Double Slit Experiment Mathematics

In summary, the conversation discusses the use of Taylor series and Euler's formula in obtaining a final wave equation for the particles passing through two slits. The book suggests using the binomial theorem and Euler's identity to simplify the calculations, but the individual is having trouble understanding the algebra. They also mention using a substitution to make the calculations easier.
  • #1
James Brady
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Electrons are shot thru two slits separated by a distance s at a screen a distance ##z_0## away. The wave function for the particles is proportional to ## e^{ik \sqrt{(x-s/2)^2+z_0^2}} +e^{ik \sqrt{(x+s/2)^2+z_0^2}}##

Taking the first one, we can manipulate the square root algebraically ##z_0\sqrt{1 + (x-s/2)^2/z_0^2}##, this is where I run into issues, according to the text we can use the fact that ##\sqrt{1+a} = 1 + a/2## in a power series expansion to get ##z_0 + (x-s/2)^2/2 z_0## for small values of x.

I don't know how they got that. When I do a Taylor series expansion on ##z_0\sqrt{1 + (x-s/2)^2/z_0^2}## I get ##z_0\sqrt{1 + (s/2)^2/z_0^2} - \frac{-s/2}{z_0*\sqrt{\frac{-2/2}{z_0^2}+1}} *x## ...

How did they come up with such a simple answer using ##\sum_{n=0}^1 \frac{f^n(0)}{n!} x^n## ?

Am I using the Taylor series wrong?

The book goes on to use ##2cos(\theta) = exp(i \theta) + exp(-i \theta)## to obtain a final wave equation (at the screen) of ##exp(i \theta) cos(ksx/2z_0)## where k is the wave number ##\lambda = 2 \pi/k##

How did they come up with that? I don't see the algebra at all.

confused.
 
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  • #2
James Brady said:
Taking the first one, we can manipulate the square root algebraically ##z_0\sqrt{1 + (x-s/2)^2/z_0^2}##, this is where I run into issues, according to the text we can use the fact that ##\sqrt{1+a} = 1 + a/2##
That can be obtained from the binomial theorem, which is a special case of Taylor series.
James Brady said:
The book goes on to use ##2cos(\theta) = exp(i \theta) + exp(-i \theta)## to obtain a final wave equation (at the screen) of ##exp(i \theta) cos(ksx/2z_0)## where k is the wave number ##\lambda = 2 \pi/k##

How did they come up with that? I don't see the algebra at all.
Do you know Eulers identity for the complex exponential?
 
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  • #3
When I apply the Taylor series about x = 0:

##\frac{f(0)}{0!}0^0:####z_0 \sqrt{1 + (0-s/2)^2/z_0^2} = z_0 \sqrt{1 + (-s/2)^2/z_0^2}##
##\frac{f^{(1)}(0)}{1!}0^1:##
##\frac{0-s/2}{z_0 \sqrt{1 + (0-s/2)^2/z_0^2}}## = ##\frac{-s/2}{z_0 \sqrt{1 + (-s/2)^2/z_0^2}}##
Adding these values gives a different answer than the book.

I'm not exactly sure what a fractional binomial theorem looks look, I understand the whole number BT, but for square roots and what not I can't find a formula online.

...

I know Euler's identity ##exp(i \pi) + 1 = 0## I just don't see how that applies here.
 
  • #4
You need to use the Taylor series around ##x = \frac s 2##.

You can make the calculation easier by the substitution ##a = \frac{(x -\frac s 2)^2}{z^2}##, as the book indicates. Then use the Taylor series about ##a = 0##.

Try Eulers formula!
 
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1. What is the Double Slit Experiment?

The Double Slit Experiment is a classic experiment in quantum mechanics that demonstrates the wave-particle duality of light and matter. It involves shining a beam of particles, such as electrons or photons, through two slits and observing the resulting interference pattern on a screen.

2. How does the Double Slit Experiment relate to mathematics?

The Double Slit Experiment is closely related to mathematics because it involves the use of mathematical equations, such as the Schrödinger equation, to describe and predict the behavior of particles. The interference pattern observed in the experiment can also be mathematically modeled using the principle of superposition.

3. What is the mathematical formula for the interference pattern in the Double Slit Experiment?

The mathematical formula for the interference pattern in the Double Slit Experiment is known as the Young's double slit equation. It is given by I = I0cos2(θ/2), where I is the intensity of the light at a certain point on the screen, I0 is the maximum intensity, and θ is the angle between the incident beam and the screen.

4. How does the Double Slit Experiment challenge our understanding of classical physics?

The Double Slit Experiment challenges our understanding of classical physics because it demonstrates the wave-like behavior of particles, which goes against the classical view of particles as solid, discrete objects. It also shows that the act of observation can affect the behavior of particles, which is not accounted for in classical physics.

5. What are some real-world applications of the Double Slit Experiment and its mathematical principles?

The principles of the Double Slit Experiment have been applied in various fields, such as optics, electronics, and quantum computing. The interference patterns observed in the experiment are also used in diffraction gratings, which are used in devices such as spectrometers and CD/DVD players. Additionally, the mathematical principles of the experiment are crucial in understanding and developing quantum technologies.

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