# Is there a clear probability density function of the place of the photon on screen, in Young's Double-slit Experiment?

I am a Mathematics student, also interested in Quantum Physics. Recently a question about the Young's Double-Slit Experiment has attracted me in the topic: Is there a closed formula (as a function of $$x$$, a point on the screen) for the probability of a photon which is passed the slits to hit the screen at $$x$$ ?

Articles and lessons on the topic suggest this formula $$I=I_{max}\cos^2\left(\frac{\pi d\sin\theta}{\lambda}\right)\left[\frac{\sin(\pi a\sin\theta/\lambda)}{\pi a\sin\theta/\lambda}\right]^2,$$ which is the intensity at point $$x$$ (with angle $$\theta$$ from the line of the centers of the slit plane and the screen) according to the intensity at the center of the screen ($$I_{max}$$). However some of them also declared that it is not complete since it does not include some quantum properties.

Moreover, it seems that this formula can not be a probability density function (pdf) at all, for it is a non-zero continuous (periodic) function that accepts $$1$$ in its values and its Integral on the whole domain (on the whole screen) is less than $$1$$.

On the other hand, pioneers such as Feynman and some books, just obtained a general formula for the wave-function and stated that $$P_{12}(x)=\vert \Psi(x)\vert^2=\vert \Psi_1(x)+\Psi_2(x)\vert^2,$$ where $$\Psi_1(x)$$ and $$\Psi_2(x)$$ are the wave-functions of the photon passed slits $$1$$ and $$2$$, respectively. However in figures, they draw a plot for the pdf $$P_{12}$$ which is very similar to the equation$$(1)$$ mentioned above. Any comments would also help and be appreciated.

EDIT- The problem of my question is that a pdf CAN take a value greater than or equal to $$1$$.

• Dear Mathematics student, you wrote: Articles and lessons on the topic suggest this formula $$I=I_{max}\cos^2\left(\frac{\pi d\sin\theta}{\lambda}\right)\left[\frac{\sin(\pi a\sin\theta/\lambda)}{\pi a\sin\theta/\lambda}\right]^2,$$ Can you name some articles or a textbook that has the formula? As mentioned, to get a probability density function, you must first integrate this formula and then divide the result by the formula. I assume integration should be done over $\theta$ from $-2\pi$ to $+2\pi$. Please, could you specify $\lambda$, $d$, $I_{max}$ and $a$. Jan 1 at 15:55

The function $I$ can be normalised so that its integral over space (ie the screen) is $1$. It can then represent a probability density function similar to $P_{12}$ .
• Thanks for your attention, I think I messed up the two concepts probability (that can not be more than $1$) and probability density function (that can take a value greater than $1$). Your answer made me review the concepts! I put my edit separately in the question to save the answer and this comment meaningful. Hope to prevent others form such mistake! Oct 3, 2017 at 17:09