A few quantum mechanics questions

[ppyadof]

I was thinking about QM a few days ago and these are some questions which I finding difficult to answer based on what I already know about the subject (which is really the semi-classical ideas and mathematics of Bohr and the others before him, the Schrödinger Equation and the uncertainty principle).

Firstly, from spectral lines, it is said that an electron going around a nucleus can not be in between energy levels, in that it must be in one or another of these allowed orbits, but if it moves from one energy level to another, there must be a small amount of time in which it is in between these energy levels. If that is true (which it must be because of relativity), how long does it spend in between these levels, or is it impossible to determine that to a very high degree of accuracy because of the uncertainty principle?

Secondly, if the uncertainty principle is considered, then it implies that the energy levels around a nucleus are regions of much higher probability density than the areas between them, thus meaning that there is a probability associated with finding an electron between energy levels (much smaller than that associated with the energy levels, but there nevertheless). Am I correct in thinking that?

Lastly, from the Schrödinger Equation (either form of it), how would one go about modelling a real atom, for example hydrogen since it is the simpliest?

By the way, why is it that some places the uncertainty principle is given as $\delta x. \delta p > \hbar$ and others give it as $\delta x.\delta p > \frac{\hbar}{2}$ (I know there should be an equals line of the inequality, but I don't know how to put it on)? I know that the factor of 2 doesn't make much difference, but why is there that difference?

Thank you very much.

 

[Einstein Mcfly]

I was thinking about QM a few days ago and these are some questions which I finding difficult to answer based on what I already know about the subject (which is really the semi-classical ideas and mathematics of Bohr and the others before him, the Schrödinger Equation and the uncertainty principle).

Firstly, from spectral lines, it is said that an electron going around a nucleus can not be in between energy levels, in that it must be in one or another of these allowed orbits, but if it moves from one energy level to another, there must be a small amount of time in which it is in between these energy levels. If that is true (which it must be because of relativity), how long does it spend in between these levels, or is it impossible to determine that to a very high degree of accuracy because of the uncertainty principle?

Secondly, if the uncertainty principle is considered, then it implies that the energy levels around a nucleus are regions of much higher probability density than the areas between them, thus meaning that there is a probability associated with finding an electron between energy levels (much smaller than that associated with the energy levels, but there nevertheless). Am I correct in thinking that?

Lastly, from the Schrödinger Equation (either form of it), how would one go about modelling a real atom, for example hydrogen since it is the simpliest?

By the way, why is it that some places the uncertainty principle is given as $\delta x. \delta p > \hbar$ and others give it as $\delta x.\delta p > \frac{\hbar}{2}$ (I know there should be an equals line of the inequality, but I don't know how to put it on)? I know that the factor of 2 doesn't make much difference, but why is there that difference?

Thank you very much.

I think the major answer to your first two questions has to do with the difference between the electron "being at a distance x" and the electron being IN A QUANTUM STATE whose radial probability distribution gives and expectation value of x. In terms of the third question, what do you mean by "model"? The SE gives the correct values of energy for a "real" hydrogen atom and all other one electron atoms (He+, Li 2+ etc). What more would you like to "model"?

 

[Mr Virtual]

Firstly, from spectral lines, it is said that an electron going around a nucleus can not be in between energy levels, in that it must be in one or another of these allowed orbits, but if it moves from one energy level to another, there must be a small amount of time in which it is in between these energy levels. If that is true (which it must be because of relativity), how long does it spend in between these levels, or is it impossible to determine that to a very high degree of accuracy because of the uncertainty principle?

The electron does not "travel" from one energy level to another. It "jumps". What I mean by that is: it disappears from its present shell, and appears in the new shell. You can't observe it "travelling". Why? For the simple reason that QM forbids electron to be present in between these shells, and traveling means that the electron has the probability if being found in this forbidden space, which is wrong. I don't know "how" this jump occurs. I think no scientist has ever found out anything about it.

warm regards
Mr V

 

[Mr Virtual]

Secondly, if the uncertainty principle is considered, then it implies that the energy levels around a nucleus are regions of much higher probability density than the areas between them, thus meaning that there is a probability associated with finding an electron between energy levels (much smaller than that associated with the energy levels, but there nevertheless). Am I correct in thinking that?

No, you are wrong. There is exactly zero probability that the electron will ever be found in the spaces between the energy shells. Zero-probability regions like these are called "nodes".

 

[FrogPad]

Lastly, from the Schrödinger Equation (either form of it), how would one go about modelling a real atom, for example hydrogen since it is the simpliest?

It is a rather involved calculation. Check out https://www.amazon.com/dp/0073104647/?tag=pfamazon01-20

or here was a quick google search:
http://spiff.rit.edu/classes/phys315/lectures/lect_5/lect_5.html

 

[JesseM]

No, you are wrong. There is exactly zero probability that the electron will ever be found in the spaces between the energy shells. Zero-probability regions like these are called "nodes".

True, but you should clarify what you mean by "between the energy shells" -- the nodes are usually infinitely thin, so there would only be a discrete set of precise distances from the nucleus where the probability is zero.

 

[Mr Virtual]

True, but you should clarify what you mean by "between the energy shells" -- the nodes are usually infinitely thin, so there would only be a discrete set of precise distances from the nucleus where the probability is zero.

Oh! I am sorry. Nodes are actually zero-probability zones in case of orbitals, not shells.

So, what I actually meant was that the electron has zero probability to be found in the region between shells. So, it just "jumps" from one shell to the other, without actually traveling the distance between the shells. I am not sure, but this phenomenon is (perhaps) called quantum tunneling.

regards
Mr V

 

[JesseM]

Oh! I am sorry. Nodes are actually zero-probability zones in case of orbitals, not shells.

So, what I actually meant was that the electron has zero probability to be found in the region between shells.

Yes, but again, only in an infinitely thin region is the probability exactly zero. And according to http://www.bcpl.net/~kdrews/mtas/modern2.html an "orbital" is usually arbitrarily defined as a region where 90% of the probability is concentrated, so there is a 10% chance of finding the electron outside the orbital, just not at the precise position of the node.

 

[StatusX]

Firstly, from spectral lines, it is said that an electron going around a nucleus can not be in between energy levels, in that it must be in one or another of these allowed orbits, but if it moves from one energy level to another, there must be a small amount of time in which it is in between these energy levels. If that is true (which it must be because of relativity), how long does it spend in between these levels, or is it impossible to determine that to a very high degree of accuracy because of the uncertainty principle?

You have to keep in mind, QM says that it is meaningless to speak of the position, energy, etc of a particle. There's just the outcomes of various measurements, and the best we can do is quantify the probabilities of various such outcomes.

In this case, an electron in a stationary state around an isolated atom can reasonably be said to have a certain energy because a measurement of this energy is guaranteed to produce a certain value. But as soon as the electron is allowed to interact with the external world (eg, the radiation field), so that it may change state, this is no longer the case. There is only a certain probability distribution for the various energies, which changes over time. If we observe it some time later and find it has a new energy, it's meaningless to ask when it switched levels.

To reiterate, all there is is states with various probabilities of producing certain results when measured. There is no little electron dot moving around at a certain speed with a certain energy. We have this picture in our minds because for things like rocks or baseballs, the states have such a narrow range of possible positions, velocities, etc, that our brain models them as having exact values, but this is not a property of the actual world.

 

[Mr Virtual]

Yes, but again, only in an infinitely thin region is the probability exactly zero. And according to this page an "orbital" is usually arbitrarily defined as a region where 90% of the probability is concentrated, so there is a 10% chance of finding the electron outside the orbital, just not at the precise position of the node.

Yes, I agree with you. It was my mistake. So an electron does have a probability to be found in the spaces between the shells, although the probability is very low. In this situation, though, the electron is quite unstable and, thus, it again comes back to its 90% probability region.
The node is actually that plane/point of space where the unstability of the electron is so high that the probability is zero.
But I have a question here. If the probability of finding an electron in a node is exactly zero, then the electron should never be found in the nucleus. However, QED says that there is a small probability that the electron may be found in the nucleus. Does that mean that even the node does not possesses 0 probability?

regards
Mr V

 

[jtbell]

If the probability of finding an electron in a node is exactly zero, then the electron should never be found in the nucleus.

The electron's probability distribution does not have a node at the nucleus, in general.

 

[Mr Virtual]

I don't know. Maybe you are right. But if you inspect http://www.bcpl.net/~kdrews/mtas/modern2.html , you will notice that it says that a nucleus is a node too.

regards
Mr V

 

[Moridin]

Look up electron capture. It is possible because electrons in the lower energy levels have a certain probability of being at the nucleus according to solutions to the Schrödinger equation.

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html#c3

 

[Feynman diagram]

Bear in mind that the energy levels in an atom are not nice little spheres surrounding the nucleus as it is tempting to think of it. The shells of different energy levels have different geometries and overlap in many instances, allowing the electron to transit from one shell to another in a 'classical' way (i.e. without tunneling from one shell to another), although of course tunneling occurs nonetheless in most situations.

 

[jtbell]

if you inspect http://www.bcpl.net/~kdrews/mtas/modern2.html , you will notice that it says that a nucleus is a node too.

That's wrong, as a general statement. Look at the actual hydrogen wavefunctions given on this page (scroll down to the sections labeled "Normalized Hydrogen Wavefunctions"). The wavefunctions for 2p and 3d states do indeed have nodes at the nucleus, because their values are zero at r = 0. However, the wavefunctions for 1s, 2s, 3s and 3p states do not have nodes at the nucleus, because their values are nonzero at r = 0.