Probability amplitude

We associate a wave $\psi$ with an electron such that the $\vert\psi(x,t) \vert^2$ gives that probability of finding the electron at x. The probability is mathematically equivalent to the intensity of the wave.

At any point on the screen

$$\psi= \psi_1 + \psi_2,$$ (18.5)

where $\psi_1$ is the contribution from slit $1$ and $\psi_2$ from slit $2$. $\psi_1$ and represents the two alternatives by which the electron can reach the point $x$ on screen B. In quantum mechanics the two alternatives interfere.

The probability of finding the electron on any point on the screen is

$$P_{12}= \vert\psi_1 + \psi_2 \vert^2 = P_{_1} + P_{_2} + 2 \sqrt{P_{_1} P_{_2}} \cos \delta.$$ (18.6)

Although the wave is defined everywhere, we always detect the electron at only one point and in whole.

The wave function $\psi(x,t)$ is the probability amplitude which is necessarily complex.

What happens if we try to determine through which slit the electron reaches screen B.

This can be done by placing a light at each slit so as to illuminate it. The electrons will scatter the light as it passes through the slit. So if we see a flash of light from slit $1$, we will know that the electron has passed through it, similarly if the electron passes through slit $2$ we will get a flash from that direction.

If we know through which slit the electron reaches screen B then

$$P_{_{12}}= P_{_{1}}+ P_{_{2}},$$ (18.7)

ie it came through either slit $1$ or slit $2$.

It is found that once we determine through which slit the electron passes, the two possibilities no longer interfere. The probability is then given by equation (18.7) instead of (18.6).

The act of measurement disturbs the electron. The momentum is changed in the scattering, and we no longer have any information where it goes and hits the screen.

There is a fundamental restriction on the accuracy to which we can simultaneously determine a particle's position and momentum. The product of the uncertainties in the position and the momentum satisfies the relation,

$$\Delta x \Delta p \ge \hbar/2,$$ (18.8)

known as Heisenberg's uncertainty principle.