Energy conservation in two-slit experiment

[jf117]

I would like, once again, trouble this community with "the only mistery" (as written by Feynman) of quantum mechanics, the double slit experiment. More specifically, I am exploring it in the recent version described by T. Marcella (Eur. J. Phys., 23 (2002)).

For the benefit of those of you who haven't followed previous arguments on this forum, I will synthetically describe the experimental setting. A quantum particle (photon ,electron, it really doesn't matter) is flying with constant momentum p=hbar*k and energy E=P^2/(2m) along the z direction towards a plate of equation z=0, containing two narrow perforations at (0,-d/2,0) and (0,d/2,0). As Marcella explains, the plate+perforations constitute a state preparation system which forces the particle wave function to be the sum of deltas of Dirac. The extreme localization of the particle in the initial state, expecially for x and z, but also for y, implies that its momentum is free to assume nearly all values of the momentum space, with higher probability on certain planes of equation py=const.

We can, therefore, find with equal probability the particle's momentum as, for instance, (a,py0,c) and (a',py0,c'), where p0y is a y-component with high probability value, and a,c,a',c' are arbitrary real numbers. What is now interesting is that in the two cases just exemplified the energy takes different values, E=(a^2+pyo^2+c^2)/(2m) and E'=(a'^2+py0^2+c'^2)/(2m), and also different from the initial value E=p^2/(2m). There is here a clear violation of the conservation principles, both for momentum and energy.

In his paper Marcella by-pass this point without any explanation, by imposing the energy conservation principle (in fact the momentum magnitude is maintained unaltered, while the diffraction angle is the only responsible for the variation of the y component), and thus violating the uncertainty principle.

I have been trying to search through various literature to better understand the whole matter, but I have had no success so far. I hope I'll get help by some of you.

Many thanks,

james

(P.S. Don't worry too much about E=P^2/(2m) for the photon case, where m=0. There you can consider E=hw, etc. This difference does not change the problem, because E=pc for a photon)

 

[ZapperZ]

We can, therefore, find with equal probability the particle's momentum as, for instance, (a,py0,c) and (a',py0,c'), where p0y is a y-component with high probability value, and a,c,a',c' are arbitrary real numbers. What is now interesting is that in the two cases just exemplified the energy takes different values, E=(a^2+pyo^2+c^2)/(2m) and E'=(a'^2+py0^2+c'^2)/(2m), and also different from the initial value E=p^2/(2m). There is here a clear violation of the conservation principles, both for momentum and energy.

In his paper Marcella by-pass this point without any explanation, by imposing the energy conservation principle (in fact the momentum magnitude is maintained unaltered, while the diffraction angle is the only responsible for the variation of the y component), and thus violating the uncertainty principle.

I have been trying to search through various literature to better understand the whole matter, but I have had no success so far. I hope I'll get help by some of you.

Many thanks,

james

There are a couple of issues here, so let's see if I can explain them as clearly as what I have in my head:

1. Invoking the conservation of energy would require that the particle in question changes direction while maintaining the same speed. Thus, the z-component of the momentum has changed, while the kinetic energy (or in the case of the photon, the "pc") remains constant. I don't think this is a problem.

2. What appears to be more of a problem is the conservation of momentum perpendicular to the original direction, i.e. in the y-direction, since the particles originally did not have any y-component of the momentum. This is where the physical existence of a FIXED slit is required. The particles interact with the "measuring" device. In this case, the slit is a "position measurer", since at the moment the particle passes through the slit, you know where the particle is up to an uncertainty equivalent to the width of the slit. The recoil momentum in the y-direction is taken up, in principle, by the slit.

Note that if you have, instead of a rigid slit, only freely floating particles arranged in the shape of the slit opening, you would not be able to maintain such momentum transfer and you would not be able to get the same diffraction pattern.

Zz.

 

[jf117]

Dear Zz, thanks for your swift reply.
I am not troubled by the change in momentum direction because this is what should happen when a diffraction occurs. What worries me is really the momentum magnitude, p, which is basically equivalent to the kinetic energy. When you say that energy conservation implies a change in direction you are perfectly right. But the uncertainty principle applied to the z component means that pz can take any values, including values greater than the initial p^2/(2m). This, in turn, implies that the momentum magnitude, py^2+pz^2 is greater than the original p^2. This is why both momentum magnitude and energy change after interaction with the slit.

I take your point, though, that the energy/momentum surplus can be absorbed by the infinitely massive plate. This, therefore, re-establishes energy/momentum conservation.

What still remains to be understood is why Marcella uses |py>=|psin(theta)> after the interaction, thus implicitly stating that p (or equivalently the energy) has not changed. Indeed, for what I've said above, one could have an infinite number of new values, p', for the momentum magnitude. This way the probability P(theta) should, more appropriately be indicated as P(py), thus implying no dependency on z (and x), i.e. the probability of two different momentum vectors, with same y component, but different x and z components, should be the same.

james

 

[seratend]

What still remains to be understood is why Marcella uses |py>=|psin(theta)> after the interaction, thus implicitly stating that p (or equivalently the energy) has not changed. Indeed, for what I've said above, one could have an infinite number of new values, p', for the momentum magnitude. This way the probability P(theta) should, more appropriately be indicated as P(py), thus implying no dependency on z (and x), i.e. the probability of two different momentum vectors, with same y component, but different x and z components, should be the same.
james

You still have to describe an interaction of the photon with the double slit (how things occur).
If you want to view it with the particle filter, you have the momentum+energy conservation that describes the interaction between the slit and the photon as with particles scattering.
Now, if you assume, [one possible model for the slits: first order approximation], that the energy of the photon is conserved (elastic scattering: no energy exchange between the scatterer and the scattered, just momentums), you recover your marcella formula: momentum norm is conserved (energy is conserved) but not the momentum direction (momentum exchange with the slits.

If you do not like the particle filter, but rather the wave filter view, you may note that the quantum propagator of a free photon (or an electron) is, formally, the same as the propagator (the green function of helmotz equation) of the classical electromagnetic waves. Therefore, you just need to choose one formal model of the classical em waves diffraction result to deduce the QM result (as well as the energy+momentum conservation).

Hope this helps you.

Seratend

 

[jf117]

That is quite interesting and helpful.
I have no much recent knowledge of quantum scattering theory, but what you are saying about using approximations to some order, in order to describe energy and momentum exchange, sounds like a good explanation to justify Marcella's choice of an elastic scattering.

Thanks!

james

 

[ZapperZ]

Dear Zz, thanks for your swift reply.
I am not troubled by the change in momentum direction because this is what should happen when a diffraction occurs. What worries me is really the momentum magnitude, p, which is basically equivalent to the kinetic energy. When you say that energy conservation implies a change in direction you are perfectly right. But the uncertainty principle applied to the z component means that pz can take any values, including values greater than the initial p^2/(2m). This, in turn, implies that the momentum magnitude, py^2+pz^2 is greater than the original p^2. This is why both momentum magnitude and energy change after interaction with the slit.

I take your point, though, that the energy/momentum surplus can be absorbed by the infinitely massive plate. This, therefore, re-establishes energy/momentum conservation.

What still remains to be understood is why Marcella uses |py>=|psin(theta)> after the interaction, thus implicitly stating that p (or equivalently the energy) has not changed. Indeed, for what I've said above, one could have an infinite number of new values, p', for the momentum magnitude. This way the probability P(theta) should, more appropriately be indicated as P(py), thus implying no dependency on z (and x), i.e. the probability of two different momentum vectors, with same y component, but different x and z components, should be the same.

james

There is no restriction on the z-components because there aren't any kind of measurement made along z. The slit has a width along y. Thus, the y-position and the uncertainty in the y-position is determined by the width of the slit. When this occurs, the uncertainty in the y-component of the momentum kicks in. The slit makes no determination of the position along z or the uncertainty in that position. So there is no restriction on the z-component of the momentum as of yet. This is what I gather from the Marcella paper.

Zz.

 

[jf117]

Exactly!
I have no problems with that. pz is free to take any value, as the uncertainty principle prescribes, and thus to make the energy after the interaction larger/smaller than the initial energy. Basically the uncertainty on the z component brings in an uncertainty on the energy.

But, as you and seratend have suggested, the interaction with the plate needs to be looked at more carefully. The incoming particle undergoes all sorts of interactions with the plate/slits. Marcella considers only interactions leading to elastic scattering.

james

 

[seratend]

Exactly!
I have no problems with that. pz is free to take any value, as the uncertainty principle prescribes, and thus to make the energy after the interaction larger/smaller than the initial energy. Basically the uncertainty on the z component brings in an uncertainty on the energy.

But, as you and seratend have suggested, the interaction with the plate needs to be looked at more carefully. The incoming particle undergoes all sorts of interactions with the plate/slits. Marcella considers only interactions leading to elastic scattering.

james

Yes, that's what we do in the first order approximations of classical as well as quantum double slit diffraction patterns. It is sufficient for almost all practical purposes.
If you look deeper inside with the formal analogy of the free propagator of the particle and the free propagator of the classical em waves, you will recover all the results without having to do the mathematical calculation.

And one important result is that the momentum pz cannot be set to what we want to get the diffraction pattern due to the geometry of the slits (wavelength limit). The transversal momentum "py" component of the wave function, in the vicinity of the slits, must be compatible with the transversal size of the slits (~local quantization of the py momentum, e.g. quantum well py>py_mini).
Due to the conservation of the energy (the approximation), if py^2_mini/2m > Energy (=p^2/2m) of the incoming particle, we have a total reflection (if we neglect the tunelling effect). This is the case if the longitudinal momentum pz is too small (wavelength to large) respectively to the slits transversal size. When this occurs, the slits may be viewed as closed slits (no slits) like in classical em (total reflection due to the slits "cavity geometry").

Seratend.

P.S. I am using the formal analogy with the classical em waves propagation, just to underline how we can quickly get numerical results in the QM case without solving the complete SE as in em (through some adequate “justified” approximations).

 

[jf117]

There is no restriction on the z-components because there aren't any kind of measurement made along z. The slit has a width along y. Thus, the y-position and the uncertainty in the y-position is determined by the width of the slit. When this occurs, the uncertainty in the y-component of the momentum kicks in. The slit makes no determination of the position along z or the uncertainty in that position. So there is no restriction on the z-component of the momentum as of yet. This is what I gather from the Marcella paper.

Zz.

Sorry, I just got the sense of your reply now! What you are saying is that no
measurement at all is made on the z (or x) component, because the slit width is only along y. We could therefore shift backward and forward the plate along z, or shift upward or downward the slit along x, and nothing would change for the probability density, which would only depend on py. In addition to that, for each value of py we could select the appropriate values, px0 and pz0, which would ensure conservation of energy. This we are now able to do because x and z are completely non-localized.

Here there is still room to allow for decisions regarding elastic or inelastic scattering. We are still free to modify Px0 and py0 ad hoc to produce a variation in energy.

Yes, I'm quite happy with this!

Cheers,

james

 

[seratend]

Well, now I am lost. Except in the case of the slit screen with a null width (in the z axis I assume to be the direction of propagation), the initial particle can only cross the slits if its initial energy is greater than a given amount of energy that depends on the width of the slits (orthogonal direction of propagation). Therefore, we need |p_before slits|> p_mini.
Where p_mini can be evaluated as the lowest energy state of a quantum well (the transversal slits aperture, coarse approximation with the energy conservation approximation).

So,how can you say that you have no constraint on the pz momentum?

Seratend.

 

[jf117]

Well, now I am lost. Except in the case of the slit screen with a null width (in the z axis I assume to be the direction of propagation), the initial particle can only cross the slits if its initial energy is greater than a given amount of energy that depends on the width of the slits (orthogonal direction of propagation). Therefore, we need |p_before slits|> p_mini.
Where p_mini can be evaluated as the lowest energy state of a quantum well (the transversal slits aperture, coarse approximation with the energy conservation approximation).

So,how can you say that you have no constraint on the pz momentum?

Seratend.

When we use the plate/slit for the system preparation we are in fact reducing the particle's state to certain, definite values. In our specific case, slit with a width along y, the initial position ket will be |x,y0,z+>, where x means anywhere along x, y0 means a single, specific value on y, and z+ means anywhere between the plate and the screen, which is supposed to be placed at a considerable distance from the plate. So, after the state preparation, we have a perfect localization of the particle only for the y component. We cannot say much for the z component and we have a complete ignorance of the particle's position along x. So, after the state preparation, the particle momentum will be represented by the ket |px0,py,pz+>, where px0 indicate a single, constant number, py include an infinite set of real numbers, each one with a certain probability assigned to it, and pz+ is another set of infinite numbers where the probability associated to each number, this time, is different from zero only in a narrow interval centred on, say, pz0.

Supposing, now, a diffraction event with energy conservation, for each value of py we can fix px0 and pz+ so to satisfy (px0)^2+(py)^2+(pz+)^2=p^2, where p is the momentum's magnitude before state preparation. For each value of initial energy we are free to fix px0 and pz+, without violating the uncertainty principle, in order to obtain the same energy after diffraction. It works for those cases where scattering is non-elastic. For those cases we will have a different value of energy, E', after the diffraction, and will be able to adjust px0 and pz+ to obtain exactly that energy. The missing energy will have been absorbed by the rigid plate.

So, I'm not saying there's no constraint on pz. Pz is constrained to assume a precise range of values, those giving the correct value for the energy.

james

 

[seratend]

When we use the plate/slit for the system preparation we are in fact reducing the particle's state to certain, definite values. In our specific case, slit with a width along y, the initial position ket will be |x,y0,z+>, where x means anywhere along x, y0 means a single, specific value on y, and z+ means anywhere between the plate and the screen, which is supposed to be placed at a considerable distance from the plate. So, after the state preparation, we have a perfect localization of the particle only for the y component. We cannot say much for the z component and we have a complete ignorance of the particle's position along x. So, after the state preparation, the particle momentum will be represented by the ket |px0,py,pz+>, where px0 indicate a single, constant number, py include an infinite set of real numbers, each one with a certain probability assigned to it, and pz+ is another set of infinite numbers where the probability associated to each number, this time, is different from zero only in a narrow interval centred on, say, pz0.

Supposing, now, a diffraction event with energy conservation, for each value of py we can fix px0 and pz+ so to satisfy (px0)^2+(py)^2+(pz+)^2=p^2, where p is the momentum's magnitude before state preparation. For each value of initial energy we are free to fix px0 and pz+, without violating the uncertainty principle, in order to obtain the same energy after diffraction. It works for those cases where scattering is non-elastic. For those cases we will have a different value of energy, E', after the diffraction, and will be able to adjust px0 and pz+ to obtain exactly that energy. The missing energy will have been absorbed by the rigid plate.

So, I'm not saying there's no constraint on pz. Pz is constrained to assume a precise range of values, those giving the correct value for the energy.

james

In the following, we assume that the incident direction of the wave packet (z direction) is perpendicular to the plate with slits (it avoids non useful complications in the formulas).

Waow! Therefore, you are saying that a plate with sub wavelength holes is not able to stop an electromagnetic wave. If this were true, it would be very hard to get an electronic device compatible with the FCC or EC rules!

More seriously, you must admit the plate has a sufficient thickness so that the incoming waves are stopped (the evanescent waves within the plate vanishes before exiting the plate) otherwise you have a semi transparent plate. Once you admit this assumption (required to get a non-transparent plate) you also admit that the thickness of the slits are sufficient to be considered locally as a quantum well (y direction, we assume the height of the slits, x direction, are large enough to neglect energy quantification problems).
Once you are forced to admit that the slits are quantum wells to a first order approximation (locally), you must admit that the energy and the momentum are quantified in the y direction (transversal to the direction of the wave propagation). Therefore you know that the momentum in the y direction should be greater that a minimum momentum. This approximated minimum is given by the quantum well approximation:
E_mini= hbar^2/2ma^2 => outside the well, py_mini= hbar/a <=> lambda_y_mini= a is the y width size of slit, for the case of massive particles, easily transposed to the photon case (i.e. em wave in a cavity).

This coarse approximation allows to say that in order to get a transmission of the particle through the slits in the case of the energy conservation approximation, we need (we assume, for simplicity, that the incident wave propagation direction is perpendicular to the screen) as you have already written in your previous post:

(px+)^2+(py+)^2+(pz+)^2=(pz-)^2

(where sign + is used for the momentums just after the plate slits and sign - for the values just before the slits).
Therefore, we have the condition for the wave transmission (coarse approximation):

|pz-| > |py+|>= py_mini to allow a transmission through the slits.

Therefore for incoming momentum values lower than py_mini, the plate with slits is almost equivalent to a plate without slits (we have a reflection of the incoming wave).

We thus rediscover a known result: the transmission of a wave (or wave packet) requires that its longitudinal wavelength (z direction) should be compatible with the transversal geometry of the slits (y direction).

Seratend.

 

[jf117]

I don't understand what is wrong in assuming an ideal plate with no thickness along the z direction. We are using it to prepare the state for our particle. Thanks to this plate/slit system, which acts as a filter, we have, for all times immediately after t=0, partial localization for the y space-component and very little, or none, localization across the z and x components.

Perhaps I could understand a little more of your argument if you explained me, in terms of particle and probability, what a thickness along z would change to the position probabilities for x,y and z.

james

 

[seratend]

Just take any QM course at the one dimension section and look at the quantum wall and quantum well examples. You got all the information you need to understand we need, at least, a non null thickness.
Do not forget that the momentum py is discrete (as well as the momentum px, but due to the dimensions of the slits delta px << 1) due to the space localisation of the wavefunction within the slits. This also explains the source of the diffraction (the necessity to get a momentum py=/=0).

Seratend.