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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2021

Paper 341

The equation can be written
$$
u_t=iu_{xx}-iVu.
$$
For the squared <L2 norm>, periodic <integration by parts> and the reality of $V$ give
$$
\frac d{dt}\int_{-1}^1|u|^2\,dx
=2\operatorname{Re}\int_{-1}^1
\left(iu_{xx}-iVu\right)\overline u\,dx
=2\operatorname{Re}\left[
-i\int_{-1}^1|u_x|^2\,dx
-i\int_{-1}^1V|u|^2\,dx
\right]=0.
$$
\b[Thus the <Schrodinger equation> generates a unitary flow and $\|u(\cdot,t)\|_2$ is constant.]

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 341 / 3 / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 341 / 3 / a / Solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 341 / 3 / a / Solution

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