Would it be possible for an electron to tunnel back and forwards between the slits as it travels though them?

Would it be possible for an electron to tunnel back and forwards between the slits as it travels though them?
No. Quantum tunnelling between two regions occurs because the potential barrier between them is finite. In the doubleslit experiment, the potential barrier from one slit to the other through the material forming the slits (i.e. disregarding paths around the material forming the slits) is effectively infinite.
Thanks for that, no tunnelling. It maybe makes the transition to one of the empty states coming out of the other slit after it leaves the slit then...?
I don't see how the transverse momentum can become quantised and produce the interference pattern unless there are transitions between the states separated by the distance between the slits. How can probabilities which are not real interfere in reality? The maths is the same, but you would be forced to conclude we live in a computer simulation which I am not happy with.
The particle going through both slits can be regarded as a superposition of the states corresponding to the particle going through either one of the slits. By the time the total state has reached the display screen, the two component states are not orthogonal and therefore interfere. But any attempt to measure which slit the particle passed through orthogonalises the two component states and they no longer interfere. Even if one orthogonalises the states by rotating the polarisation plane of one of the states, there will be no interference even if no attempt is made to actually measure which slit the particle passed through.
Sorry KJW, I'm not really getting this either. Are you saying that states interfere or their probability amplitudes interfere?
The latter. However, a quantum state can be regarded as a vector in an abstract space known as a Hilbert space. The two components to which I refer add vectorially to produce a single vector. If these two component vectors are not orthogonal, then the vector sum corresponds to interference, whereas there is no interference if the two vectors are orthogonal. This is a consequence of Pythagoras' theorem in the Hilbert space.
When a quantum state is measured, the state vector is decomposed into a set of orthogonal component vectors, and the result of the measurement will be one of the component vectors, chosen randomly from the perspective of the observer. This is why measuring which slit the particle passed through destroys the interference pattern: because that measurement makes the component vectors corresponding to each slit orthogonal, and orthogonal vectors don't interfere.
The orthogonality of the component vectors produced by the measurement process is also why only one of the component vectors is observed: because being orthogonal to the observed component, the unobserved components cannot make their existence manifest.
Thank you. Now I think I understand that and the inner product will be zero if they are orthogonal. Hilbert space must have many dimensions indeed (infinite?) for the states to be all orthogonal to each other. It's a useful mathematical construct. But how does that relate to the physical observes such the quantisation of the transverse momentum? Spherical confinement leads to quantisation of angular momentum. Would transverse confinement not create the quantisation of transverse momentum in the same way?
That's a lovely way of putting it. Is that why spin is always found to be along the axis you're measuring it?
I'm afraid that we have to write out the mathematics to get this anywhere near right. The mathematical language is a great simplification once you have seen the manipulations once, but they can be very confusing before that. What we have here is a state that has two parts: one spreading outward from the left slit and one spreading outward from the right. Hence
= _{L}+_{R}
In other words, the probability amplitude for finding the particle is given by the sum of the part coming from the left slit and the part coming from the right slit. This is the state of the system in the space beyond the slits. The probability is then the square of this wave function,
^{*} = (_{L} + _{R})^{*} (_{L} + _{R})
= _{L} ^{*} _{L} + _{R} ^{*} _{R} + _{L} ^{*} _{R} +_{R}^{ *} _{L}
The last two of these give the interference pattern. The left and rightslit parts of these terms are not changed; they are just multiplied by each other and therefore depend on the particle having gone in some sense through both slits. Since these parts of the wavefunction have not changed in any way beyond the slits, there is no need for any kind of tunneling between the two parts of the state. What is actually happening is that a part of the wavefunction has been removed by being blocked by the material between the slits, while _{L} and _{R} have not changed.
It is perhaps worth noting that the two parts of psi are quite complicated and do not have one definite value of momentum parallel to the direction from one slit to the other.
Hilbert spaces often are infinitedimensional. Each possible outcome of a measurement moreorless corresponds to a dimension of the corresponding Hilbert space. And if there are multiple particles, then the dimensionality of the multiparticle Hilbert space is the sum of the corresponding singleparticle Hilbert spaces. It is worth noting that the higher the dimensionality of a space, the more likely two arbitrarily chosen vectors in that space will be close to orthogonal, and this provides an explanation of why classical states do not interfere and therefore why only one classical state is observed.
Mvb, I appreciate the effort you are making to lead me onto the right track! I can see how mathematically a filtering of the position states would lead to a change on the momentum space distribution (Fourier transform.) I can also see that only the last terms would cause the interference. You make a good point that psi is not straightforward as it leaves the slits. This is true and also the inference pattern is somewhat blurred. It maybe that Psi is a good mathematical shorthand for an alternative interpretation being transitions between states. The amplitude being proportional to the inner product. I lean towards this interpretation perhaps because of my chemistry background apologies!
The usual situation in chemistry is actually quite different than the situation in the doubleslit experiment. Both the part of the state spreading out from the left slit and that spreading out from the right slit are stable states by themselves; they do not change into something else with time. There is no way to raise the size of the rightslit part by transferring something from the leftslit part or viceversa. On the other hand, if an atom can be in a ground state or an excited state, the excited state can normally transform itself into the ground state in one way or another, such as giving off a photon if its energy is high enough. Then afterwards the two resulting parts of the wavefunction could in principle interfere with each other to give really weird results. The state does consist of two parts, but one of them is continually getting smaller and the other bigger as the decay becomes more and more likely. [Interference might then mean, among other possibilities, a nonexponential time dependence of the decay process]. Here there is a transformation between the two parts of the state (the two psi's) before the interference; in the doubleslit experiment there are two parts of the wavefunction but they are stable in size.
In quantum mechanics, the difference between the two situations shows up in the character of the timedependence of the two parts. The calculation you do with the spatial parts of the wavefunction is the same in the two cases, but handling the time dependence of the functions is more complicated in the case of decay than it is in the doubleslit interference experiment.
If the electron were moving back and forth between the two parts of psi, there would be an alternating electric current which would generate an alternating magnetic field. No such field is detected, so the electron must be in two places at once. That is, the physics is in the statement that there is a probability of its being in either place at any given time.
Another point is that if the electron were always in one place or the other, the _{L}^{*} _{R} term in my earlier post would be zero at any one given time, and no interference pattern would show up. And in fact, if you determine experimentally which side the electron is on, the interference goes away. The world is set up with probability amplitudes, not definite locations. In fact, the world is set up with amplitudes, not with definite probabilities. It is that is determined by the equations, not the probability ^{*} .
Needless to say, it is not easy to get reconciled with this situation. However, we have no choice in the matter.
I can see that it were actually moving between the two positions though space it would generate alternating fields. I don't think quantum transitions involve classical motion! (But perhaps classical motion is comprised of a series of quantum transitions?) If the location is not definite until it's measured would it be possible for two different observers agree on it? The interference pattern is the easiest part to explain. The amplitude for the transition would be zero unless p x delta x = nh where p is the momentum perpendicular to the main direction of travel. The interference pattern just arises due to the quantisation of the transverse momentum.
You can keep setting up rationalizations and I can keep saying that isn't what happens, but there is no way out. The wave function is the physicallydetermined quantity, and what we see is governed by the probabilities determined by the wave function. There is no underlying classicaltype motion. Granted, even Schrodinger didn't like that [see Schrodinger's cat] but that is the way the world works.
I guess it come down to the interpretation in the end. A wavefunction has no solution and a particle no physical property without boundry conditions (interactions or measurements). If that is all there is then we really are living in a computer simulation. An idea I'm not comfortable with but am going to reconcile myself to.
No. A wavefunction is a definite property, just not the property we seem most interested in (position). Consider an audio waveform. It is not possible to simultaneously identify both a single time and single frequency for that waveform. This is because the mathematical relationship between time and frequency disallows these two observables from having simultaneously definite values. The problem isn't with the electron, it is with the observables we are trying to attribute to the electron.
I like this analogy as a spent a summer when I was younger performing fast Fourier transforms on music. Frequency and time cannot be determined at once. However I believe the analogy is not the same as position can be determined at a definite time. Please correct me if I am wrong in this assumption.
OK! Here's something fun to think about.... Imagine I was an electron headed for the double slits. How would it experience it? Would the slits appear to jump around keeping the same distance apart or would they jump about randomly and sometimes cross over so if I timed it just right I could pass though both at the same time?
You appear to be missing the point. A wavefunction isn't just a representation of probabilities of definite states. A wavefunction is a definite state in its own right. It is just not a definite position state. But position is not the only domain in which reality can be described. An audio waveform which is a description in the time domain can be Fouriertransformed to the frequency domain, and the frequency domain description of the waveform is every bit as valid a description of the waveform as the time domain description. Only if the position is measured will the electron have a definite position. But that is also true for any observable that we measure. The measurement of any observable produces a definite value for that observable. It is the mathematical relationship between the observables themselves that is why the measurement of one observable for a particle that is in a definite state of another observable may lead to a statistical outcome. What it means is that an electron cannot be viewed as a classical particle. The electron is its wavefunction.
The conjugate variable of spatial position is momentum. The conjugate variable of time is energy. Momentum is proportional to wavenumber. Energy is proportional to frequency. Thus, you are mixing up space and time. But otherwise, the Heisenberg uncertainty principle is exactly for the same reason that an audio waveform cannot have both a definite time and definite frequency.
Are you saying that position cannot be established at a definite time then? I would have thought you could have uncertainty in momentum without having uncertainty of energy (take an electron in a atom for example) so I don't see where the energy v time uncertainty comes into it.
No, I'm not saying that. I'm saying that a particle with definite momentum (a definite wavenumber) cannot have a definite position. And conversely, a particle with a definite position cannot have a definite momentum (or wavenumber). Energy and time only came into it because the audio waveform analogy is a timefrequency domain example. Note that wavenumber is the spatial form of frequency, so the audio waveform analogy fits even though it is in the timefrequency domain and for the electron we are considering the positionwavenumber domain.
This question has classical overtones because asking where is the energy is moreorless the same as asking where is the particle.
One further point that should be noted is that when a superposition of states is measured and a single state is selected as the outcome, all the other states of the superposition that were not selected disappear from having ever existed as far as the observer is concerned. That is, the state behaves as if it is and always was the measured result and no other result. Thus, the measurement of one particle of a quantum entangled pair of particles effects the measurement of its entangled partner even though that particle was not physically measured. But if a superposition of states is never measured, then all the states of the superposition exist in a way that is distinguishable from each particle being only one of the components of the superposition. Thus, the interference pattern distinguishes particles that have each passed through both slits from each particle passing through one or the other slit, for which there is no interference pattern.
Not exactly. For any individual measured particle, there is no way of knowing that it didn't have that value to begin with. It is only by considering the unmeasured particles that we infer that the measured particle was in a superposition of states prior to measurement. The manyworlds interpretation provides a neat explanation: the superposition of particle states interacts with the measurement device to produce a superposition of measurement device states, and when the observer observes the superposition of measurement device states, is placed in a superposition of observer states. Each observer state only sees one measurement device state, which has responded to only one particle state. Because the components of the superposition of states are orthogonal, and because quantum mechanics is linear, each component of the superposition of states behaves as if the other components don't exist. But when the superposition of particle states is not measured, the observer observes the superposition of states.
If the Jilans observe a quantum experiment for which the possible outcomes have different probabilities, then the proportion of Jilans who observe each outcome will differ. It is worth noting that the manyworlds interpretation removes the indeterminism from the objective reality (the multiverse), and that the statistical nature of the outcomes is about which versions of the observer observe which outcomes. That is, the probability of a given outcome is the probability of being an observer who observes that outcome.
Could you interpret these universes to form a continuum along a secondary time axis that runs orthogonal to the one we are used to? I could see then how I couldn't interact with the other Jilans. It would be just the same way as I cannot interact with the me that existed 5 seconds ago.
Do you say that because the universes need to be orthogonal in a dimensional sense? Could it be possible that they would need to be orthogonal in a QM sense only (so could be phase shifted just slightly) so that <u1u2> would be zero? If time were a vector having phase as well as modulus an infinite number of universes could then sit quite happily round the circle being oblivious to each other.
I personally don't like the multiverse interpretation at all, but for a different reason. Once another universe has split off, nothing that happens there can possibly affect our world. Nothing there needs to be known to predict what has taken place in our universe, because the other one didn't exist separately before the split. So all those multiverses are irrelevant to our ability to tell what is going on in our universe. There is no need for them, and traditionally things that aren't necessary in that sense are discarded from science and from philosophies of science.
All that the multiverses do is to provide a way for us to view quantummechanical probabilities as frequencies. We can do that visualization without believing that the other universes really exist.
Basically, what I am saying (philosophically) is that the universe is one gigantic wavefunction which allows you to calculate all that you can know about what has and will happen. This is pretty close to, if not a copy of, what is called the "shutup and calculate" interpretation of QM.
I read about the splitting thing and it doesn't sound right me me either. Starting from a initial event like the emission of a particle you would need an infinite number of universes to be created at once, like a a radial expansion in time, with the universes all slightly phase shifted around the circle in order to cope with probabilities that differed from 50:50. A quantum particle might exist in a slightly different place in each universe, thereby being in two places at the same (same radial displacement from the centre) a measurement with a nonquantum device would just determine which axis or phase angle you are at or which universe you are in. You would either detect the particle or you wouldn't. I guess it might be like when you measure spin it defines the axis in space, so measuring position could determine the axis in time?
Where would be the fun in shutting up and calculating?
Yes, a particle would certainly be in different places in each universe, but only one of those places count. All the others are gone forever, because you can never go back to those universes.
The jocular name "Shut up and calculate" is meant, I am sure, to refer to the expectation that the physics itself will have plenty of interesting things to investigate, so there isn't time to worry about the nature of what might have been, especially since you can't observe that which might have been.
Yes I can see that would be the case after the position is measured and the axis is determined, a new t=0 would be established as entropy will have increased. But before it is could they possibly interfere if the wavefunctions overlapped sufficiently around the circle allowing transitions between the universes? This might explain how some things appear to happen instantaneously i.e. Not through radial time as we know it. Also would it be possible to observe something in two states at once if you had a special type quantum measuring device that did not set the preferred axis?
No, not in the usual manyworlds interpretation. The other worlds don't exist until the measurement is made and are inaccessible afterwards. So in fact they have no effect on our universe at all.
Edit: What I am arguing is that "manyworlds" cannot survive the application of Okham's razor.
Last edited by mvb; 10302013 at 08:54 PM. Reason: Additional thought
Maybe I'm looking at this the wrong way round. From the particle's point of view the world might appear to be jiggling about and the particle might conclude that there were many universes slightly different to each other. It might also conclude that it was travelling between these universes pretty quickly, but when measurement is made it fixes which one it is in. In that case the many worlds exist before the measurement and actually become only one following the measurement.
In that case, what you have is essentially the Copenhagen interpretation, which is fine if an explanation of quantum mechanics is unimportant. Personally, I see the Copenhagen interpretation as a mere description of quantum mechanics from the perspective of the experimenter, and not an explanation. It does not explain how the particular outcome is selected. In particular, it does not explain how the outcome is randomly selected. The manyworlds interpretation does explain the randomness by it being from the viewpoint of the observer rather than a physical process. The absence of wavefunction collapse as a physical process has the significant consequence that the macroscopic realm is no longer qualitatively different to the microscopic realm, and this is perhaps the most important lesson to be learned from the manyworlds interpretation. The application of Occam's razor to the unobserved alternative worlds would be valid if the many worlds were simply a statistical ensemble, but the existence of observed effects of quantum superposition requires the many worlds to exist for the macroscopic realm to be essentially the same as the microscopic realm.
The Copenhagen interpretation suggests that a statistical selection process occurs at the time of measurement. The problem with this is the failure to explain the various experiments pertaining to quantum entanglement and Bell's theorem. By contrast, because the statistical nature of the manyworlds interpretation is purely from the perspective of the experimenter and not a consequence of the physical measurement process, counterfactual indefiniteness becomes a natural consequence and explains the various experiments pertaining to quantum entanglement and Bell's theorem without invoking "spooky action at a distance".
Finally, although the alternative worlds are unobservable, their assumed existence may provide a mathematical foundation for the fundamental laws of physics.
I don't think that either interpretation is really needed to get all the experimental results. There is a very instructive calculation in one of the early editions of Schiff's Quantum Mechanics. One way to "localize" an electron is for it to have enough energy to ionize an atom. The atom's location is much better known than that of the electron, because of the effect of its larger mass on x p >= hbar. Therefore the ionization of an atom determines rather well the location of the electron. In the Copenhagen interpretation, the electron's position has now been set and if the electron ionizes a second atom, the direction from the first atom to the second must be along the electron's momentum. Schiff showed that if you simply used normal secondorder perturbation theory to calculate the probability of two different atoms being ionized by the electron, not using any localization process after the first ionization, the probability you got was negligible unless the direction between the atoms was indeed along the electron's momentum. So the wavefunction alone carries the information, and you have no need of localizing the electron in the first process.
While I haven't done the calculation, I am convinced by looking at the simpler cases that the calculation of processes involving quantum entanglement using conventional wavefunction methods alone would work fine. If so, the multiple universes aren't adding any explanatory power to the calculation of the effects of entanglement. This makes sense, since no one would be surprised if the calculation of the correlation coefficient between all the results of the two measurements in entanglement situations came from the wavefunction alone. The multiverse interpretation makes the situation easier to follow, but the Schrodinger equation gives the answers.
And Schiffs calculations borne out by experiments?
I would imagine this would be the case as an atom would be a good quantumlike measuring device. I expect it's only the larger devices that freeze the axis (see post 52).
Last edited by Jilan; 10312013 at 09:04 PM. Reason: A bit slow tonight!
The part about the atoms having to be lined up properly certainly is. I expect the details are, but I haven't really looked for data.
Apologies if I am not reading this correctly but are you saying that the alternative realities are observable or not? Don't we now have experiments that can actually observe two states at once? I agree about the macroscopic realm being equivalent to the microscopic realm due to the principle of relativity (see post #55).
However, the main difference between them may be to do with reversibility, increase in entropy and the arrow of time. The multiple universes perhaps only exist temporarily until a non reversible process occurs in one of them, then the arrow of time is reestablished and they join back up.
I think part of the problem here is the term "alternative realities". I think the term is incorrect strictly speaking and my use of the term has been somewhat of an oversimplification. More correctly, we have an infinitedimensional space and orthogonal directions in that space behave independently without any interaction. When we measure some observable, the different possibilities are orthogonal to each other, so only one possibility is observed. But if the directions in the infinitedimensional space are not orthogonal, then we can observe interference between them. This is the case of the interference pattern of the doubleslit experiment, where if the particular slit the particle passed through is not measured, the components corresponding to the particle passing through each slit are not orthogonal. But if the particular slit the particle passed through is measured, then these components become orthogonal and there is no interference pattern, each particle being observed to pass through one or the other slit, mutually exclusively.
Because the dimensionality of a multiparticle quantum state is the sum of the dimensionalities of the singleparticle states, the dimensionality of a macroscopic state is much higher than of a microscopic state (this is a quantitative difference between the macroscopic and microscopic realms, not a qualitative difference). Because arbitrary directions in a higherdimensional space are more likely to be close to orthogonal, the macroscopic realm tends to behave "classically", while the microscopic realm tends to exhibit quantum effects.
This is not how I see the multiverse. I see it as a configuration space of every possible reality. As such, it is fixed and completely determined. In the same way that probabilities are observer effects rather than physical, entropy and the second law of thermodynamics are also observer effects. The universal wavefunction evolves "unitarily" (reversibly).
When I said that the "components corresponding to the particle passing through each slit are not orthogonal", I don't mean that these components have a separate existence. The actual state of the particle is the vector sum of these two components. That is, it is a single direction in the infinitedimensional space. Mathematically decomposing this direction into the sum of the directions corresponding to the particle passing through each slit is a convenient way to calculate the interference pattern. But the particle still passes through both slits at the same time. That is, the particle wavefunction is a twopeaked function, and this is the state of the particle.
It should be noted that when the particle is actually detected, the particle state is decomposed into its position eigenstates. Thus, each particle is detected at only a single position (the position eigenstates being orthogonal). Therefore, to observe the interference pattern requires that it be statistically accumulated by the detection of many particles.
I think I can see what you are saying, but isn't that saying that the particle isn't real? If it were real it couldn't pass though BOTH slits at once as that would violate conservation of energy? I can see how it would pass though one or the other in different universes though. Isn't the "state" a mathematical representation of our uncertainty of what universe we are in? I apologise in advance if I have missed something in what you have already gone to great pains to make clear!
I don't see how any of this could possibly lead to the Schrodinger equation for a wave function which does not itself give the probability of an occurrence, but rather has to be squared to give the probability. I don't even see how the Schrodinger equation could even be used to approximate the probabilities. If you can't in fact get the Schrodinger Equation from these ideas, how can the argument be real? Am I missing anything?
My maths isn't great so please excuse me if I am not a big help here . Rather than squaring the amplitude it is multiplied by its complex conjugate. I know that multiplying exp(iwt) by exp(iwt) eliminates the phase factor . The normalisation constant is actually set in such a way to give the correct result that the sum of all probabilities =1. In order to understand how the MWI interpretation leads to the Schroedinger equation the Path Integral Formulation of QM is useful as you can derive the equation this way. If one is trying to represent a two dimensional time in just one dimension I fear that this equation is going to be as close as one can get.
A key aspect that was implicit in what I said way back in this thread but needs to be made more explicit, is that we are taking about a symplectic manifold. This is the technical term for what is better known as a "phase space". This the natural setting for Hamiltonian mechanics, which is the classical theory that underpins quantum mechanics. In phase space, points are coordinated by generalised position coordinates and their corresponding conjugate momenta. In quantum mechanics, the generalised position and corresponding conjugate momenta are related by a Fourier transformation, which can be viewed as a "rotation" of the basis vectors of the Hilbert space in which the quantum state vector exists. Roughly speaking, the state vector can be viewed from many different angles, and it is the choice of observable that we measure that determines the viewpoint from which we viewing the state vector. The key point to the manyworlds interpretation is that a state vector interacts with the measurement device in an initial state to produce a measurement device state vector that depends on the state vector being measured, such that each orthogonal component of the state vectors act independently. Linearity is a key aspect, which means that:
Quantum wavefunctions are complexvalued because the Fourier transformations that relate position to momentum operate in the complex number domain. It is the complexvalued nature of quantum mechanics that gives rise its wave character.
The manyworlds interpretation is deterministic. A pure quantum state is a definite state without uncertainty when considered within the multiverse as a whole. In other words, the statistical nature of quantum mechanics is an artefact of the state vector nature of the observers themselves.
It may be possible to arrive at the same result without necessarily have to work with infinite dimensions! Would not just one (or two)extra time dimensions do the trick just as well? I found this article today which is pretty much in line with the situation as I see it. I didn't write it but I wish I could have done! I would be very much interested to hear what you make of it....
http://arxiv.org/pdf/quantph/0501034v2.pdf
Thanks. I had a brief look at it, but it's a bit much to digest in a sitting. However, I must say at the outset that I've never been a fan of KaluzaKlein type theories. Fourdimensional spacetime has unique properties compared to other dimensional manifolds. Also, the brief read of the article suggests to me that they've missed the point of what quantum mechanics is about. I actually like the manyworlds interpretation because it seems to tie everything together in what I regard as a natural way.
This one has less maths in it, better for the weekend maybe.....
http://arxiv.org/pdf/quantph/0505104v2.pdf
Last edited by Jilan; 11022013 at 04:58 PM. Reason: Less maths in this one
OK, I see what is going on now. I must say I'm not very happy with treating the extra dimesions as being potentially real, since that doesn't seem to be their role in the mathematics. However, I can see why that could appeal to people. For me, it is easier to think of one 3+1 dimensional world than to have an infinite number of unreachable dimensions in my mental pictures. I would rather get resigned to probabilities being fundamental.
But that's my point exactly! You don't need an infinite number, just a couple of extra ones..
This is quite good and readable and easy to visualise....
http://arxiv.org/pdf/quantph/0505104v2.pdf
As for being real, it might be that they are only real for quantum objects until they interact with something macroscopic and there is only one of the universe at large. I love QM!
Last edited by Jilan; 11022013 at 06:30 PM. Reason: Afterthought
One thing that I perhaps should make explicit:
Consider a vector in an dimensional (Euclidean) space. Suppose one has a transformation matrix that transforms vectors to other vectors. For example, the transformation matrix could be a rotation matrix that rotates vectors. Then the transformation is represented by the following equation:
Now consider a function of a single variable , a transformation function of two variables , and consider the integral transform:
Note the parallel. The difference is that in the first case, we are dealing with a finitedimensional space, and in the second case, the number of dimensions is infinite (continuum cardinality). That is, the value of at each value of is a component of the vector in the infinitedimensional space.
Thus, the quantum wavefunction is a vector in an infinitedimensional space. For example, the complex exponential functions are momentum basis vectors in the position domain, and the Fourier transformation of the wavefunction is a decomposition of the quantum state vector into its momentum components.
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