Quantum Mechanics; Sakurai Correlation Amplitude

Question

On page 79 Sakurai (2nd edition) states:

"As we sum over many terms with oscillating time dependence of different frequencies, a strong cancellation is possible for moderately large values of $t$. We expect the correlation amplitude that starts with unity at $t=0$ to decrease in magnitude with time."

Question:

• Why is after equation $(2.1.68)$ written that a strong cancellation is possible for moderately large values of $t$? How can I see that mathematically?

• Why should the correlation amplitude decrease in magnitude?

As far as I understand, the oscillation increases with the constant $t$, because we are in the "energy-space" and energy is the variable.

Answer 1

• For $t=0$ the oscillating term $\exp(- i \Delta E t /\hbar)$ is equal to one for all $\Delta E$. Hence, all oscillators are in-phase for $t=0$.
• However, if $t$ increases, the phase of each oscillator is given by $\Delta E t /\hbar$. Hence, as $t$ increases the phases of the oscillators become "scambled" -- they behave as if they were random.

The following plot shows the sum $$ y = \sum_{i=1}^{100} \cos(2\pi f_i t) $$ where $f_i = i \cdot f_0$ and $f_0$ is once $0.01Hz$ and the second time $0.02Hz$.

We see, that this function behaves as advertised: At $t=0$ the oscillators are in phase. Thus, they are correlated and we obtain a peak, which is independent of $f_0$. However, as time increases the oscillators are not synchronised. Therefore, the amplitude decreases. The "decay" is faster for the larger frequency, $f_0 = 0.02Hz$.