How Do Physicists Deal with Interference?

Abstract

So, we have seen one of the two “impossible things that we have to believe before breakfast” namely things being apparently in two different states before being measured or before one looks at them.

Appendices

Appendix

4.A The Wave Function

The first precision to be made about the wave function is that $\Psi (x)$ is in general a complex number and, to be correct, one should have written everywhere $|\Psi (x)|^2$ instead of $\Psi (x)^2$ in Sects. 4.1 and 4.2, where, for a complex number $z=a+ib$, $|z|^2= a^2+ b^2$.

The fact that the total area under the curve in Fig. 4.3 is equal to one means that $\int _{{\mathbb {R}}}|\Psi (x)|^2 dx =1$. This ensures that the probability of finding the particle somewhere is equal to one, as it should!

The probability of finding the particle in a region A is therefore $\int _A |\Psi (x)|^2 dx$, see Fig. 4.3.

Finally, in order to keep that constraint, each of the collapsed wave functions, after a measurement, must also satisfy $\int _{{\mathbb {R}}}|\Psi (x)|^2 dx =1$. In Fig. 4.5, the situation is symmetric and, since the regions where $\Psi _1(x)$ and $\Psi _1(x)$ are non-zero do not overlap, one has $\int _{{\mathbb {R}}}|\Psi (x)|^2 dx= \int _{{\mathbb {R}}}|\Psi _1(x)|^2 dx + \int _{{\mathbb {R}}}|\Psi _2(x)|^2 dx=1$ and thus $\int _{{\mathbb {R}}}|\Psi _1(x)|^2 dx = \int _{{\mathbb {R}}}|\Psi _2(x)|^2 dx=\frac{1}{2}$. So, the collapsed wave function is not $\Psi _1(x)$ or $\Psi _2(x)$, as we said in Sect. 4.2, but rather $\sqrt{2} \Psi _1(x)$ or $\sqrt{2} \Psi _2(x)$, that satisfy $\int _{{\mathbb {R}}}|\sqrt{2} \Psi _1(x)|^2 dx = \int _{{\mathbb {R}}}|\sqrt{2} \Psi _2(x)|^2 dx=1$.

To illustrate the phenomenon of constructive and destructive interferences, consider Fig. 4.13, where the three curves represent the functions $\Psi _1 (x)^2$, $\Psi _2(x)^2$ and $(\Psi _1 (x)+ \Psi _2(x))^2$ (we suppress the variable t here). We chose those functions (with the x axis drawn horizontally), so that they resemble (qualitatively) the wiggly blue curve on the right of Fig. 4.10. We note that the function $(\Psi _1 (x) +\Psi _2 (x))^2$ may vanish at points x where neither $\Psi _1 (x)^2$ nor $\Psi _2 (x)^2$ vanish. It can also be larger than the sum $\Psi _1 (x)^2+ \Psi _2 (x)^2$ for other x’s. In the latter case, one says that the waves interfere constructively and in the former one that they interfere destructively.