Quantum tunneling and time/energy conjugacy

[KipIngram]

So if a quantum system is confined to a potential well of the right form it can "tunnel" to the other side with some probability. I've seen examples where that's expressed as the probability curve for finding the system in various states falls off as you move into the barrier, but still has a non-zero value on the other side.

 

But if I've interpreted what I've read correctly, there would be no tunneling with an infinite potential barrier, because the curve goes to zero at the well-side of the barrier. Does this wind up having to do with the random energy fluctuations that arise from energy and time being conjugate variables? I could also imagine the system "climbing over" the potential barrier as a result of getting the right sort of random, brief energy boost.

 

Is this just two ways of looking at the same thing, or are then entirely different things?

[Mordred]

In the classical system the infinite potential barrier. You are correct no tunnelling. In the QM scenario there is still a probability due to the HUP.

 

The position is uncertain the particle could be on the other side. In essence.

 

The energy is uncertain the particle may have enough energy to tunnel.

 

Whether its described as mounting the crest or fluctuations arising on the other side is in essence equivalent. In essence you will often come across a term called quantum current and even the infinite barrier will have a quantum probability current.

Edited May 23, 2017 by Mordred

[KipIngram]

Perfect - thank you.

[Mordred]

Your welcome

[studiot]

Are you aware of the two different meanings of quantum probability?

[KipIngram]

Hmmm. Maybe but I'm not sure. I just take it to be the probability that an interaction will occur. What's the other one?

[Mordred]

I'm not quite sure myself but the definition I'm familiar with is Von Neumann algebra "non commutative probability space" where as classical probability is commutative and satisfies the axiom if x and y are two real or complex random values then xy and yx are the same random variable.

[studiot]

We talk of (the modulus of the square of) the wave function as being the probability of finding the particle at a particular point.

But this type of probability is classical probability.

It is mutually exclusive.

When we look , we see the particle is here, or there, but not both, and each location with its own p-value.

This type of probability requires more than one experiment to encompass all the possible probability values at different locations.

That is each instance or experiment offers a particular value.

The sum of many individual experiments establishes a distribution pattern for the probabilities, but at any one time only one is in play.

This is of course because we are regarding the particle as a localised entity that moves about the region of interest.

 

The field approach says, hey wait a minute, the particle is actually non localised or smeared out in proportion to the localised p values in the region of space of interest.

This is how a field works.

 

In both cases the p values are the same and when normalised add up to 1

[Mordred]

k thats what I figured I described it by the commutations. However in essence the same

[studiot]

k thats what I figured I described it by the commutations. However in essence the same

 

This has implications both for your field view thread and the umpteen youngs slit threasd we see.

 

You need the field interpretation to develop the field as all the values are always present and real (all though they may be changing).

 

In the moving particle version only one value is ever real at a time, all the rest are potential values that are not realised.

[Mordred]

Exactly thats an excellent example. Made worse in that there is multiple values changing at each coordinate of the complex field which is composed of embedded fields example different overlapping Hilbert space. Another example being the electromagnetic, Higgs, strong force,weak force overlaps.

 

Each being a seperate field, each with their own values at every coordinate.

 

So if you measure one particular wavefunction you collapse that particular wavefunction without affecting other quantum wavefunctions of the particle.

Edited May 23, 2017 by Mordred

[studiot]

A better name for the 'probability' in the field interpretation is

 

The probability density.

 

There is a connection here with mathematics in as far as the field interpretation is a continuous function of probability density over the region of the field.

 

The particle interpretation is an impulse type function, everywhere zero except at the point of the particle.

 

Compare this with a mass density function in a gas v point function particle in classical mechanics.

Edited May 23, 2017 by studiot

[KipIngram]

So if you measure one particular wavefunction you collapse that particular wavefunction without affecting other quantum wavefunctions of the particle.

 

But each interaction / measurement has to cause at least one quantum's worth of action change to the involved fields, right? And I read Hobson to imply that that each quantum of action would come from a distinct mode of the field? And that the action comprising that quantum is actually spread out over all of space?

Saying the above in a different way: The fundamental "unit" of interaction is the transfer of one quantum of action from one field to another field, which affects both fields everywhere in space.

[Mordred]

correct little side note in QFT all operators fulfill the quanta of action. If you have a wavefunction or fluctuation/VP these use Propogators. So when studying the mathemstics bear this in mind.

 

Now connect that with Studiots comment regarding Normalized units.

 

c=g=h=1. Then look at a Feynman diagram. The external legs are operators. Your internal squiggly line propogators.

 

PS I have a lesson I plan on writing on how Feyman diagrams work on S matrix. I have all the details I need. Just need to simplify it

Edited May 24, 2017 by Mordred

[KipIngram]

What do you mean by "fulfill"? Are you noting that there's no concept of a wave evolving "by itself"? That in any evolution there has to be something playing the role of the operator (such as the propogator)?

[Mordred]

Bad wording meet the quanta of action requirement

[KipIngram]

Ok. So I think it's time again for me to try to tackle an introductory QFT book. I've tried before, but found myself "no sufficiently ready." I don't think the math was the problem; I think I was just totally missing the right way of thinking about it all. Maybe I can do better this time. Can you recommend any particular online reference?

[Mordred]

QFT lectures are a bit tricky to find but here is a series of lectures with notes posted on arxiv.

https://www.physics.harvard.edu/events/videos/Phys253

54 lectures if I recall.

Sydney R Coleman with Harvard University. though unfortunately poor video quality. is the

 

Here is the course notes.

 

https://arxiv.org/pdf/1110.5013.pdf

 

there is also this series

 

https://ocw.mit.edu/courses/mathematics/18-238-geometry-and-quantum-field-theory-fall-2002/lecture-notes/

 

A Simple Introduction to Particle Physics
Part I - Foundations and the Standard Model

 

 

https://arxiv.org/pdf/0810.3328.pdf

 

David Tong: Lectures on Quantum Field Theory

http://www.damtp.cam.ac.uk/user/tong/qft.html

 

PS past posters have usually enjoyed David Tongs series

Edited May 24, 2017 by Mordred

[KipIngram]

Ok, so I have a question that's nagging at me, but I'm not sure I know how to express it cleanly. I can picture electrons changing energy levels, quanta either going through slits or not, etc., but all of that imagery causes me to think of energy doing this. Which would mean it's energy residing (or not residing) in the modes of the fields. I can also think in terms of action when I set up an action minimization problem to find a trajectory. Thinking in terms of action is easy enough, because that integral definition of action is right there staring at me. But you talk constantly as though it's action that transfers from field to field in these interactions, and thus would be associated with the modes of the fields and so on, rather than energy. I know the definition of action, but I'm fuzzy on how it manifests in this picture I'm trying to build.

 

I do see that when I set up and solve an action problem I'm making reference to an initial state and a final state, and the action is associated with some possible trajectory between them. And in the quantum problems we're talking in terms of an initial state and a final state, and I sort of think we're not supposed to imagine the intervening "trajectory points" as even existing in a real way. We just start here and finish there. is that the right way to look at it? We start in this state, and end in that state, and in getting from here to there we have an integral number of action quana moving from "these modes in these fields" to "those modes of those fields"?

So, do we talk about action instead of energy because the possible paths from initial state to final state have action associated with them?

I'm watching the first of the Harvard lectures right now, btw - you're right; 70's video quality sort of sucked.

[Mordred]

Action is your coordinate displacement. In essence any interaction you can describe by a set of coordinates can be described by action.

 

"In physics, action is an attribute of the dynamics of a physical system from which the equations of motion of the system can be derived. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result"

 

Energy is simply the ability to perform work. It doesn't exist on its own but simply a property much like mass is the property resistance to inertia change.

 

The nice thing about action is it has the dimensions energy times time or momentum times length. So in several regards its a time saver in calculations as you've already factored in the f=ma relations.

 

This allows us additional flexibility in designing symmetry relations. Particularly under field treatments.

 

As a side note the relations between kinetic energy and potential energy in the action principle is incredibly useful and even when your not working under action is used in cosmology for example.

 

https://en.m.wikipedia.org/wiki/Equation_of_state_(cosmology)The scalar field equation applies these relations to describe the energy density and pressure. The numerator being pressure the denominator the energy density.

Even GR applies these relations for geodesic motion. (principle of least action) so its not restricted to QFT but also in classical fields.

Spin foam in Loop quantum gravity also uses action.

When you adopt the field philosophy to describe any and all interactions action is incredibly flexible and versatile but it is just a function. It is a mathematical methodoly not an entity.

 

Under coordinate changes its incredibly useful or kinematic motion particularly since all interactions are describable via differential geometry.

 

Just a side note I used to worry about what is fundamental. After years of study the only thing I could name was "change"

 

Everything else energy,mass,time,pressure, fields, etc etc are all devices to describe change. Beyond that well I simply don't have that answer.

Edited May 24, 2017 by Mordred

[KipIngram]

So is this generally on the right track? I'm trying to apply Feynman's path integral approach in the context of what we're talking about. We have the universe in some assumed initial state. So we 1) identify all the possible changes to that state that can occur by moving one quantum of action. Then 2) the probability of each of those outcomes is found by summing over all of the possible ways that change can occur. So now we have a set of possible next states and the probability of each, and that's all we can say, right? In step 2 we have to consider the cases where sub-quantum virtual fluctuations occur as well?

[Mordred]

Fairly accurate I don't see any mistakes in that descriptive.

[KipIngram]

Oh, maybe #1 should be "identify all the possible changes to that state that can occur by moving any integral number of action quanta (including zero)"? I assume "no change" would have some probability as well, and so should be included?

 

How does time enter into that? When the quanta move, they move instantaneously according to Hobson. I could see a logical path if I evaluated the probabilities every Planck time or something like that, but it's hard for me to see what to do with instantaneous changes in the context of a continuous time variable.

[Mordred]

I will have to reread the Hobson paper this eve to see what context he describes as instantaneous but it never truly is.

[KipIngram]

He stated that pretty strongly, I believe, by invoking the idea that if it took any time at all then there would be some time during when you had fractional quanta.