Questions on Electron Orbitals, Wave Packets, and Coherent States

[logic smogic]

I'm trying to interpret a research paper on single photon ionization using extreme UV attosecond laser pulses, and I realized I have some very basic questions concerning electrons in an atom. Technically, the answers to these questions are found online and in the literature, but the wording always tends to be the same - and it's just not doing it for me. A short answer to any of these in your own words would be greatly appreciated!

1.) Do bound electrons in an atom follow a closed Keplerian orbit around the nucleus?

2.) Can/should an electron (bound or unbound) always be described as an "electron wave packet"? Is a bound electron in an atom always a "Gaussian wave packet" with minimum uncertainty that is nondispersive?

3.) Say I shine laser light on a Hydrogen atom.
a.) If the frequency of the light is too low, the electron will need to absorb many photons to gain enough energy to tunnel through the Coulomb barrier and ionize the atom. How is this explained in the wave-packet formalism? Is there a relationship between coherent states and whether or not the electron overcomes the Coulomb barrier?

b.) If the frequency is high enough (extreme UV range), then the electron may absorb enough energy to immediately escape the Coulomb potential barrier. How will the electron wave-packet after ionization be different in this case than in the case of part (a)?

Any answers (even partial) to these questions would be extremely helpful!

 

[logic smogic]

Alright - it seems I didn't search through enough threads to arrive at some of these already-addressed questions. Here are some of the answers I've found so far.

1.) Do bound electrons in an atom follow a closed Keplerian orbit around the nucleus?

"Electrons DO NOT ORBIT the nucleus. If they orbited, the acceleration of the charge would produce photons making the electrons lose energy and fall to the nucleus."

-yourdadonapogostick

2.) Can/should an electron (bound or unbound) always be described as an "electron wave packet"? Is a bound electron in an atom always a "Gaussian wave packet" with minimum uncertainty that is nondispersive?

"The electrons are in stationary states --- there is no change in their probability distribution with time. Motion, especially trajectory, is not a defined concept in quantum mechanics."

-genneth

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Strangely enough, I was told today by a prof that electrons "do" move on a picosecond timescale - which is why femto- or attosecond laser pulses are needed to cause ionization.

He also mentioned something about "slamming" the electron into the "ion-core", and observing the coherent state or resulting electron. Any idea what he may have been getting at in light of the context?

Thanks.

 

[lbrits]

1.) No, their position is undetermined but the probability density is described by (the square of the) wavefunction, which assigns a complex number to every point in space.
2.) A wave-packet is an extremely general type of wave-function. Essentially, any wavefunction that you can normalize (square, integrate over all space) but usually just means that it is localized. In other words, the electron is most likely to be found "near" one spot. The minimum uncertainty gaussian wavepacket is very special and most wavefunctions can be quite complicated.
2.b) I don't recall the Gaussian wave packet being non-dispersive
3.) The "coulomb barrier" gives rise to a number of discrete energy levels, and a bound electron could (and generally does) have a definite energy. For the electron to absorb a photon, the energy of the photon has to equal (or be very close to) the difference between two energy levels (otherwise it will just flop back and forth between them). This is best understood in terms of perturbation theory. Now, the electron can undergo a number of transitions to get out of the energy well, or it can do it all at once. You seem mainly interested in how the wavefunction changes... Well, you can solve the Schrodinger equation and see what the wavefunction looks before hand. You then do perturbation theory to see how the wavefunction changes with time...

Take a look here: http://www.falstad.com/qmatomrad/

When the electron completely escapes, it will probably have a much better defined momentum since it's position is no longer very localized, whereas if the electron stays bound, both momentum and position have some uncertainty to them.

 

[lbrits]

"The electrons are in stationary states --- there is no change in their probability distribution with time. Motion, especially trajectory, is not a defined concept in quantum mechanics."
mmm... Only applicable to electrons in stationary states, and not to electrons in general (e.g. undergoing transitions, interacting with light, etc.). Even for electrons in stationary states, this only means that their energy is well defined.

---
Strangely enough, I was told today by a prof that electrons "do" move on a picosecond timescale - which is why femto- or attosecond laser pulses are needed to cause ionization.
----
Well, ordinary light causes ionization. I think the point of the high speeds is to get high resolution. I'm not sure, though.

 

[logic smogic]

1.) No, their position is undetermined but the probability density is described by (the square of the) wavefunction, which assigns a complex number to every point in space.
2.) A wave-packet is an extremely general type of wave-function. Essentially, any wavefunction that you can normalize (square, integrate over all space) but usually just means that it is localized. In other words, the electron is most likely to be found "near" one spot. The minimum uncertainty gaussian wavepacket is very special and most wavefunctions can be quite complicated.
2.b) I don't recall the Gaussian wave packet being non-dispersive
3.) The "coulomb barrier" gives rise to a number of discrete energy levels, and a bound electron could (and generally does) have a definite energy. For the electron to absorb a photon, the energy of the photon has to equal (or be very close to) the difference between two energy levels (otherwise it will just flop back and forth between them). This is best understood in terms of perturbation theory. Now, the electron can undergo a number of transitions to get out of the energy well, or it can do it all at once. You seem mainly interested in how the wavefunction changes... Well, you can solve the Schrodinger equation and see what the wavefunction looks before hand. You then do perturbation theory to see how the wavefunction changes with time...

Take a look here: http://www.falstad.com/qmatomrad/

When the electron completely escapes, it will probably have a much better defined momentum since it's position is no longer very localized, whereas if the electron stays bound, both momentum and position have some uncertainty to them.

Thanks for the response and the link, lbrits.

When you say "the energy of the photon has to equal (or be very close to) the difference between two energy levels (otherwise it will just flop back and forth between them)" - it brings to mind my confusion over coherent states.

If the electron is in a coherent state, is it a superposition of two energy levels (caused, perhaps, by the absorption by the electron of a photon with energy that does not correspond exactly to the difference between two energy levels) that corresponds to a simple harmonic motion of the spatial atomic orbital cloud? In other words, can I think of those "balloon" like orbital clouds as moving slowly with simple harmonic motion if the electron is in a coherent state?

Also, why does the mere action of the electron escaping cause it's position to be less localized? Is it just because when it is bound, we know it must exist in a certain volume, and when it has escaped, we have no idea which direction it left the atom in, and hence the volume that it could exist in is now much larger? Or does it have to do with the physical motion of the wave packet through space in an unbounded system?

 

[logic smogic]

Strangely enough, I was told today by a prof that electrons "do" move on a picosecond timescale - which is why femto- or attosecond laser pulses are needed to cause ionization.
----
Well, ordinary light causes ionization. I think the point of the high speeds is to get high resolution. I'm not sure, though.

Ordinary light would normally cause ionization only through multiphoton ionization, right? You would need very high frequency light (extreme UV) to ionize an atom in "one kick", without intermediate virtual states - right? Just want to make sure I'm understanding all of this.

 

[lbrits]

Ordinary light would normally cause ionization only through multiphoton ionization, right? You would need very high frequency light (extreme UV) to ionize an atom in "one kick", without intermediate virtual states - right? Just want to make sure I'm understanding all of this.

Yes.

With regards to your other question, I'm not sure what you mean by coherent state in this context.

As to your other question, I'm sorry I kind of "intuited" this and I guess it is bad. I assumed that the atom is under the influence of monochromatic radiation with definite momentum, and kind of went from there. But I guess you could cook up some other scenario. It's just that the numbers that you usually treat as well defined, energy and momentum, will remain well defined, and so the position will be poorly defined.

 

[logic smogic]

Okay, my understanding on this topic has developed quite a bit this evening, and below are the posts I made documenting that development. Skip to the bottom to see my most recent remarks on the topic.Post #1-

With regards to your other question, I'm not sure what you mean by coherent state in this context.

I suppose I was referring to the definition as used in these articles,
http://en.wikipedia.org/wiki/Coherent_state
http://webphysics.davidson.edu/Projects/AnAntonelli/node46.html

It's not clear to me why we could ever assume that by shining light on an atom, the electron would assume a coherent state - unless what I said about it being a superposition of states corresponding to a certain energy is true - in which case I can see how one might tune the incident laser light to force a coherent state.

But then, why would this be at all useful in ionizing electrons? Perhaps it's not, but it's useful in controlling the location of the electron, should you wish to "videotape" it, as the authors of this paper are trying to do (referring to my original post).

I suppose I'm mostly confused on why coherent states are ever "useful" for anything. Sure, they correspond in some sense to classical harmonic oscillators - but in the world of quantum mechanics, where we don't think about trajectories, etc, and just probabilities and observables, how does this simplify things? I don't expect anyone to really answer these questions (they're mostly ramblings at this point), but they hint at the source of my confusion.Post #2-
Okay, after a bit more online research (and a little thinking), I've compiled all of this into the example of the Hydrogen atom. Below I've restated some conclusions from the above posts, and what I've just learned about coherent states.

A Hydrogen atom contains a nucleus (proton) and a bound electron. The electron is in a stationary state, where its probability distribution does not evolve with time. This is because the electron is bound by the Coulomb potential, which gives rise to a number of discrete energy levels. Occupation of any of these energy levels corresponds to a stationary state.

If the electron absorbs a photon, its state will change depending on the energy of the absorbed photon. If the photon’s energy exactly corresponds to the distance to a higher allowed energy level of the electron, then there is a certain finite probability that the electron will jump to that higher energy state. If the photon’s energy is anything else, then the electron will assume a superposition of energy eigenstates. Certain (special) superpositions correspond to “coherent states.”

A free electron wave packet will spread out over time because the different momentum components move at different speeds. When an electron is in a coherent state, it is because the potential (in this case, the Coulomb potential) is somehow keeping the wave packet together, that is, keeping it a minimum uncertainty wave packet at all times. Physically, a coherent state is left unchanged by the particle’s detection. This is why it is experimentally useful.Post #3-
An excerpt from the article of interest is given below, followed by at least one question I have from the text:

"...Further control of the electron dynamics requires that the creation and acceleration of the electron wave packets (EWPs) are decoupled; this is not possible using tunneling ionization since the same laser field governs both events. Decoupling can be achieved by using extreme UV (XUV) attosecond pulses to create temporally localized EWPs through single photon ionization at a well defined phase of a synchronized IR field which drives the dynamics from that point forward. These attosecond EWPs are distinctly different from their tunnel ionization counterparts. They are born at the center of the potential well with properties that are directly inherited from the XUV pulses, which can be tailored in time and frequency. They can also have a nonzero velocity, and their subsequent dynamics can be controlled by choosing the phase and amplitude of a synchronized IR field appropriately. In particular, the laser field needed to drive these EWPs back to the potential is usually weaker than the laser field needed to form tunnel EWPs, leading to much less distortion of the properties to be studied.

Here, we demonstrate an attosecond quantum stroboscope capable of capturing electron motion on a subfemtosecond time scale. This technique is based on a sequence of identical atttosecond pulses which are synchronized with an IR laser field. The pulse to pulse separation in the train is tailored to exactly match the optical cycle of the laser field, and the electron momentum distributions are detected with a velocity map imaging spectrometer (VMIS). This technique has enabled us to image the coherent scattering of electrons that are driven back to the ion by the laser field following their ionization. We envision that coherent electron scattering from atoms, molecules, and surfaces captured by the attosecond quantum stroboscope will complement more traditional scattering techniques since it provides high temporal as well as spatial resolution."

-J. Mauritsson, et al, “Coherent Electron Scattering Captured by an Attosecond Quantum Stroboscope”, Physical Review Letters 100, 073003

Question: Any ideas on what might be meant by “coherent scattering” in this case? Am I to imagine that we are only interested in electrons that are “bounced” off of the ion-core and that exit the atom with energy equal to the XUV laser (i.e., coherent scattering)?Post #4-
Further, here is an excerpt from a summary of the article in February 2008's edition of Physical Review Focus:

"...The pulses in the train were just three hundred attoseconds (10-18 seconds) long. The researchers synchronized the pulse train with the oscillations of a relatively weak infrared laser, so that their cloud of helium atoms received a strong, ionizing "kick" at a precise time during each laser cycle. Each attosecond pulse released a few electrons, some of which were thrown back against their atoms before being pushed sideways and detected."

The question here is: what is important about throwing the electron back against the atom before it is whisked away for detection?Post #5, Final-
My interpretation of the excerpt shown above:
1. A weak IR-frequency laser is shown on a cloud of atoms. This causes the electrons to "wiggle" back and forth in the direction of the laser's E-field at at the laser's frequency, but is not enough to ionize the atoms or excite the atoms to higher states.
2. A train of evenly-spaced XUV attosecond laser pulses are directed at the cloud of atoms.
3. The pulse-spacing is matched to the frequency of the IR laser so that an electron in a given atom will be ionized (via single photon ionization) at a crest or trough of the E-field oscillation.
4. Depending on the timing in part (3), the electron will be driven back through the ion-core and towards a detector.
5. The electron's initial momentum after being ionized is well-defined and based on parameters of the XUV laser, and the electron's momentum is also recorded on a detector after ejection from the atom.

Anything seem wrong or missing in that analysis?