5.4: Harmonic Perturbation

This is generally useful since by Fourier analysis we can decompose any periodic perturbation into harmonic components.

Let the perturbing potential be $V ({\bf r}, t) = V ({\bf r}) \cos \omega t$

If the initial state at $t = 0$ is $k$, and the final state $m$ then

$$c_m \approx \frac{−i}{\hbar} V_{mk} \int^t_0 e^{i\omega_{mk}t} \frac{1}{2} (e^{iwt} + e^{−iwt} ) dt = \frac{V_{mk}}{2\hbar} \left( \frac{e^{i(\omega_{mk}−\omega )t} − 1}{\omega_{mk} − \omega} + \frac{e^{i(\omega_{mk}+\omega )t} − 1}{\omega_{mk} + \omega} \right) \nonumber$$

where $V_{mk}$ is the time independent part of the matrix element $\langle m|\hat{V} |k \rangle$. This function is dominated by the first term in the region around $\omega_{mk} = \omega$, so we can consider only the first term to obtain an estimate for the transition probability:

$$|c_m(t)|^2 = \frac{V^2_{mk} \sin^2 [(\omega_{mk} − \omega )t/2]}{\hbar^2 (\omega_{mk} − \omega )^2} = \frac{1}{4\hbar^2} V^2_{mk} f(t, \omega_{mk} − \omega ) \nonumber$$

Where the function $f$ is the same as we encountered earlier. Thus an external perturbation at a given frequency most strongly induces transitions between energy levels separated by $\hbar \omega$.

This is another manifestation of an uncertainty principle. If the potential is electromagnetic, the most probable transition is the absorption of a $\hbar \omega_{mk}$ photon as the system changes energy by $\hbar \omega_{mk}$. But if the transition happens very fast, the peak is broad and the photon could have a wide range of energies, contrariwise, if the transition occurs after a long time the photon frequency is well defined: $\Delta E\Delta t \geq \hbar/ 2$. This uncertainty gives rise to the ‘natural linewidth’ of a particular transition, and causes a limit to the accuracy of certain experiments. There is a slight difference from the Heisenberg Uncertainty in non-relativistic quantum mechanics because time is not an operator so one cannot define the commutator of time with the Hamiltonian.

Note the extraordinary result that the transition probability at small times is $\left( V^2_{mk}/4\hbar^2 \right) t^2$. Consider what happens if the state is measured frequently compared to if measurements are made infrequently: frequent measurement tends to inhibit the transition!