 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

2022

Paper 4

Insert the stated expansion into the <Time-dependent Schrödinger equation>
$$
i\hbar\frac d{dt}|\psi(t)\rangle
=(H_0+v(t))|\psi(t)\rangle.
$$
When the derivative acts on each free phase, its energy factor is cancelled by the action of $H_0$ on the corresponding <eigenstate>. The remaining terms obey
$$
i\hbar\left[
\dot a(t)e^{-iE_0t/\hbar}|0\rangle
+\int_0^\infty e^{-iE_kt/\hbar}
\bigl(\dot b_k(t)|k,+\rangle+\dot c_k(t)|k,-\rangle\bigr)\,dk
\right]
=-F e^{-i\omega t}x|\psi(t)\rangle.
$$
Division by $i\hbar$ gives
$$
\boxed{
\dot a(t)e^{-iE_0t/\hbar}|0\rangle
+\int_0^\infty e^{-iE_kt/\hbar}
\bigl(\dot b_k|k,+\rangle+\dot c_k|k,-\rangle\bigr)\,dk
=\frac{iF}{\hbar}e^{-i\omega t}x|\psi(t)\rangle}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 33A / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 33A / a / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 33A / a / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)