- #1

Ad VanderVen

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- TL;DR Summary
- A certain function ##f(x)## represents a probability density function. This means that the integral over the definition range of ##x## must be at least equal to a constant. However, I cannot solve the integral.

In an article written by Richard Rollleigh, published in 2010 entitled

"For something to be predictable, it must be a consistent measurement result. The positions at which individual particles land on the screen are not consistent: each particle could land in any bright fringe. Positions are not predictable. What is consistent is the probability of each particle’s landing at any position, i. e. the probability density function (pdf) of each particle’s position. The pdf of position is just the double slit interference pattern illustrated in Figure 3. It is reproduced any time you repeat the experiment and it is predicted by Equation 2."

Equation 2 reads as follows:

$$\displaystyle I={\frac {I_{{\max}}{\lambda}^{2}}{{\pi }^{2}{a}^{2} \left( \sin \left( \theta \right) \right) ^{2}} \left( \cos \left( {\frac {\pi \,d\sin \left( \theta \right) }{\lambda}} \right) \right) ^{2} \left( \sin \left( {\frac {\pi \,a\sin \left( \theta \right) }{\lambda}} \right) \right) ^{2}}$$

If equation 2 represents a pdf, then the following integral:

$$\displaystyle \int_{-\infty }^{\infty }\!{\frac {{\lambda}^{2}}{{\pi }^{2}{a}^{2} \left( \sin \left( \theta \right) \right) ^{2}} \left( \cos \left( {\frac {\pi \,d\sin \left( \theta \right) }{\lambda}} \right) \right) ^{2} \left( \sin \left( {\frac {\pi \,a\sin \left( \theta \right) }{\lambda}} \right) \right) ^{2}}\,{\rm d}\theta$$

must be equal to a constant. I just can't solve this integral. Who can help me?

*The Double Slit Experiment and Quantum Mechanics*, he argues as follows:"For something to be predictable, it must be a consistent measurement result. The positions at which individual particles land on the screen are not consistent: each particle could land in any bright fringe. Positions are not predictable. What is consistent is the probability of each particle’s landing at any position, i. e. the probability density function (pdf) of each particle’s position. The pdf of position is just the double slit interference pattern illustrated in Figure 3. It is reproduced any time you repeat the experiment and it is predicted by Equation 2."

Equation 2 reads as follows:

$$\displaystyle I={\frac {I_{{\max}}{\lambda}^{2}}{{\pi }^{2}{a}^{2} \left( \sin \left( \theta \right) \right) ^{2}} \left( \cos \left( {\frac {\pi \,d\sin \left( \theta \right) }{\lambda}} \right) \right) ^{2} \left( \sin \left( {\frac {\pi \,a\sin \left( \theta \right) }{\lambda}} \right) \right) ^{2}}$$

If equation 2 represents a pdf, then the following integral:

$$\displaystyle \int_{-\infty }^{\infty }\!{\frac {{\lambda}^{2}}{{\pi }^{2}{a}^{2} \left( \sin \left( \theta \right) \right) ^{2}} \left( \cos \left( {\frac {\pi \,d\sin \left( \theta \right) }{\lambda}} \right) \right) ^{2} \left( \sin \left( {\frac {\pi \,a\sin \left( \theta \right) }{\lambda}} \right) \right) ^{2}}\,{\rm d}\theta$$

must be equal to a constant. I just can't solve this integral. Who can help me?