Space-Time — Momentum-Energy uncertainty relations

In summary, complementarity is the idea that classical concepts such as space-time location and energy-momentum cannot be combined in quantum mechanics. This is reflected in the formalism, where a localized wave-packet does not have a definite energy-momentum. The concept of time is treated differently in different approaches, as relativity demands treating it as a 4-vector while the canonical formulation of quantum mechanics treats it separately. While there are inequalities relating energy distributions to time dependence, there is no widely accepted energy-time indeterminacy relation. This is seen in the double-slit experiment, where the wave-fronts of probability do not advance in a circle-fashion with time, but rather reach the screen simultaneously or at different points randomly, causing the interference pattern
  • #1
etamorphmagus
75
0
From the wikipedia page on http://en.wikipedia.org/wiki/Complementarity_(physics)" :

In a restricted sense, complementarity is the idea that classical concepts such as space-time location and energy-momentum, which in classical physics were always combined into a single picture, cannot be so combined in quantum mechanics. In any given situation, the use of certain classical concepts excludes the simultaneous meaningful application of other classical concepts. For example, if an apparatus of screens and shutters is used to localize a particle in space-time, momentum-energy concepts become inapplicable. This is reflected in the formalism in the fact that a localized wave-packet is a superposition of plane waves, and therefore does not have a definite energy-momentum. This reciprocal limitation in the possibilities of definition of complementary concepts corresponds exactly to the limitations of the classical picture, where any attempt at the localization of a particle through objects such as slits in diaphragms introduces the possibility of an exchange of momentum with those objects, which is in principle uncontrollable if those objects are to serve their intended purpose of defining a space-time frame. Another famous example is 'Heisenberg's microscope', using which Heisenberg first discovered his uncertainty relations.

As I recently noticed in the double-slit experiment there is a lot of time-uncertainty, not only position-momentum uncertainty. That is shown in the fact that the probability wave does not reach the screen gradually, but instantly, for distribution purposes, to create interference with single-particles at a time, rather than a beam. So from this article I wonder, is there a relation on space-time uncertainty? Is there a relation of time-energy relation (in the double slit?)? And time-momentum etc? Instead of a pair of variables, quad-uncertainty??

I know this is always true: [tex]\Delta x \Delta p\approx h[/tex]
And this is true for particle decays: [tex]\Delta E \Delta t\approx h[/tex]

Is there something like this [tex]\Delta t \Delta x\approx h[/tex] or [tex]\Delta E \Delta t\approx h[/tex] or even this [tex]\Delta E \Delta t \Delta x \Delta p\approx h[/tex] ?!


Seems to be that there are 4 uncertainty pairs, anyhow:

[PLAIN]http://img228.imageshack.us/img228/4797/89768410.gif
 
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  • #2
Please, this is a really frustrating technical part of QM I face. From a different thread I understood that time-uncertainty is implied in the double-slit experiment as well as other uncertainties - so this is a decent question.

I'll appreciate any attempt to reply. Thank you.
 
  • #3
Yes it is a decent question. I've read through the suggested previous threads on the same topic and there are lots of good answers, but unfortunately no one knock-down answer. So we must end up staying a bit confused.

The reason is that there are several levels of quantum formalism and we require a deeper answer at each stage as we go from

quantum mechanics -> relativistic quantum mechanics -> relativistic quantum field theory -> quantum gravity.

My only advice would be that when reading answers, ask which formalism is being assumed. Also beware that when you get to the final level (quantum gravity) there is no one answer because there is no completed quantum gravity theory.

Also beware that even within the basic quantum mechanics formalism there are several equivalent pictures of the same thing (e.g. Schrodinger picture, Heisenberg picture, operator approach, path integral approach) and time is treated a bit differently in each.
 
  • #4
I understand, the different approaches and levels make it very complicated.

Is it safe to say though that time-location-momentum-energy uncertainties are inherent in QM and always there in every experiment? Which will manifest in not knowing the time-travel of a particle in the double-slit experiment, and not being able to produce EM-radiation of one wavelength ideally, because energy-momentum uncertainty is always there - coupled with location uncertainty and probably time too.

In the different approaches they treat time differently, is it because time is not used classically, and use a unified space-time (makes sense) - which is not intuitive for us anyways so it doesn't matter, the methods are all mathematical?
 
  • #5
etamorphmagus said:
Is it safe to say though that time-location-momentum-energy uncertainties are inherent in QM and always there in every experiment? Which will manifest in not knowing the time-travel of a particle in the double-slit experiment, and not being able to produce EM-radiation of one wavelength ideally, because energy-momentum uncertainty is always there - coupled with location uncertainty and probably time too.

In the different approaches they treat time differently, is it because time is not used classically, and use a unified space-time (makes sense) - which is not intuitive for us anyways so it doesn't matter, the methods are all mathematical?

I guess it is safe to say all that.

Yes: the basic conflict of how to treat time, is that relativity demands we treat position and time on an equal footing as 4-vectors, however the canonical formulation of quantum mechanics requires we treat time seperately. That's roughly it.
 
  • #6
Have a look at Ballentine section 12.3 for a readable discussion on
energy-time indeterminacy relations from a modern perspective.

Here's a quote from his concluding paragraph at the end of that section:

[...] There is no energy-time relation that is closely analogous to the
well-known position-momentum indeterminacy relation. However, there
are several useful inequalities relating some measure of the width of the
energy distribution to some aspect of the time dependence. But none
of these inequalities has such a priority as to be called the
energy-time indeterminacy relation.
 
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  • #7
strangerep said:
Have a look at Ballentine section 12.3 for a readable discussion on
energy-time indeterminacy relations from a modern perspective.

Here's a quote from his concluding paragraph at the end of that section:

I know, the energy-time relation for particle decays - it explains the short life-span of heavy particles.

But I talk about general time-indeterminacy, like in the double slit experiment where you can't know how long it takes for the particle to travel from gun to screen, or the time it take the probability wave to advance. Taking all the particles into account that create interference patten the only explanation is that averagely the wave-fronts reach the screen at the same time, or maybe different points reach first, randomly - Point is, the wave is not advancing in a circle-fashion with time, if it would, you couldn't really explain the interference distribution of the screen. I would draw it for you if you like, it is hard to explain with words.
 
  • #8
Just thought about something more.

Say an electron moves towards the slits with 100% momentum in the x direction, meaning 0 uncertainty on y-momentum. After it passes the slits the y-location is localized so there is uncertainty in y-momentum, which gives is a total momentum which is a combination of x- and y-momentum. It means it has higher momentum than before the slits = higher energy.

If this is so, how can the 2 slits be coherent sources? If the wavelength is different for different directions, it wouldn't create an interference pattern. Maybe this wouldn't matter if the y-momentum is very small, but I don't think this is the case. What happens to energy conservation here? Is the answer is that y-momentum increases and x-momentum decreases so direction is different but total energy is the same? But then what decreases x-momentum? It's not the uncertainty principle - the x-location is completely unknown.

I think the real answer to this conundrum is coupling time and energy uncertainties somewhere to the location-momentum uncertainties. So what do you think about the problem with the change in overall momentum in the slits?
 
  • #9
Hi.

#1
I would like to comment on your interesting topics of slit. In optics or electron beam diffraction at slits or crystals does not change frequency or wavelength of light i.e. energy.

#2
In standard QM, space(location x,y,z), momenta in x,y,z direction and energy are all operators but time is not an operator but a parameter. Uncertainty relation between time and energy should be interpreted not the same way as that of coordinate and momentum. Some scientists do not like it and have been trying to constract time operator but I have not heard their general success.

Regards.
 
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  • #10
sweet springs said:
Hi.

I would like to comment on your interesting topics of slit. In optics diffraction at slits does not change frequency or wavelength of light i.e. energy.

Regards.

Ok, but optics is a classical approximation, how does it sit with the uncertainty principle?

I think I have an explanation:
The wave-function interacts with the slit, it places some boundaries on the probability-wave, so it might be that y-momentum increases but x-momentum decreases so that only the direction changes, not the magnitude. This way energy conservation remains, and momentum conservation too, by the wall itself moving in the opposite direction (the wall with the slits).

I still wonder at the mechanism of the momentum-change. Should we even treat it this way? The uncertainty principle is a classical treatment of particles in the slits, when in fact the correct treatment is of a wave, a wave does interact with the wall.

Also, when talking about different uncertainties in this experiment, there is no energy uncertainty after all - we get the same "packet" every time.

In standard QM, space(location x,y,z), momenta in x,y,z direction and energy are all operators but time is not an operator but a parameter. Uncertainty relation between time and energy should be interpreted not the same way as that of coordinate and momentum. Some scientists do not like it and have been trying to constract time operator but I have not heard their general success.

Yeah, this strengthens the fact that QM is very incomplete, but BruceG said that there is a relativistic QM, surely, this treats time differently?
 
  • #11
etamorphmagus said:
Say an electron moves towards the slits with 100% momentum in the x direction, meaning 0 uncertainty on y-momentum. After it passes the slits the y-location is localized so there is uncertainty in y-momentum, which gives is a total momentum which is a combination of x- and y-momentum. It means it has higher momentum than before the slits = higher energy.

I'm pretty sure you'll find that energy is conserved. :-)

To analyze this stuff properly, one must think of an initial state (corresponding to
a plane wave), and the screen with 2 slits as an operator acting upon the initial
state to produce a new state. Then compute matrix elements between the new state
and position eigenstates corresponding to an array of detectors.

I.e., trying to mix classical reasoning and QM is a minefield. Better to stick
with hard core QM and work things out properly - computing both the
expectations and variances.

It wasn't clear from your reply whether you've studied Ballentine in detail,
but if you haven't then it would almost certainly be a profitable investment.
 
  • #12
strangerep said:
I'm pretty sure you'll find that energy is conserved. :-)

To analyze this stuff properly, one must think of an initial state (corresponding to
a plane wave), and the screen with 2 slits as an operator acting upon the initial
state to produce a new state. Then compute matrix elements between the new state
and position eigenstates corresponding to an array of detectors.

I.e., trying to mix classical reasoning and QM is a minefield. Better to stick
with hard core QM and work things out properly - computing both the
expectations and variances.

It wasn't clear from your reply whether you've studied Ballentine in detail,
but if you haven't then it would almost certainly be a profitable investment.

So basically the notion of explaining diffraction with the uncertainty principle is an approximation of physics-classes, to show that narrowing the slit increases diffraction efficiency, but really you can't understand it like that. I'm talking about Lewin's lectures from MIT.

And I'll take Ballentine in mind, however I'm only starting bachelor's.
 
  • #13
The lecture I talked about:

Heisenberg's Uncertainty Principle by Walter Lewin
link: http://www.youtube.com/watch?v=IcOJHJJpd0w&feature=player_embedded
time: about 33:00

He speaks about uncertainty in location and momentum- BUT NOT AROUND A SPECIFIED AXIS. Is it wrong to say that? because then you could say that certainty in y-location results in uncertainty in x- and y- momentum, so that the direction changes, not the magnitude of the momentum.
 
  • #14
etamorphmagus said:
The lecture I talked about:

Heisenberg's Uncertainty Principle by Walter Lewin
link: http://www.youtube.com/watch?v=IcOJHJJpd0w&feature=player_embedded
time: about 33:00

Unfortunately, I'm stuck at the end of a slow link which makes it impractical
for me to download video.

He speaks about uncertainty in location and momentum- BUT NOT AROUND A SPECIFIED AXIS. Is it wrong to say that? because then you could say that certainty in y-location results in uncertainty in x- and y- momentum, so that the direction changes, not the magnitude of the momentum.

Consider the operators [itex]x, p_x, y, p_y[/itex]. There's a nontrivial uncertainty
relation between [itex]x, p_x[/itex] because they don't commute. There's also a similar
nontrivial uncertainty relation between [itex]y, p_y[/itex].

In contrast, [itex]x[/itex] and [itex]p_y[/itex] do commute, hence you don't get
a nontrivial uncertainty relation between these two.
 
  • #15
strangerep said:
Unfortunately, I'm stuck at the end of a slow link which makes it impractical
for me to download video.
Consider the operators [itex]x, p_x, y, p_y[/itex]. There's a nontrivial uncertainty
relation between [itex]x, p_x[/itex] because they don't commute. There's also a similar
nontrivial uncertainty relation between [itex]y, p_y[/itex].

In contrast, [itex]x[/itex] and [itex]p_y[/itex] do commute, hence you don't get
a nontrivial uncertainty relation between these two.

O.K. but then there's a problem:
[itex]p_{i}[/itex] initial momentum magnitude
[itex]p_{f}[/itex] final momentum magnitude

Initial state: [itex]p_x, p_y=0[/itex] , so [itex]p_{i} = p_x[/itex]
At slit: [itex]p_x, p_y[/itex], and [itex]p_{f}[/itex] is a composition of [itex]p_x, p_y[/itex]

So, [itex]p_{f} > p_{i}[/itex]

Explanation?
 
  • #16
etamorphmagus said:
O.K. but then there's a problem:
[itex]p_{i}[/itex] initial momentum magnitude
[itex]p_{f}[/itex] final momentum magnitude

Initial state: [itex]p_x, p_y=0[/itex] , so [itex]p_{i} = p_x[/itex]
At slit: [itex]p_x, p_y[/itex], and [itex]p_{f}[/itex] is a composition of [itex]p_x, p_y[/itex]

So, [itex]p_{f} > p_{i}[/itex]

The last statement doesn't follow. Momentum is a vector, but you said
[itex]p_f, p_i[/itex] were "magnitudes".

Like I said before, one must track through the QM math carefully.
If you haven't got that far yet, I guess maybe you'll have to be patient
until you're a bit further into your degree. I doubt I can help much
more on this until then. Cheers.
 
  • #17
strangerep said:
The last statement doesn't follow. Momentum is a vector, but you said
[itex]p_f, p_i[/itex] were "magnitudes".

Like I said before, one must track through the QM math carefully.
If you haven't got that far yet, I guess maybe you'll have to be patient
until you're a bit further into your degree. I doubt I can help much
more on this until then. Cheers.

I meant that [tex]P_{i}=P_{x}[/tex] and [tex]P_{f}=\sqrt{P^{2}_{x}+P^{2}_{y}}[/tex] so [tex]P_{f}>P_{i}[/tex]. I was talking about magnitudes after all the vector operations. Momentum not conserved? How QM deals with simple diffraction, I mean — professors of physics demonstrate the HUP with laser diffraction. So what is the problem, even generally speaking, please?
 
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  • #18
Sorry for bringing this up again, but maybe someone has some response regarding the last message?
 

Related to Space-Time — Momentum-Energy uncertainty relations

1. What are space-time and momentum-energy uncertainty relations?

Space-time and momentum-energy uncertainty relations are fundamental principles in quantum mechanics that describe the limits to which we can simultaneously measure pairs of physical quantities. The space-time uncertainty relation states that there is a fundamental limitation to how precisely we can measure the position and time of a particle. Similarly, the momentum-energy uncertainty relation states that there is a limit to how precisely we can measure the momentum and energy of a particle.

2. How are these uncertainty relations related to Heisenberg's uncertainty principle?

Heisenberg's uncertainty principle is a more general concept that encompasses both the space-time and momentum-energy uncertainty relations. It states that the more precisely we know one physical quantity, the less precisely we can know its conjugate quantity. In other words, the uncertainty in one variable is inversely proportional to the uncertainty in its conjugate variable.

3. What is the significance of these uncertainty relations in quantum mechanics?

These uncertainty relations are significant because they fundamentally challenge our classical understanding of the universe. In the classical world, we can measure the position and momentum of a particle with arbitrary precision. However, in the quantum world, these measurements are limited by the uncertainty relations. This highlights the probabilistic nature of quantum mechanics and the limitations of our knowledge of the physical world.

4. Are there any exceptions to these uncertainty relations?

There are no exceptions to the space-time and momentum-energy uncertainty relations. They are fundamental principles that hold true for all particles in the quantum world. However, there are ways to minimize the uncertainties in one variable at the expense of increasing the uncertainty in its conjugate variable, as described by Heisenberg's uncertainty principle.

5. How do these uncertainty relations affect our understanding of the universe?

These uncertainty relations have significant implications for our understanding of the universe, particularly in the field of cosmology. They play a crucial role in the study of the early universe, where the uncertainty in the position and momentum of particles was large, and the effects of these uncertainty relations were more pronounced. They also have implications for the behavior of black holes, where the extreme gravitational forces can cause significant uncertainties in the position and momentum of matter.

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