0
$\begingroup$

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.

Figure in Feynman lectures on physics

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 $1$.

$\endgroup$
0
$\begingroup$

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 $P_{12}$ .

$\endgroup$
  • $\begingroup$ 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! $\endgroup$ – Toughee Oct 3 '17 at 17:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.