# Relativistic energy equation applied to a double-slit experiment

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SEYED2001
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My question: How do the values for the velocity, momentum and energy of an electron in a double-slit experiment are altered by the observation?
My question:

How do the values for the velocity, momentum and energy of an electron in a double-slit experiment are altered by the observation?

Probably,energy is altered. Given that energy is a function of momentum and velocity, either or both of these must have been changed. However, I am not sure how accurate this prediction is. Does the electron change its velocity and/or mass when it's observed? Or was I wrong assuming that the energy of the electron changes as it's observed? After all, was I wrong in applying the relativity to QM?

Seyed

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Your question is implicitly mixing classical and quantum thinking and that's leading to some confusion. Here I'll try to describe the process in purely quantum terms.

You start with an electron with well defined momentum and energy (it's wave function is a planar wave moving toward the double slits). This means it is in a superposition over a continuum of positions, it's "state" description is a vector sum of a continuum of positions. Each of those definite position components that one is adding up in the superposition forming the definite momentum electron are themselves in a superposition of a continuum of momenta and energies but with this addition process of Hilbert space vectors these all cancel out except for the one component momentum indicated which is reinforced.

Along comes the double slit. It selects out two (or two small bands) of lateral position. What this means is that you have a position dependent interaction (being absorbed or reflected vs traveling past the double slit). So afterwards the electron is in a superposition of not having been transmitted with a reduced range of superpositions of positions, and having been transmitted with the complementary* superposition of narrow double bands of position, [*complementary as in set complement].

Let me qualify here that by saying "the electron's are in a superposition" this is not a description of the states of the electron but of the states of our asserted knowledge about the electron. Until we again observe the electron its energy and momentum (or position at given times) are not defined.

Finally the electron eventually reaches an array of detectors each of which will interact with the electron if it is in the same position as the given detector. The array may detect no electron implying (in an ideal setting) the electron never made it through the double slits, or the array may detect an electron at a particular position indicating that it was in a corresponding superposition of momenta and energies.

Since the original momentum and energy was known but the final momentum and energy are not known we cannot say whether the momentum and energy changed or not. Such a statement contradicts the Heisenberg uncertainty principle as it implies simultaneous knowledge of the momentum and position of the electron (at the detector). We can say that it might have occurred and sometimes occurs but never that if definitely did occur for a particular electron. Only in the aggregate can we say that for many many experiments there will be an aggregate average interaction for e.g. cases where the electron ended up at a particular sensor position.

Note that any attempt to observe the recoil of the slit as the electron passed through should in principle disrupt the interference pattern of the classic double slit when you observe many experiments.

Finally I don't see that relativity plays any significant role in this analysis.

SEYED2001
Thank you very much for your detailed response. I see your point about the uncertainty principle and its connection with my question. However, I think the principle can be avoided in this context. Let me rephrase my question in this way (into three smaller ones):

a) If a moving electron is observed, is its momentum changed? In theory, we must be able to measure the momentum of the electron just before the observation and just after it, and see if they're different or not.
b) If a moving electron is observed, is its velocity changed? In theory, we must be able to measure the velocity of the electron just before the observation and just after it, and see if they're different or not.
c) If a moving electron is observed, is its energy changed? In theory, we must be able to measure the energy of the electron just before the observation and just after it, and see if they're different or not.

By "after observation", it is meant at the time that the observation has occurred and has affected the electron (i.e. at the instant that the wavefunction is collapsed).

This way, no relativity is involved: pure quantum!
Hence, I am just asking for the effect of observation on the above-mentioned three observable; i.e. energy, momentum, and velocity.

Now, to avoid the uncertainty principle, we can postulate that the change of each of these parameters is universal. Hence, if energy changes as the result of the act of observation, it is true for the energies of all quantum systems and observation. In other words, the energy always change as the result of an observation. The same can be postulated for the other two variables. So, I can theoretically say about a particular electron if both the momentum and energy of it have changed due to the observation or not, by extrapolating the result of each individual measurement.

In other words, i am not interested in the exact value of the parameters, so I don't need to deal with the uncertainty principle I believe. I just want to know how these parameters change with respect to their previous, before-observation value: same or different.

Is it still impossible to figure out the answer due to the uncertainty principle? If yes, would yo please explain me how my method to avoid the principle was flawed? If no, would you please tell me what happens for each of the above-mentioned observables; energy, momentum, and velocity; when an observation is taken place? Do they stay the same or become different?

Seyed

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I don't see where your scheme circumvents the uncertainty principle as it applies. But you are breaking things down to their essentials well. One note, is that typically a velocity is not considered an observable per se. Rather the momentum (and velocity is inferred from the fixture of the mass). This may sound counter intuitive in that classically the opposite is typically done. But momentum is directly measurable by invoking the wave-like properties of a quantum particle, e.g. it is reflected by a diffraction grating at a specific angle.

I think the issue at hand is the fact of making the interim observation is itself fundamentally, via the uncertainty principle, precluding knowing whether or not the complementary quantities have changed. You can say "Measure X, then Measure Y, then Measure X again" and if [X,Y] ##\ne 0## then the nature of measuring Y precludes being able to sensibly say whether a the X observable will or will not change. It's not just a matter of disturbing the classically defined values. It is a matter of invalidating the meaning of them as a valid quantity while the Y observation is going on.

I can only answer your questions about whether X will change or not by saying "sometimes". The amount of change will be fundamentally uncertain and ill defined to the point that you may just as easily think of the particles at each stage being distinct. Specifically in the momentum->position->momentum sequence you can sat that the previous definite momentum particle is via the position measurement annihilated creating a definite position particle which then propagates with uncertain momentum until a later momentum measurement annihilates it creating another definite momentum particle. The fact that, if these measurements are made to ultimate precision the prior and latter momentum will be totally uncorrelated makes it hard to say it is the same quantity changing rather than a new value being defined.

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Thank you very much for your detailed response. I see your point about the uncertainty principle and its connection with my question. However, I think the principle can be avoided in this context. Let me rephrase my question in this way (into three smaller ones):

a) If a moving electron is observed, is its momentum changed? In theory, we must be able to measure the momentum of the electron just before the observation and just after it, and see if they're different or not.

This makes no sense. Measuring a particle is an observation. There is no concept in QM of well-defined "before" and "after" values. You measure the momentum of a particle and you get an outcome. It makes no sense to talk about how much the momentum changed by observing it. Quantum objects do not behave classically and cannot be described classically.

If you are going to study QM, you need to adapt your thinking to the prinicples of QM. In particular, you can't avoid the UP (uncertainty principle) or dismiss it as an inconvenience. The UP is at the heart of QM.

SEYED2001
Thank you all for your help.

So, consider the question: does the momentum of an electron change when it's observed? By your answers, did you mean that:
1. "change of momentum" is not defined, hence the question can never be answered.
2. "change of momentum" is defined, so the momentum does change or dosn't change (or maybe both or neither); however, we cannot experimentally find out which one was the case: if it changed or not (or maybe more complicated answers(?)).
Which one of the two above-mentioned statements do you mean?
Thank you once again

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Thank you all for your help.

So, consider the question: does the momentum of an electron change when it's observed? By your answers, did you mean that:
1. "change of momentum" is not defined, hence the question can never be answered.
In general, in QM it is not accurate to say that a measurement changes a dynamic quantity. Instead, there is no well-defined dynamic quantity until a measurement is made. There is, therefore, no well-defined value for it to change from.

Talking about the precise momentum of an electron before measurement doesn't make sense in QM.

• SEYED2001
SEYED2001
In general, in QM it is not accurate to say that a measurement changes a dynamic quantity. Instead, there is no well-defined dynamic quantity until a measurement is made. There is, therefore, no well-defined value for it to change from.

Talking about the precise momentum of an electron before measurement doesn't make sense in QM.

Thank you very much. I think I get it better now. But, if we consider the probability disstribution of, let's say, momentum of an electron, then does the act of observation change this probability distribution? Or the probability density function of momentum is not changed due to the act of measurement?

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Thank you very much. I think I get it better now. But, if we consider the probability disstribution of, let's say, momentum of an electron, then does the act of observation change this probability distribution? Or the probability density function of momentum is not changed due to the act of measurement?
Yes, a measurement in general changes the wave-function (to an eigenstate of the observable). This is often called the "collapse" of the wave-function.

In general, immediately after a measurement (and before the system has had time to evolve significantly) you have a much tighter distribution of possible momenta.

For other observables, like spin, which have a discrete spectrum of possible measurement results, the situation is conceptually much clearer. Position and momentum (being observables with a continuous spectrum) are not so conceptually clear-cut.

• SEYED2001
SEYED2001
Yes, a measurement in general changes the wave-function (to an eigenstate of the observable). This is often called the "collapse" of the wave-function.

In general, immediately after a measurement (and before the system has had time to evolve significantly) you have a much tighter distribution of possible momenta.

For other observables, like spin, which have a discrete spectrum of possible measurement results, the situation is conceptually much clearer. Position and momentum (being observables with a continuous spectrum) are not so conceptually clear-cut.

Thank you for your rsponse. So the wavefunctions of energy probability and mass probability density functions, not unlike the momentum's, collapse into an eigenstate of the observable... . But I still don't see something: if the momentum is in a superposition of many values, then my question is ill defined in the context of QM, since there is no definite momentum to change, as you mentioned earlier. However, what if the momentum has a definite value before the measurement is made (but we just don't know it)? I don't ask if this realist viewpoint is true; I just say if it's true, then what happens to this definite value for momentum? Does the act of masurement change it?

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However, what if the momentum has a definite value before the measurement is made (but we just don't know)? I don't ask if this realist viewpoint is true; I just say if it's true, then what happens to this definite value for momentum? Does the act of masurement change it?
This is where spin is conceptually cleaner. Technically, momentum eigenstates are not physically realisable. You always have a spread of possible momenta. For a free electron, at least. This leads to the concept of a "wave-packet".

Again, a qusetion like: "does a measurement of a electron with a definite momentum change that momentum - to a different definite value" is the sort of question that reveals you have not yet begun to understand QM! I'm sorry to say.

If we take spin, then repeatedly measuring the spin about a certain axis never changes the result from the first measurement. But, if you change the axis about which you measure the spin, you have a probability of getting both spin values about the different axis.

This is a different sort of uncertainty.

If you know the spin about the z-axis (say), then the spin about the x-axis or the y-axis is not well-defined.

If, now, you measure the spin about the x-axis, then you know the spin about that axis, but now the spin about the y and z-axes is not well-defined.

These measurements change the spin state of the electron. But, because you don't know the spin before and after, can't say that it changed from plus to minus or vice versa.

This is why it's so important in QM to start thinking about the state of a particle (i.e. the wave-function) and not about a set of well-defined quantities that you would have in classical mechanics.

In general everything is probabilistic.

SEYED2001
If we take spin, then repeatedly measuring the spin about a certain axis never changes the result from the first measurement.

Thank you very much for your reply. So, if I measure the spin of a specific axis, this act of masurement doesn't change the state. I wonder why this cannot be generalised to all observables, not just spin but also momentum or energy? So, why can't we conclude: measuring a momentum of an electron doesn't change it, for the same reason that measuring the spin over x-axis doesn't change the x-spin?

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Thank you very much for your reply. So, if I measure the spin of a specific axis, this act of masurement doesn't change the state. I wonder why this cannot be generalised to all observables, not just spin but also momentum or energy? So, why can't we conclude: measuring a momentum of an electron doesn't change it, for the same reason that measuring the spin over x-axis doesn't change the spin?

That's because spin has realisable eigenstates and the dynamics of position/momentum are different. You have the UP, which applies to position and momentum for all states that a particle can possibly be in. This is a very general UP for position and momentum. It actually implies that neither momentum not position can ever be totally definite.

If a system is in an eigenstate of an observable and you repeat a measurement of that observable then the state does not change. But, as explained above, there are no physically realisable eigenstates for position and momentum.

Note I didn't say the spin didn't change. I said that a measurement of spin about a given axis always gives the same result if you repeat it. That's what it means to be in an eigenstate.

I have to say that you are just repeating the same error now. You have to change your thinking and your language. Otherwise, 50 posts from now you'll still be asking "does the momentum of an electron change when you measure it?" and I'll still be saying that that question makes no sense in QM.

SEYED2001
Thank you very much sir for your help tonight (or maybe today for you, depending on where you live!). The only thing that I haven't understood yet is:
if I assume the realist viewpoint to QM, so the momentum of an electron has a definite, well-defined value, p, just before the measurement (that we don't know it and perhaps that's why we assign the momentum valus some probabilities); and then if I do the measurement, then the well-defined momentum value at the instant of the measurement would be q. Is p=q?
Please note that I am not assuming the orthodox viewpoint, or the Copenhagen interpretation (even if they're actually correct). I just assume that the probabilities that we assign to the diffrenet momentum valus are due to the incompleteness of QM, and hence the existence of some hidden variables. In other words, it is assumed that nothing is probabilistic (even if they actually are), and there IS a definite, well-defined value for p and q.
Therefore, if realist viewpoint is postulated, does the measurement of the momentum of an electron alter the well-defined value of the momentum of the electron?
I hope that the way that I wrote my question is now more accurate in terms of the common terminology in QM. After-all, I see that I learned a lot of QM in these hours! Much faster than my months of irregular studying.
Thank you once again for your help.
Seyed

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if I assume the realist viewpoint to QM, so the momentum of an electron has a definite, well-defined value, p, just before the measurement

... then you are not doing QM.

SEYED2001
... then you are not doing QM.
Isn't realism one of the viewpoints to QM?

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Isn't realism one of the viewpoints to QM?
No. There is Bohmian mecahnics, but that is a different ball-game altogether.

SEYED2001
No. There is Bohmian mecahnics, but that is a different ball-game altogether.
Oh I see... . I thought that the hidden variable theory of QM is still one of the options.

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I want to add a few comments if I may (and if you find it helpful). The problem with understanding quantum mechanics is that there is a substantial shift in the scope of the semantics. Classically we think of a particle as an object with an objective state and thus speak of that state without needing to refer to the empirical act of verifying that state. The world is a world of objects. (An "object" in this context is a system with a complete set of properties, i.e. what might be observed and, critically, each of these properties has a specific value defining the system's state.)

Since, in classical mechanics, the evolution is deterministic at the fundamental scale we do not need to distinguish between a specific instance of say a (classical) electron or the class of electrons with the same objective state. How one behaves all behave and we can be sloppy with the language. The language we use for classical systems is ontological (what is). We can imagine the classical object's state as a point in a "state space" (typically phase space) spanning the range of a complete set of observable values from which all other observables can be expressed as functions. Now we do refer to classes of classical systems when we begin describing uncertain knowledge about the system state and for this, we utilize probability distributions over the state space.

Now with quantum mechanics, we suspend the assumptions stated above. Quantum systems (or simply quanta) also have a complete set of properties (potential observables) but we only acknowledge values for these properties at the point of observation. When we casually say "an electron with momentum p and (z component of) spin ##s_z##" i.e. when we write down the wave-function for such an electron we are in point of fact expressing the class of such electrons. Insofar as we define a measurement as actually having occurred, we know that an immediate subsequent measurement of the same properties will yield the same values. So, we know the measurement is meaningful. We can even predict future compatible measurements after dynamic evolution of the system.

At this point it is important then to make a distinction between a single instance of a quantum and a class of quanta. This is because without classical determinism our description must expand to probabilistic predictions. A singular system (both classical and quantum) will have a singular sequence of measured values. It doesn't have a probability distribution. A class of (either type of) systems may have a predicted distribution of future behaviors and we can either write down a classical probability distribution over the state space, or with quanta a wave function or other Hilbert space vector, or more generally a density operator.

So, in the examples described as "wave function collapse" we must understand that one observes a given instance of a quantum, then based on that measurement, we classify that system and write down its wave-function expressing the class to which it belongs. This description comes with a system of probabilistic predictions about its future behavior. If we then later observe that specific quantum, we will observe one of those possible predicted outcomes but having done so we update our class description. It is no different than the classical Bayesian updating of probability distributions given new information, but we are not using classical probability distributions. The term "collapse" is unfortunate as it is rather more a "jump". The discrete jumping of descriptions simply reflects the fact that we are carrying out a discrete sequence of observations.

The "mystery" of "wave function collapse" comes when we over reify the wave-function thinking it is an ontological representation of the system rather than a description of the framework of predictions for the class of systems into which we categorize a given observed quantum. This mystery evaporates when one understands the description as being pragmatic and not representative. (This is the essential framework of the Copenhagen Interpretation to which I ascribe.)

Of course there are countless books and reams of articles exploring all this and alternative interpretations and I'm already posting a very long comment but let me add one last point. We can directly argue via to Bell's inequality violations that the probabilities associated with quantum predictions are inconsistent with a probability measure over some state space (even including hidden variables). The additivity of probabilities just doesn't work the same. The invocation of locality in the various discussions is a red herring IMNSHO. Attempts to reconcile classical ontological descriptions necessarily invoke either phenomena which allow future actions to alter past (hidden) state variables which is just a novel way to invalidate the use of an objective description of a system... or invoke the "all realities are equally real" which renders meaningless the statements that some phenomenon did or did not occur. They are attempts to fit a square peg into a round hole by banging on it until the hole becomes square and they declare the peg is really round. Again, this is my opinion based on my Copenhagenist understanding of QM.

• SEYED2001
SEYED2001
Thank you sir for your detailed, deep explanation of the nature of the stochasticity as seen in QM. So, by QM we are actually going back to the time that people believed that humans are at the center of the universe, but with the difference that it is not achieved ontologically, rather we just accept that we're a part of this great, complicated system of universe and we only know what we measure from it.

Mentor
I thought that the hidden variable theory of QM is still one of the options.
Yes, hidden variable theories are possible. So far nobody has come up with one that works, but that doesn't mean it won't happen - there's no theoretical or experimental reason to say that such a theory is impossible.

What is impossible is a hidden variable theory in which the hidden variables are both "local" and "real". I've put quotes around those words because they are generally used as a convenient verbal shortcut for certain mathematical properties that the hypothetical hidden variables might or might not have. You have to look at the math and not just the words to understand exactly what the constraints are... but the quck summary is that any theory of the sort that you're considering in this thread is precluded. Any successful hidden variable theory will be at least as weirdly counter-intuitive as quantum mechanics itself.

You can google for "Bell's theorem" to get started, and https://www.drchinese.com/Bells_Theorem.htm is a good start.

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• SEYED2001
SEYED2001
Yes, hidden variable theories are possible. So far nobody has come up with one that works, but that doesn't mean it won't happen - there's no theoretical or experimental reason to say that such a theory is impossible.

What is impossible is a hidden variable theory in which the hidden variables are both "local" and "real". I've put quotes around those words because they are generally used as a convenient verbal shortcut for certain mathematical properties that the hypothetical hidden variables might or might not have. You have to look at the math and not just the words to understand exactly what the constraints are... but the quck summary is that any theory of the sort that you're considering in this thread is precluded. Any successful hidden variable theory will be at least as weirdly counter-intuitive as quantim mechanics itself.

You can google for "Bell's theorem" to get started, and https://www.drchinese.com/Bells_Theorem.htm is a good start.

Thank you sir.