Uncertainty Principle and Single-Slit Diffraction

In summary, the conversation covers the topic of calculating the uncertainty in momentum and direction resulting from the certainty in position caused by a small slit, as a function of the slit width. The Heisenberg Uncertainty Principle is used as a starting point, with the slit width and angle of the photon being variables. The calculation involves solving the Schrödinger equation or using Huygen's principle. It is noted that the magnitude of momentum is always given by h/lambda, but the direction can vary due to Heisenberg Uncertainty. The concept of standard deviation is also introduced. The conversation concludes with a discussion about the applicability of Huygen's principle at different slit widths.
  • #1
peter0302
876
3
I'm trying to calculate the uncertainty in momentum/direction resulting purely from the certainty in position caused by a small slit, as a function of the slit width. Can anyone tell me if I'm doing this right?

Starting with the HUP:
dX * dP >= h/4pi

dX = the slit width w

P = h/lambda * sin(theta)
where theta is the angle that the photon might be traveling after the slit relative to the center of the slit

so

dP = 2 * h/lambda * sin(theta)
A factor of 2 because the uncertainty can extend in either direction from the center of the slit

so

w * 2 * h/lambda * sin(theta) >= h / 4pi
sin(theta) >= lambda / (8pi * w)

Is that right?

My only concern with the calculation is that "w" could be so small that sin(theta) could be greater than 1, which is of course impossible, so that's why I think I made an error somewhere.

Thanks for any ideas!
 
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  • #2
I guess the resolution to the problem is the assumption that
[tex]|p| = \frac{\hbar}{\lambda}[/tex]
after the diffraction after the diffraction. In fact, for the particle to pass through the narrow slit, it's momentum is not a single value, but given by a probability distribution. The width of that distribution is what you'd call [tex]\Delta p_x[/tex]. If the slit is made very narrow, [tex]\Delta p[/tex] can grow arbitrarily large, and so [tex]\langle \sqrt{|p|^2}\rangle[/tex] can grow arbitrarily large.
 
  • #3
Ok .. so how do I calculate the probability distribution after going through the slit?
 
  • #4
Either solve the Schrödinger equation or use Huygen's principle. Incidentally, in the far field region, the spatial wave-function is well approximated by the spatial Fourier transform of the aperture. The uncertainty principle applies to Fourier transforms also.
 
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  • #5
Show me how to solve the Schrodinger equation for a photon.

Also I thought Huygen's principle doesn't work when the slit is smaller than a wavelength. When a slit is larger than a wavelength, you get single-slit diffraction which is not what I want.

In fact, for the particle to pass through the narrow slit, it's momentum is not a single value, but given by a probability distribution.
The magnitude of the momentum is always given by h/lambda. It's only the vector components, i.e., direction, that change due to Heisenberg Uncertainty. Otherwise light would change color after going through a slit and all lasers would be white.
 
  • #6
Ah, can I just use the Fraunhofer diffraction formula:

[tex]\psi[/tex]([tex]\theta[/tex])=sinc([tex]\pi[/tex]a[tex]\theta[/tex]/[tex]\lambda[/tex])

?
 
  • #7
peter0302 said:
Show me how to solve the Schrodinger equation for a photon.

Also I thought Huygen's principle doesn't work when the slit is smaller than a wavelength. When a slit is larger than a wavelength, you get single-slit diffraction which is not what I want.


The magnitude of the momentum is always given by h/lambda. It's only the vector components, i.e., direction, that change due to Heisenberg Uncertainty. Otherwise light would change color after going through a slit and all lasers would be white.

Momentum is always given by h/lambda, but lambda is sometimes not well defined. Far away from the slit waves will be nearly straight, so the momentum distribution will be delta(p-p0). But this is not true close to the slit. The momentum operator is defined by -i*2*pi*h* grad. As soon as you have more than one point source of waves, the sum of gradients can have different magnitude on diferent points.

The uncertainity principle alone can prove that the magnitude of p has a distibution with non-zero width when the uncertainity in position aproaches zero (since uncertainity in momentum aproaches infinity and becomes greater than the average momentum).
Actually your calculation was quite usefull for a slit with w>>lambda. I think you could also omit the number 2 in dp, since the uncertainity is defined by average distance from the center of distribution.
 
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  • #8
Ok, so I just learned for the first time that in the familiar uncertainty relation
dX dP >= h_bar / 2

dX and dP are not literally the range of values X and P can have, but are the standard deviation in those values. Glad I know that now.

So the Fraunhoffer formula should provide the right wavefunction for the photon going through a single slit.
 
  • #9
peter0302 said:
Show me how to solve the Schrodinger equation for a photon.
Cute.
 
  • #10
Huygen's principle works just fine at all wave-lengths... it just needs to be applied properly.
 
  • #11
sin(theta) becomes greater than 1 when the slit width is less than the wavelength. So it simply doesn't work. Fortunately, we can make the slit width larger than a wavelength and still get uncertainty effects.
 

1. What is the Uncertainty Principle?

The Uncertainty Principle, also known as Heisenberg's Uncertainty Principle, states that it is impossible to know both the exact position and momentum of a particle at the same time. This is because the act of measuring one quantity will inevitably disturb the other, resulting in uncertainty.

2. How does Single-Slit Diffraction occur?

Single-Slit Diffraction is a phenomenon that occurs when a wave, such as light or sound, passes through a narrow opening or slit. The wave spreads out and creates a pattern of interference, resulting in a diffraction pattern.

3. What is the relationship between the Uncertainty Principle and Single-Slit Diffraction?

The Uncertainty Principle and Single-Slit Diffraction are closely related because they both deal with the behavior of waves and particles. The Uncertainty Principle explains the limitations of measuring the position and momentum of a particle, while Single-Slit Diffraction demonstrates the wave-like nature of particles and their tendency to spread out when passing through a narrow opening.

4. How does the size of the slit affect the diffraction pattern?

The size of the slit directly affects the diffraction pattern. The narrower the slit, the wider the diffraction pattern will be, and vice versa. This is because a narrower slit will cause more diffraction or spreading of the wave, resulting in a wider pattern.

5. Can the Uncertainty Principle be applied to macroscopic objects?

The Uncertainty Principle is typically only observed at the microscopic level, as quantum effects become negligible at larger scales. However, some scientists argue that the Uncertainty Principle can still be applied to macroscopic objects, but the effects are too small to be detected by current technology.

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