The problem looks very simple. We have a time-dependent Hamiltonian:(adsbygoogle = window.adsbygoogle || []).push({});

$$H(t) = B(t)H_0$$,

where ##B(t)## is a numerical function, and matrix ##H_0## is time-indpendent.

Let us consider:

$$B(t) = \begin{cases}

1,&\text{for $0\leq t\leq t_0$}\\

A,&\text{for $t>t_0$.}

\end{cases}$$

Also, let us consider that ##H_0## has an eigenstate ##|n\rangle## :

$$ H_0|n\rangle = \varepsilon_n |n\rangle$$.

The problem is to find solution ##|\psi(t)\rangle## of the Schrodinger equation

$$i\frac{\partial}{\partial t}|\psi(t)\rangle = H(t)|\psi(t)\rangle ,$$

when ##|\psi(0)\rangle = |n\rangle##.

My solution is:

$$|\psi(t)\rangle = e^{ -i\varepsilon_n\int_0^t B(t')dt'}|n\rangle$$

So, there are no transitions to other levels from the ##|n\rangle## state.

Some people disagree with me because of "jump function makes exact solution not possible".

Is my solution right or not?

Thanks!

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# A Is my solution of time-dependent Schrodinger equation right?

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