Hi everyone,

I have some doubts in a exercise from chapter 3 of Quantum Mechanics (Cohen-Tannoudji). It is number 3 from that chapter. It defines a wave function of a free particle at time t = 0 as:

\[\phi (x,0)\ =\ N\int_{-\infty}^{\infty}e^{{-\mid k \mid}/k_o} e^{i k x} dk\]

where \(k_o\)and N are constants.

It first asks for the probability of a measurement of the momentum, performed at time t = 0, which yields a result included between \(-p_1\ \)and \(p_1\). That is easily done.

Then, it asks the same for a measurement performed at time t. Now, I arrived at the same probability, which makes sense because it is a free particle.

After that, it asks for the form of the wave packet at time t=0. And that is the following:

\[\phi (x,0)\ =\ \frac{2N/k_o}{{1/k_o}^2+x^2}\]

I wanted to show the plot but for some reason I was not able to upload it. Its shape as similar to a gaussian wave packet. It then asks to calculate \(\Delta X \Delta P\ \)for that time t which yields \(\hbar/2\). Finally it asks to describe qualitatively the wave packet evolution in time. My reasoning is the following:

As it looks similar to a gaussian wave packet, a first guess would be that it behaves similarly as well. Therefore, the wave packet should spread as it evolves through time. That means \(\Delta X\ increases\ and\ \Delta P\ decreases.\) But if that is true, then there probability of our measurement should increase also and not stay constant.

I would be really grateful if anyone could elucidate this doubt.

Best Regards,

Frantic Undergrad