\psi^*+\psi\psi_t^*
=\frac{i\hbar}{2m}(\psi^*\psi_{xx}-\psi\psi_{xx}^*)=-j_x.
$$
The potential terms cancel precisely because $V$ is real. Thus \b[$\rho_t+j_x=0$], the <continuity equation> for probability. Integrating over an interval says that its probability changes by the incoming current minus the outgoing current.

Solution

Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 3 / 8B / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 3 / 8B / a / Solution

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Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 3 / 8B / a / Solution

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