# Double-Slit Experiment Explained

Quantum physics is a mathematical theory whose equations yield a complex (as in “complex numbers”) mathematical function denoted by $\Psi$, which contains information about the behavior of a physical system. Quantum physics is based on two assumptions about $\Psi$:

• The square of (the absolute value of) $\Psi$ is the probability of the behavior of the system. That is why $\Psi$ is called the probability amplitude or probability wave.
• If there are two paths for the system to develop, the total $\Psi$ of the system is the sum of the $\Psi$s for each path. This is called the superposition principle.

Using these two assumptions, we can easily explain the spooky behavior of photons in a double-slit experiment. Let $\Psi_1$ be the probability amplitude for the first slit and $\Psi_2$ for the second one. Then the total amplitude is $\Psi_1+\Psi_2$, and the total probability is $\left |\Psi_1+\Psi_2\right |^2$.

If only the first slit is open, the total amplitude is just $\Psi_1$ (because $\Psi_2=0$), and the probability is $\left |\Psi_1\right |^2$. This gives rise to a single bright image on the photographic plate. The same result holds if only the second slit is open, except that now the probability is $\left |\Psi_2\right |^2$; but this probability is identical to $\left |\Psi_1\right |^2$ if the two slits are identical. When both slits are open, the probability is $\left |\Psi_1+\Psi_2\right |^2$, which is not just the sum $\left |\Psi_1\right |^2+\left |\Psi_2\right |^2$, as the reader recalls from high school algebra: the square of the sum of two quantities is the sum of the squares of the two quantities plus twice the product of the two quantities. It is this last term that gives rise to the dark bands as the following discussion shows.

If you are familiar with complex numbers and trigonometry you can appreciate the following mathematical derivation, which shows precisely where the dark regions come from. It turns out that one can write the complex probability amplitudes in polar form as $\Psi_1=A e^{i\phi_1}$ and $\Psi_2=A e^{i\phi_2}$, where $A$ is a real number, $i=\sqrt{-1}$, and $\phi_1$ and $\phi_2$, also reals, are called the phase angles of the two probability amplitudes and depend on the distance from the slits to the point of interest on the photographic plate, and therefore, on the location of that point on the plate. Then

$\left |\Psi_1\right |^2=\Psi_1\Psi_1^*=A e^{i\phi_1}A e^{-i\phi_1}=A^2e^{(i\phi_1-i\phi_1)}=A^2e^0=A^2,$

where * denotes complex conjugation. Similarly, $\left |\Psi_2\right |^2=A^2$. On the other hand,

$\left |\Psi_1+\Psi_2\right |^2=\left(A e^{i\phi_1}+A e^{i\phi_2}\right)\left(A e^{i\phi_1}+A e^{i\phi_2}\right)^*$

or performing the complex conjugation,

$\left |\Psi_1+\Psi_2\right |^2=\left(A e^{i\phi_1}+A e^{i\phi_2}\right)\left(A e^{-i\phi_1}+A e^{-i\phi_2}\right)$

Multiply out the last two parentheses to get,

$\left |\Psi_1+\Psi_2\right |^2=A^2\left(e^{i(\phi_1-\phi_1)}+e^{i(\phi_1-\phi_2)}+e^{i(\phi_2-\phi_1)}+e^{i(\phi_2-\phi_2)}\right)$

or since $e^0=1$, the first and last term add up to 2, and we obtain

$\left |\Psi_1+\Psi_2\right |^2=A^2\left(2+e^{i(\phi_1-\phi_2)}+e^{-i(\phi_1-\phi_2)}\right).$

Recalling that for any angle $\theta$, $\cos\theta=\dfrac{e^{i\theta}+e^{-i\theta}}{2}$, we finally get

$\left |\Psi_1+\Psi_2\right |^2=2A^2\left[1+\cos(\phi_1-\phi_2)\right].$

As you move on the photographic plate, $\phi_1-\phi_2$ changes. When it is zero or a multiple of $2\pi$, $\cos(\phi_1-\phi_2)=1$ and the total probability will be $4A^2$, corresponding to bright bands. When $\phi_1-\phi_2$ is an odd multiple of $\pi$, $\cos(\phi_1-\phi_2)=-1$ and the total probability will be zero, corresponding to dark bands. There is no better “explanation” than this! Anyone giving a non-mathematical explanation such as Gary Zukav, who assigns consciousness to photons, either does not understand QT, or if (s)he does it purposefully, a charlatan!