For a harmonic, time dependent perturbation in QM, how is energy conservation imposed?

Question

I am currently reading Sakurai's Modern Quantum Mechanics, and in the section on time dependent perturbation theory, he derives the first order coefficient for an energy state n at time t in the presence of an oscillating potential:

$$ c_n^{(1)} = \frac{1}{\hbar} \left [ \frac{1-e^{i(\omega+\omega_{ni})t}}{\omega+\omega_{ni}}V_{ni} + \frac{1-e^{i(\omega_{ni}-\omega)t}}{-\omega+\omega_{ni}}V^\dagger_{ni} \right] $$

And then he states that the transition probability $|c^{(1)}_n|^2$ is only appreciable if

$$ \omega_{ni} + \omega \approx 0 \quad \text{or} \quad E_n \approx E_i -\hbar\omega \\ \\ , \\ \\ \omega_{ni} - \omega \approx 0 \quad \text{or} \quad E_n \approx E_i +\hbar\omega $$

But I don't understand how he imposes these energy conditions. Just from the first formula, if it was true that $\omega_{ni} + \omega \approx 0$, couldn't it still be the case that Energy was gained by the system, and the state got boosted up an energy level?

Is he simply just manually imposing conservation of energy here? Or is there something I am missing?

Thanks.

Answer 1

The approximate energy conservation in form $E_n - E_i = \pm\omega$ is not imposed manually but is a consequence of the expression for $c_n^{(1)}$. Actually, for arbitrary time-dependent external potential, the system can gain or lose any amount of energy. But for harmonic perturbation, the transitions obeying the conservation law $E_n - E_i = \pm\omega$ have the largest probability.

For small perturbation and in the absence of resonance ($\omega$ equals neither $\omega_{ni}$ nor $-\omega_{ni}$) the transition amplitude from $|i\rangle$ to $|n\rangle$ remains small at all $t$. For $\omega \approx \pm\omega_{ni}$, as Joseph Takach noted, the denominators approach zero, which leads to large transition amplitude at large times. For example, at $\omega \approx \omega_{ni}$, one can expand the second exponent as $e^{i(\omega_{ni} - \omega)t} \approx 1 + i(\omega_{ni} - \omega)t$ and get \begin{equation} c^{(1)}_n = \frac{1}{\hbar}\left[-iV^\dagger_{ni}t + \frac{1 - e^{i(\omega+\omega_{ni})t}}{\omega+\omega_{ni}}\right] \approx -\frac{i}{\hbar}V^\dagger_{ni} t. \end{equation} Growing value of the transition amplitude is consistent with the energy conservation law. However, the system can still undergo a non-resonant transition to some other state $|n'\rangle$ which violates energy conservation in form $E_n - E_i = \pm\omega$.