Solving an Integral involving a probability density function

In summary, Richard Rollleigh argues that for predictability, consistency in measurement results is necessary. However, in the double slit experiment, the positions where individual particles land on the screen are not consistent, but the probability of each particle's landing at any position can be predicted using a probability density function (pdf) represented by Equation 2. This equation also shows that the integral of the pdf must be equal to a constant. While the integral may be difficult to solve, it is independent of the width of the slits and the intensity of the particles can be affected by the wavelength and slit separation. The range of theta in the experiment is between -π/2 and π/2, and using substitution, it can be shown that
  • #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 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?
 
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  • #2
Ad VanderVen said:
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?
Try Wolfram Alpha?
 
  • #3
PS The range of ##\theta## is ##[-\dfrac{\pi}{2}, +\dfrac{\pi}{2}]##. The substitution ##u = \dfrac{\pi\sin \theta}{\lambda}## is probably a good start. Then try Wolfram Alpha.
 
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  • #4
PeroK@ Can you tell me why the range of ##\theta## is ##\displaystyle [- \frac{\pi}{2}, \, + \frac{\pi}{2}]##?

Integration by substitution doesn't work because the derivative of ##u## is again a function of ##\theta##.
 
  • #5
Ad VanderVen said:
PeroK@ Can you tell me why the range of ##\theta## is ##\displaystyle [-\pi /2,\,+\pi /2]##?
Because ##\theta## is the diffraction angle, with ##\theta = 0## representing the line directly between the slits.
Ad VanderVen said:
Integration by substitution doesn't work because the derivative of ##u## is again a function of ##\theta##.
It looks tricky. My guess is that the integral is independent of the width of the slits (##a##), but that ##I_{max}## depends on the slit separation (##d##) and the wavelength (##\lambda##).

The next try would be a numerical computation.
 
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  • #6
PeroK said:
My guess is that ... that ##I_{max}## depends on the slit separation (##d##) and the wavelength (##\lambda##).
It must.

For large separation you get two single slit patterns - and the formula may even break down.

For large ##\lambda## you get almost no diffraction and two points of high intensity.
 
  • #7
\begin{align*}
\displaystyle \int_{-\pi/2 }^{\pi/2 }&\;{\frac {{\lambda}^{2}}{{\pi }^{2}{a}^{2} \left( \sin \left( \theta \right) \right) ^{2}} \left( \cos \left( {\frac {\pi \,b\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\\[10pt]
&\stackrel{u=\sin \theta\, , \,\omega =\pi/\lambda }{=}\dfrac{b\omega }{a\omega }\int_{-1}^{1}
\dfrac{\cos^2(b\omega u)}{b\omega u}\dfrac{\sin^2(a\omega u)}{a\omega u} \dfrac{du}{\sqrt{1-u^2}}\\[10pt] &\stackrel{v=a\omega u}{=} 2(a\omega )^2\int_{0}^{a\omega }\cos^2\left(\dfrac{b}{a}v\right)\dfrac{\sin^2(v)}{v^2\sqrt{(a\omega)^2-v^2}}\,dv\\[10pt]
&= 2(a\omega )^2 \int_0^{a\omega } \underbrace{\dfrac{\cos^2\left(\dfrac{b}{a}v\right)\sin(v^2)}{v^2}}_{\text{continuous}} \underbrace{\dfrac{1}{\sqrt{(a\omega)^2-v^2}}}_{\text{integrable and }\geq 0}\,dv\\[10pt]
&=2(a\omega )^2 \,\dfrac{\cos^2\left(\dfrac{b}{a}\xi\right)\sin(\xi^2)}{\xi^2} \,\int_0^{a\omega }\dfrac{dv}{\sqrt{(a\omega)^2-v^2}}<\infty \text{ for some }\;\xi\in [0,a\omega ]
\end{align*}
 
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  • #8
There should be another factor of ##u^2## on the denominator there. I set ##k = \frac{\pi a}{\lambda}## and ##u = k\sin \theta##, so that ##d\theta = \frac{du}{\sqrt{k^2 - u^2}}##. We need to show that the following integral converges:
$$\int_0^k \frac{\cos^2\big (\frac d a u \big )}{\sqrt{k^2 - u^2}} \frac{\sin^2 u}{u^2} \ du$$Now, the following function is bounded (by ##1##):
$$\cos^2\big (\frac d a u \big )\frac{\sin^2 u}{u^2}$$And:
$$\int_0^k \frac{1}{\sqrt{k^2 - u^2}} = \frac \pi 2$$And that's all we need.
 
  • #9
The simpler version of the integral can be done using Wolfram Alpha, using:
$$\int_0^1 \frac{\cos^2(kx)}{\sqrt{1-x^2}} dx = \frac \pi 4 [J_0(2k) + 1]$$Where ##J_0## is a Bessel function of the first kind.

The more complicated version times out on the free version (perhaps I should invest in one of these maths engines?):
$$\int_0^1 \frac{\cos^2(kx)\sin^2(lx)}{x^2\sqrt{1-x^2}} dx = ?$$
 
  • #10
PeroK said:
There should be another factor of ##u^2## on the denominator there. I set ##k = \frac{\pi a}{\lambda}## and ##u = k\sin \theta##, so that ##d\theta = \frac{du}{\sqrt{k^2 - u^2}}##. We need to show that the following integral converges:
$$\int_0^k \frac{\cos^2\big (\frac d a u \big )}{\sqrt{k^2 - u^2}} \frac{\sin^2 u}{u^2} \ du$$Now, the following function is bounded (by ##1##):
$$\cos^2\big (\frac d a u \big )\frac{\sin^2 u}{u^2}$$And:
$$\int_0^k \frac{1}{\sqrt{k^2 - u^2}} = \frac \pi 2$$And that's all we need.
Yes, I wrote it in the editor here and missed the ##u## in the trig terms. I have corrected it for the sake of completeness, thanks.

But we have a singularity in both factors, one at each end of integration. How can we get boundedness or continuity?
 
  • #11
fresh_42 said:
But we have a singularity in both factors, one at each end of integration. How can we get boundedness or continuity?
The singularity in ##\dfrac{sin \ u}{u}## is removable. And ##\dfrac{1}{\sqrt{1 - u^2}}## is integrable on ##[0, 1]##. That's enough, surely?
 
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  • #12
PeroK said:
The singularity in ##\dfrac{sin \ u}{u}## is removable. And ##\dfrac{1}{\sqrt{1 - u^2}}## is integrable on ##[0, 1]##. That's enough, surely?
Yes, that is sufficient. (See problem 27 in my challenge thread of last month).
 
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  • #13
Since I was mostly into complex analysis, I would straightaway substitute the complex identities for sine and cosine and se what happens. Exponentials are much easier to handle than trig functions.
 
  • #14
Svein said:
Since I was mostly into complex analysis, I would straightaway substitute the complex identities for sine and cosine and se what happens. Exponentials are much easier to handle than trig functions.
That won't save you. You will end up in exp / polynomial.
 
  • #15
Svein said:
Since I was mostly into complex analysis, I would straightaway substitute the complex identities for sine and cosine and se what happens. Exponentials are much easier to handle than trig functions.
Even the simplified integral throws up a Bessel function.
 
  • #16
PeroK said:
The simpler version of the integral can be done using Wolfram Alpha, using:
$$\int_0^1 \frac{\cos^2(kx)}{\sqrt{1-x^2}} dx = \frac \pi 4 [J_0(2k) + 1]$$Where ##J_0## is a Bessel function of the first kind.

The more complicated version times out on the free version (perhaps I should invest in one of these maths engines?):
$$\int_0^1 \frac{\cos^2(kx)\sin^2(lx)}{x^2\sqrt{1-x^2}} dx = ?$$

Using trig identities yields[tex]
\begin{split}
\int_0^1 \frac{\cos^2(kx)\sin^2(lx)}{x^2\sqrt{1-x^2}} dx
&= \int_0^1 \frac{\sin^2((k+l)x) + \sin^2((k-l)x) + 2\sin((k+l)x)\sin((k-l)x)}{4x^2\sqrt{1-x^2}}\,dx \\
&= \frac{f(k+l,k+l) + f(k-l,k-l) + 2f(k+l,k-l)}{4} \end{split}[/tex] where [tex]
f(a,b) = \int_0^1 \frac{\sin(ax)\sin(bx)}{x^2\sqrt{1-x^2}}\,dx.[/tex] WolframAlpha can do the case [itex]a = b[/itex] with a result in terms of Bessel and Struve functions, but cannot do the
[itex]a \neq b[/itex] case. However, it does give [tex]
\int_0^1 \frac{\sin(ax) \sin(bx)}{\sqrt{1-x^2}}\,dx = \frac{\pi}{4} (J_0(a-b) - J_0(a+b))[/tex] and differentiating [itex]f[/itex] twice with respect to [itex]a[/itex] or [itex]b[/itex] would put the integrand into that form.
 
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  • #17
The integral can be done analytically to obtain an infintie series of Bessel functions. It's best not to substitute [itex]x =\sin\theta[/itex] but to start from
[tex]
\sin(a\sin\theta) = 2\sum_{k=0}^\infty J_{2k+1}(a) \sin((2k+1)\theta)
[/tex] (Abramowitz & Stegun, 9.1.43). Hence [tex]
\begin{split}
\cos(a\sin\theta)\sin(b\sin\theta) &= \frac{\sin((a+b)\sin\theta) - \sin((a-b)\sin\theta)}{2}\\
&= \sum_{k=0}^\infty (J_{2k+1}(a+b) - J_{2k+1}(a-b))\sin((2k+1)\theta) \\
&= \sum_{k=0}^\infty A_k \sin(2k+1)\theta \end{split}
[/tex] which leaves us with [tex]
\begin{split}
\int_{-\pi/2}^{\pi/2} \frac{1}{\sin^2\theta} \cos^2(a\sin\theta)\sin^2(b\sin\theta)\,d\theta &=
\sum_{k=0}^\infty\sum_{l=0}^\infty A_kA_l \int_{-\pi/2}^{\pi/2} \frac{\sin((2k+1)\theta) \sin((2l+1)\theta)}{\sin^2 \theta}\,d\theta \\
&= 2\sum_{k=0}^\infty\sum_{l=0}^\infty A_kA_l \int_{0}^{\pi/2} U_{2k}(\cos \theta) U_{2l}(\cos \theta)\,d\theta \end{split}[/tex] where [itex]U_n(\cos\theta) = \sin ((n+1)\theta)/\sin\theta[/itex] is a Chebyshev Polynomial of the Second Kind, [tex]
U_n(x) = \sum_{m=0}^{\lfloor n/2 \rfloor} (-1)^m \frac{(n-m)!}{m!(n-2m)!} (2x)^{n-2m}
[/tex] (Abramowitz & Stegun, 22.3.7). Thus the problem reduces to integrating powers of [itex]\cos\theta[/itex].
 
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1. What is an integral involving a probability density function (PDF)?

An integral involving a PDF is a mathematical calculation used to find the area under the curve of a probability distribution. It is used to determine the likelihood of a random variable falling within a certain range of values.

2. How do you solve an integral involving a PDF?

To solve an integral involving a PDF, you must first identify the limits of integration and the equation for the PDF. Then, you can use integration techniques such as substitution or integration by parts to calculate the area under the curve and determine the probability.

3. What is the purpose of solving an integral involving a PDF?

The purpose of solving an integral involving a PDF is to determine the probability of a random variable falling within a specific range of values. This information is useful in many fields, including statistics, economics, and engineering.

4. What are some common techniques used to solve integrals involving PDFs?

Some common techniques used to solve integrals involving PDFs include substitution, integration by parts, and partial fractions. Additionally, numerical methods such as the trapezoidal rule or Simpson's rule can be used for more complex integrals.

5. How can solving an integral involving a PDF be applied in real-world scenarios?

Solving an integral involving a PDF can be applied in real-world scenarios to determine the likelihood of events occurring. For example, it can be used in finance to calculate the probability of a stock price falling within a certain range or in medicine to determine the probability of a certain treatment being effective. It can also be used in quality control to determine the probability of a product meeting certain specifications.

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