 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 313

Treat $\phi$ and $\bar\phi$ as independent fields. The <Lagrangian density> is
$$
\mathcal L=\frac i2(\phi\dot{\bar\phi}-\bar\phi\dot\phi)
+\frac12\phi_x\bar\phi_x-\frac12(\phi\bar\phi)^2
+\epsilon V(\epsilon x)\phi\bar\phi.
$$
The <Euler-Lagrange equation> for $\bar\phi$ is
$$
-i\phi_t-\frac12\phi_{xx}-|\phi|^2\phi
+\epsilon V(\epsilon x)\phi=0,
$$
or
$$
\boxed{i\phi_t=-\frac12\phi_{xx}-|\phi|^2\phi
+\epsilon V(\epsilon x)\phi}.
$$
The equation from variation of $\phi$ is its complex conjugate.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 313 / 1 / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 313 / 1 / Solution

 Ancestors (10)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 313 / 1 / Solution

 Ancestors (10)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)