Schrodingers Cat Thought Experiment

If you try to detect it taking one path, you can, you just have a 50% chance of the photon using the slit you are monitoring.

 

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Can you back this up with some evidence?

 

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It worse than that. If you detect the path of the photon AFTER it's gone through the slit, it acts like it just took a single path and doesn't interfere.


http://en.wikipedia.org/wiki/Wheeler's_delayed_choice_experiment

 

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Some pointers from the book "The Emperor's New Mind" by Roger Penrose.

1) Light is detected in discrete localised units of energy \( E \; = \; \hbar \nu \), which means "half-photons", or any fraction, are never detected.

2) Because the wavefunction has a probability amplitude, if it can take two paths it will.

3) For interference to be seen as a pattern of light and dark bands (on a detector screen) nothing can be known about any path information. Using a detector at one of the slits destroys the pattern.

4) Quantum phenomena are often thought of as being those confined to the realm of the "very small", but quantum mechanics has nothing to do with the "size" of particles, it has to do with path lengths and interactions over macroscopic and microscopic scales, respectively.

5) For interference to occur (at macroscopic scales), alternative paths must have only amplitudes, not probabilities.

6) We can't assume that a photon has a 50% probability of being in one place, and a 50% probability of being in another. Every possible position that a photon might have is an alternative, and if it's possible it has a probability which is nonzero.

 

What does that mean?
That at microscopic scales they have the opposite, or what?

 

I was talking to a guy called Good Elf about this kind of stuff a couple of weeks back. Maybe a good way to think about it is to say that detecting a photon at a slit or at the screen "forces a fourier transform" upon it. Imagine the photon as a spatially extended wave, a little simpler than the input image, which is detected as a single dot either at one of the slits or on the screen:

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http://sharp.bu.edu/~slehar/fourier/fourier.html

 

Farsight I think there is something here, but I believe you have it reversed. The photon source would be the DC term on the right, while the slits are represented by the frequency terms..!

 

Farsight, did you even read your own source? That's what the lense does, in a setup that has little to do with double-slit interference.

 

Not sure, RJ. When I think of a photon going through both slits at once I visualize this sort of thing going on:

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Caption: 3D plot of a single photon showing wave-like behaviour
See http://physicsworld.com/cws/article/news/46193.

Detection is doubtless some photon-electron interaction, essentially the photoelectric effect or Compton scattering. You can make an electron from a photon in pair production (forget the positron), then you can do the double-slit experiment with electrons, see for example this web page. So you've got one extended entity interacting with another over a extended region. But the electron isn't just some amorphous blob, it's got spin angular momentum and magnetic dipole moment, which means it's got some kind of centre like a hurricane has a centre. I'd say the dot on the screen picks this out.

For a photon source we've got something like an inverse Compton going on, where everything happens in reverse. So we could be talking about an Inverse Fourier Transform, but I can't see how a wavefunction can "uncollapse" during emission.

 

Przyk, please don't be so quick to criticize. There's something fundamental here that may be well established to Physicists and Mathematicians but it's completely new to me. I suspect someone will trivialize this by pointing out that the math behind the calculations in a real-world experiment is exactly equivalent to a Fourier transform but this could have value to me in trying to "grok" it.

 

Farsight, my response would be that there should not be a separation between emission and absorption in a time-symmetric world; it's almost like the loss of information (i.e. concerning the photon's path) is the focusing lens. Therefore, we could both be right; the transform is applied to the source and, if pathing information is obtained, reversed again.

 

No, interference is a macroscopic effect. It depends on photons or other kinds of quanta having sufficiently large path lengths that the microscopic properties are additive, but that's a sort of rule of thumb rather than anything definitive. I guess what I should say is that "sufficient path lengths" is a requirement for us to observe macroscopically, because as Heisenberg showed we can't observe microscopically--the whole concept of observation breaks down.

Understanding the microscopic "interaction space" if you will, really does require an understanding of somewhat advanced mathematics. You also need to come to grips with particles whose position is only 'localised' when measurements are done, and which have momentum, which is out of phase with position up until measurement occurs. You have to kind of abandon your usual idea of what time is too, time is in some sense, meaningless--but becomes meaningful when we measure the states.

Do you understand complex numbers, and does the phrase "probability amplitude" make you nervous?

 

I don't understand anything not covered in Dr Q.
And I only just understand that.

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Isn't the main thing that it is a wave that can collapse to a particle, and that the place where it becomes that particle is based on probability?
Or have I got that wrong?

 

Sounds good to me, RJ. Like, the electron wavefunction that interacts with the photon wavefunction is acting like a lens.

 

Sure I did, przyk. We were talking about phase space and how a fourier transform take you from position space to momentum space. This kind of stuff: UCSD Quantum Physics 130 by Jim Branson. By the way, have you ever heard of Steven Low?

 

You obviously ignored my last post. If you think a photon is literally as long as its wavelength then that means it takes \(\frac{\lambda}{c}\) seconds for an electron to emit the photon. Why don't we see such protracted emission back reactions from electrons? Why don't radio wave emissions take the microseconds your notion would imply?

Wavefunction collapse is not reversable. It's one of the fundamental things which distinguishes it from other processes/evolution mechanisms in quantum mechanics. A time evolution operator in QM is unitary, it can be reversed. A wavefunction collapse is not unitary. This is particularly important when considering entanglement because unitary operators cannot change the level of entanglement but observations, which cause wavefunction collapse, do. The Fourier transform is invertible and thus you can't associate it to the act of observing/wavefunction collapse.

As for your picture, when are you going to start producing your own pictures? Isn't 5 years enough time for you to get to the point where you can produce your own results, rather than having to use other people's pictures and argue by analogy? Or are you still unable to produce any quantitative results/predictions? Must be upsetting, being less applicable than string theory

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The state the photon is in isn't a wave, as such, at least it isn't a wave the way we usually think about waves being "in" some medium.

There is no medium as far as photons, they are the ultimate 'free' particle. But, well, really a particle is just a model. So is a vector space.

You could start with the naive approach, and say that each dot on the screen is like a number, after all, the dots you see are all distributed spatially. You could assign a number to them, something that is like a description of its position, in terms of an angle between the double slit and the screen, say.

I mean, why not assume the pattern (lets make it for a smallish number of photons or other particles) is like a spatial 'derivate' of whatever gave it momentum--usually this is called energy--and try describing it somehow? This is pretty much what wave mechanics can do, for a classical interference pattern, but you're looking for a discrete form.

The other point is that QM isn't "based on" probability. Quantum probability is roughly the expectation of seeing a particle in one place rather than another, and it's fundamentally a different class of animal than throwing dice or tossing a coin or two, because of superposition of states.

 

AlphaNumeric, you say that like it's settled science yet the DCQE suggests otherwise. Once the pathing information is gone the wave-like behavior is again expressed. If you're speaking in terms of evolution mechanisms then you are left with the explanation that the wavefunction has reversed itself after the photon chose a distinct path.

Relax your mind for a second, doesn't it at least strike you as interesting that the effects of a Fourier transform appear to have a optical analog? And if we consider the inverse Fourier transform we arrive at something extremely close to a dual-slit experiment?

If we consider the transformed image to be the source/slit system upon which we are going to apply an inverse Fourier transform, the "spatial manipulations of the transformed image" would be represented by moving the frequency terms (aka the width of the slits), and of course the photon source itself representing the DC term (both which determine intensity) should be obvious.

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The "lens" in this analogy is as I suggested: a change in knowledge of the pathing information. An inverse Fourier transform is applied when the source is arranged in such a way as to remove pathing info from our knowledge; wave-like behavior remains unless we obtain such knowledge again, at which point the wave-like colimated light is subjected to a proper Fourier transform to arrive at a single photon location.

Are you so vested in your blood-feud with Farsight that you've lost all sense of wonder?

 

Yes. You're just one long diatribe. There's just no talking to you.


RJB, this is an interesting paper: How Long Is a Photon? by Drozdov and Stahlhofen. Here's an excerpt:

"It appears more appropriate to consider a single photon like a real ultra-short pulse; the frequency is then understood as an inverse of the emission time r instead of an ambiguous wave length. It is characterized by a well defined spatial localization, related to them as λ ≈ c r. The term ”frequency” could be entirely interpreted as a parameter of the leading component of the Fourier decomposition of the pulse shape, meaning that the frequency makes sense in a purely technical, but not a physical context".

IMHO it's important to remember that \(A^{\alpha}\) is more fundamental than E or B, and thus that the typical sinusoidal electromagnetic waveform is the derivative of an \(A^{\alpha}\) pulse.

 

That assumes the collapse is done by the equipment and then reversed. If the collapse does not occur until the particle hits the screen then no 'uncollapse' occurs. The relevant measurement of the photon is not done at the slit but at the screen, as we're interested in the pattern formed at the screen.

The collapse is removal of information. For example, a general superposition might be of the form \(|\psi\rangle = \sum_{n}c_{n}|\psi_{n}\rangle\) and a collapse has the effect of removing all but one state (this is when you work in a basis of eigenstates of the relevant observable), ie \(|\psi\rangle \to |\psi_{m}\rangle\). The information pertaining to the \(c_{n}\) coefficients is lost. If the wavefunction could uncollapse that information would have to be reconstructed, meaning it was stored somewhere. This then gets into the realm of hidden variables.

The collapse is a non-reversible process.

The double slit result, in the absence of doing a slit observation, is the combination, the weighted complex sum, of two waves from the same source, leading to constructive and destructive wave interference. The Fourier transform is a weighted sum where the weights are complex trig functions, so it's not particularly surprising they look very similar to one another.

In fact in quantum mechanics momentum relates to wavelength and frequency and to convert to the 'momentum space' representation you do a Fourier transform. To convert back into 'position space' you do an inverse Fourier transform.

It's possible to make the case that quantum mechanics is basically the physical application of the mathematics pertaining to Fourier transforms. You can use them to study the Schrodinger equation as the space of solutions to the Schrodinger equation is a rigged Hilbert space, they must be Fourier transformable when you use a particular Gaussian limiting process. Thus finding a Fourier transform, particularly qualitatively, arising in something pertaining to wave behaviour in quantum mechanics is like finding puddles of water at a swimming pool.

The Fourier transform and its inverse do not remove information, only reshuffle it. If a transformation has an inverse than it cannot remove information almost by tautology. It means that the structure before and the structure after are equivalent, only written in a different way. This is why the momentum and position space formulations in quantum mechanics are used, they are entirely mathematically equivalent but in certain cases one makes a result clearer than the other. For example derivatives are transformed into multiplications by the Fourier transform, ie \(\frac{d}{dx}\psi(x) \to k \tilde{\psi}(k)\) where \(\mathcal{F}(\psi(x))(k) = \tilde{\psi}(k)\) is the Fourier transform. Hence it's easy to solve the free Schrodinger equation or wave equations in momentum space. A similar principle is used by engineers in regards to the Laplace transform, as it does the same (it's a restricted special case of the general Fourier transform).

No, I just happen to be familiar with the role Fourier transforms play in quantum mechanics, quantum field theory and Poisson/Symplectic mechanics in general. It's one thing to point out these things to an interested layperson as hints of something deeper but mathematically elaborate but Farsight argues entirely by analogy and pictures. He doesn't know how to do actual Fourier transforms or Fourier decompositions, in their application to classical mechanics, quantum mechanics or any other part of physics. Instead he takes the hard work someone else has done and uses it as an excuse to those buzzwords into his "I've explained [something], where's my Nobel Prize!?" pieces of 'work'.

I'd like to see him do more than just talk, I'd like to see him demonstrate a working understanding of the things of quotes and the articles he lifts pictures from. Most of all I'd like to see him produce any quantitative prediction but he knows he won't ever do that hence why he ignores me when I ask him to provide it.