A samll question about orbital angular momentum of an electron

Discussion in 'Physics & Math' started by Secret, Apr 20, 2014.

  1. Secret Registered Senior Member

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    We knew that in quantum mechanics, an electron can be described by a wavefunction, which via the Born probability interpretation, describes the probability of finding the electron. Thus an electron is like a probability cloud

    We also know from classical mechanics that angular momentum is mass*torque x perpendicular distance
    We also know from classical electromagnetism that a moving charge generates a magnetic field according to Faraday's law, hence magnetic moment

    I often read form various chemistry text (e.g. Housecroft and sharpe 2nd edition) that if we assume the electron orbit the electron classically, then the electron possess an orbital angular momentum and this give rise to a magnetic moment

    But if electrons are probability clouds, how can we understand how this cloud 'rotate'

    I know that electron spin is something discovered form the Stern Gelald experiment, thus there isn't much explanation on what it actually 'looks like', but for orbital angular momentum, what physically is happening to the electron cloud that give rise to the orbital angular momentum? (other than describing the electron is rotating around the nucleus, since this picture does not capture the probability cloud properties of an electron)?
     
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  3. Farsight

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    IMHO you should forget about probability and think wavefunction or waves. Remember that you can make an electron (and a positron) out of light in gamma-gamma pair production. Then in electron-positron annihilation you get the light back. In between these two processes you can diffract electrons. And in atomic orbitals (see Wikipedia) electrons "exist as standing waves". Then look to Dirac's belt wherein "a Mobius strip is reminiscent of spin-1/2 particles in quantum mechanics, since such particles must be rotated through two complete rotations in order to be restored to their original state". As for what it "looks like", think spinor and think of it as something like this. But without the outer surface. It doesn't have a surface just as a whirlpool doesn't have an edge.
     
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  5. OnlyMe Valued Senior Member

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    There you go again, Farsight! You state the above in bold as if it were something that has been directly observed rather than theorized.

    I am pretty sure I have asked you before for a direct reference that experimentally demonstrates any photon-photon interaction. Gamma pair production remains.., as a function of experience, an interaction between a gamma photon and a nucleous.

    Again, do you have some experimental reference that demonstrates any photon-photon interaction, let alone a gamma-gamma interaction?
     
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  7. Trippy ALEA IACTA EST Staff Member

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  8. CptBork Valued Senior Member

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    Actually it's mass*velocity*distance perpendicular to velocity (taken with respect to a specified point), give or take a negative sign. Just wanted to clear that up for everyone.

    You ask some very good questions. I'd like to first point out another correction, which regards your claim that the Stern-Gerlach experiment was the first indication of instrinsic electron spin. In fact, electrons were already predicted to have quantized spins a few years earlier, I think either by Wolfgang Pauli or Enrico Fermi, as an attempt to use Arnold Sommerfeld's early quantum model (an extension of Bohr's model) to explain hyperfine splitting in the hydrogen spectra and the increased pronunciation of this split when hydrogen atoms are exposed to external magnetic fields.

    As to the question itself, when envisioning orbital angular momentum, an electron with orbital angular momentum is a probability wave in which each piece of the wave has a momentum to orbit around some axis, so you can basically envision it like a rotating cloud. On the other hand, spin angular momentum is a property belonging to point particles with no length, width or depth -although they have a probability of being detected in any region of space- and quantum spin doesn't manifest itself in many meaningful ways other than with phenomena such as magnetic field interactions, or tendencies to cause or participate in certain subatomic reactions/decays over others, and in its ability to be converted to and from the more intuitive orbital form of angular momentum we're more familiar with on a classical level. So basically, QM simply allows for some strange counterintuitive stuff like spinning point particles, without providing any deeper explanation for how or why such things exist.
     
  9. lpetrich Registered Senior Member

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    I'll discuss QM angular momentum in a rather sketchy fashion.

    One gets the position of a QM particle by applying a position operator to its wavefunction: (wvfn).(position operator).(wvfn) integrated over space.

    One does likewise with a momentum operator.

    One composes an angular-momentum operator by doing (position operator) x (momentum operator), in analogy with classical angular momentum. It gets angular-momentum values like the other operators I've mentioned.

    Strictly speaking, this is orbital angular momentum. One must add spin angular momentum, something derived from the space-time structure of the elementary-particle fields. Both orbital and spin angular-momentum operators are related to rotation generators, generators of rotations of space-time and the particle fields.

    There is a very elegant formalism for getting angular-momentum states, the ladder-operator formalism. It works for both orbital and spin angular momentum, and any combinations of them.
     
  10. exchemist Valued Senior Member

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    OK I think the first thing is is to distinguish between orbital and spin angular momentum. "Spin" is something intrinsic to electrons that is not related to the probability cloud one sees in an orbital.

    Then, think of the number of phases of the wave you encounter in tracing a path once round the nucleus. In an s orbital there are no changes of phase. This signifies that there is no orbital angular momentum in an s orbital. In an s orbital the electron can be thought of as swinging like a pendulum from one side of the nucleus through it and out to the other side and back. It has done the QM equivalent of "falling into" the nucleus though, due to QM zero point energy, it has too much energy to stay in the nucleus but instead comes out the other side.

    In a p, d or f orbital you encounter phase changes as you traverse a path round the nucleus, signifying that an electron in one of these states circulates, in some undefined way, round the nucleus, so as to give it angular momentum. Notably, there is in all these states a node of zero electron probability at the nucleus, meaning that the electron cannot approach the nucleus in the way it does in an s orbital. This is due to its angular momentum. The more phase changes in a circuit round the nucleus, the more angular momentum it has.

    The classical counterpart of an s orbital is one is which the electron has fallen into the nucleus and the counterpart of a p, d or f orbital is an orbit in which the angular momentum of the electron prevents this from happening.
     
  11. Secret Registered Senior Member

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    I am not good enough on spinors to understand this one, except that If my memory serves orbital angular momentum (the question in OP) has nothing to do with spinors

    Well these only described the orbital angular momentum as something that pops out from the Scrodinger equation when you plug in the hydrogen atom coloumb potential, and the vector diagrams just representing it without saying how the angular momentum (represented by the vector) is generated by the electron cloud physically. He does put the precession of a magnetic moment as an analogy, but that's only how it interacts with the field, not how it is produced

    1. Ya, my bad, rusty memory

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    . I was thinking in terms of r x p and rmb that p=mv for non relativistic, but somehow got muddled up in the middle of the thinking and it becomes mtau x r

    2. I did not aware that, probably because my uni quantum 2nd year course skimmed through the Old Quantum Theory as all we need to know is just to be aware that it exist.

    3a. That's an interesting way to understand it. It also kinda fits the thinking of a physicist in how physicists usually understand a complex thing by taking it apart into identical small pieces, understand the individual pieces, and the big thing is just the sum of its parts

    3b. Bolded part: Probably why the quantum world is so weird. But what's even weird, is why the maths (Scrodinger equation) seemed to work perfectly for the majority of the cases, as it is (according to the 2nd year course) is basically conjectured based on De Brogile's wave particle duality hypoethesis (experimentally supported ever since it is proposed)

    3c. Bolded part: I am confused, how does spin converted into the classical angular momentum, and back?

    1. Yup I have came across these in my 2nd year. Since the integral is basically summing infinitestimal bits, and that a postulate of quantum mechanics is that a wavefunction contains all information you can know about the particle or a system of particles, is the following a good way to think about the operator maths in a physical picture? (Ps I haven't learnt the ladder operator formalism yet, unless it is the same as the operator stuff form my 2nd year)

    i.e. int_all space((wvfn)(operator)(wvfn))=For the wavefunction that spans all space, the operator extract a physical quality from every point in the wavefunction. This is then summed together to form a 'profile' of this physical quality of the waveunction (e.g. if operator is (linear) momentum operator, then what you get is basically a distribution of momenta throughout the wavefunction?)

    If this is the correct way to think about it, then I can see how the maths ties with the "rotating cloud" explanation by Cptbork

    2. My hazy memory told me that spin arises when SR is applied to quantum mechanics, though I have not learnt the maths tool to came across the details yet to understand how it works (even mathematically). As for the bolded comment, it reminds me of this
    http://www.sciencedaily.com/releases/2011/03/110319085514.htm
    But otherwise I know too little about quantum field theory to understand the bolded part

    1. yup

    2. This remind me back in 1st year, they have introduced a classical analogy between the orbital shape, azimuthal quantum number in the form of swinging a noose, and then a figure 8 hoop and then a clover shaped hoop (which classical angular momemtum increases as the hoop get more loops), but that the analogy does not reflect the details of what happens physically in the real quantum thing (the orbitals) as described by the maths

    Although I understand how your description can help picture how the orbital angular momentum is produced, one thing that puzzles me is why the electron can be thought of swinging a like pendulum in the atom, since the coloumb potential does not look like (similar) a quantum SHO thus hooke's law does not apply?
     
  12. lpetrich Registered Senior Member

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    "Spinor" is short for "spin vector". It's something like a vector, but for spin 1/2 instead of spin 1.

    A 3-space vector has 3 components, Vx, Vy, Vz.

    These can be rearranged using the angular momentum projected in the z direction:
    • Up, +1, Vx + i*Vy
    • Side, 0, Vz
    • Down, -1, Vx - i*Vy
    A spinor has only
    • Up, +1/2
    • Down, -1/2
     
  13. exchemist Valued Senior Member

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    Sure it's not SHM. All I meant was to convey was the idea of motion that interchanges potential and kinetic energy, without any net circulation around the nucleus.
     
  14. Billy T Use Sugar Cane Alcohol car Fuel Valued Senior Member

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    The Stern-Gerlach experiment showed that the already known spin of electrons also had a magnetic moment and that its "z component" was quantized. I.e. in an external magnetic field it was either "up" or "down." The experiment was actually done, first with a beam of thermal energy silver atoms but any atom with "un-paired" electrons will show that too.

    I'm sure it was Pauli who announced the exclusion principle than bears his name. The repeating nature of the "periodic table" was well know to Chemists in the 1800s. Pauli was one of the first, if not the first to explain it. I.e. an electron in the principle quantum level, n =1 , if I remember my QM, from more than 40 years ago, cannot have any anglar momentum, L, must be with that quantum designation, "s" This is following the QM rule L = l < n where l = s = 0 for n =1. If n = 2 then l can be either 0 or 1 and these states are called, s & p. For n = 3, the L can be 0, 1 or 2. and that new state in specrosocpic terminology is d. Generally speaking a great deal of atom spectra data was available before QM. Some spectral lines were nearly the same wavelengths - i.e. doublets (that is were the d name comes from) and some were singlelets (that is were the s comes from) and I forget the German word that gave birth to the p designation. The two slightly different, bright yellow lines of Sodium are still called the "sodium doublet."

    Pauli noted that hydrogen atom had in its ground state one n=1 & L = s = 0 electron and that could radiate (or absorb) with that electron falling form (or boosted to) the n=2, L = p = 1 level but not to the n=2 L= s = 0 level. I.e. there were known lots of these "radiative selection rules" known. This is the one telling that L must change by 1. I.e. conservation of angular momentum requires it as the phono has one unit of angular momentum is carries away or gives to the atom when it is absorbed.

    Chemisty and the periodic table are telling the Pauli principle (no two electrons in same atom can have all their quantum numbers the same). So the first row should have only one, hydrogen, as the ground state has n=1 s can only be 0, no angular momentum, but He is there too also with n= 1 & s=0.

    Likewise there should only be four, not 8 possible outer electrons in the second row of the periodic table (n = 2 s= 0), (n = 2, L = 1, and m = 1). (n = 2, L = 1, and m = 0), (n = 2, L = 1, and m = -1) where m is the projection of component of the angular momentum along the angular momentum axis of the orbiting electron. That quantum number, m, is quantized too.

    Etc for the third row. Pauli suggested there was yet one more quantum number, now called the spin, that could take only one of two values, now called up and down. With this guess the first row could have two elements, the second 8 etc.

    So Pauli to make his exclusion principle conform to nature, postulated that electrons have a binary valued quantum number, we now call spin. I.e. Pauli invented the concept of intrinsic spin.
     
  15. Billy T Use Sugar Cane Alcohol car Fuel Valued Senior Member

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    Now to answer, Secrete's OP question:

    Yes in some classical attempt you can consider the electron in some orbital to be a cloud, but then you are hard pressed to understand why, for example the negative cloud, strongly attracted by the positive nucleus does not shrink into it. A better view, if you must think in classical terms, is that it is spinning so fast the centripetal force balance this attraction like it does to keep Earth from falling into the sun. That works pretty well for orbit with L > 0 but the s orbit (L=0 or no angular momentum) in classical POV goes right thru the nucleus.

    Best is not to even try to ram nature into your limited (by experience) human knowledge frame. Just accept the QM descriptions that do describe nature well. I. e. the bound energy levels of electrons "tied" to positive nucleus are quantized, in energy, angular momentum, its component along that momentum axis and how it can interact with units of the high frequency EM field (photons). That electrons do have "intrinsic spin" a quantized quantity, perhaps badly named to encourage silly attempts to ram it into human directly learned concepts.
     
  16. Secret Registered Senior Member

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    Yup I know but there's some more math details about spinors which I guess I just have to be patient until I learn the subject formally
    The moebius strip model mentioned by Farsight (skipped) , which is also mentioned in Wikipedia and some other introduction of spinors in some pop science magazine and some other physics websites although can capture one of the main properties of spinors intuitively (i.e. how can a 360 deg rotation flip the sign), it failed to capture the rest of the math of spinors and how they behave

    So you mean the kinectic energy and potential energy for the electron interconvert just like that in the pendulum, and for the l>0 orbitals, part of the kinectic energy is rotational kinectic energy, sorta like as a rough analogy, a precessing pendulum (like those which is used to show that earth is rotating)?

    Thanks for the more detailed background about the history of Pauli Exclusion Principle. In our 2nd year courses in quantum physics, they pretty much skipped the pauli exclusion principle, fermions and bosons due to syllabus changes. In chemistry, they only stated the significance, but not the observations and history that lead Pauli to propose that there is something called spin and that it must have only two values

    1. Actually I never have trouble with the concept of quantized energy levels, because I, my friends and some professors often relate it to a very nice an accurate analogy of "going up and down the stairs"(1) and "energy barriers"(2), that is, for analogy (1), you cannot really stand between steps, and (2) It only jumps if you supply just the right energy, too little, then not enough to made the jump, too much and it will just pass right through and do nothing, and nature is kinda "do it or not do it" kind of entity

    2. I have no problems of these, the issue in the OP is needing an accurate and intuitive enough picture to understand HOW the orbital angular momentum arises from the electron wavefucntion (which chemist like to think of it as a cloud because of the concept of orbitals), while such explanation still contains pretty much most of the maths behind it so that it is easy to understand and also sufficiently accurate in describing what the maths is describing. (Again I am ok with the maths, cause we been through solving the scrodinger equation for the hydrogen atom back in the uni course by solving the differential eqt in spherical coord.

    so far it seems Cptbork's rotating cloud explanation is intuitive and accurate enough to explain what happened, and if I understand the operator notation properly, seems to also closely matches what the maths is describing

    IMO A good and accurate explanation is something that is not necessary classical, and might sound counterintuitive, but as long it is logical and you knows how in the explanation A leads to B, then it is good enough. so in short, I am not demanding the explanation must be classical, but it must be (trying to phrase it correctly, I know it is hard to express this) easy to follow via first principles

    e.g. Back in my 2nd year course when my professor taught about the quantum SHO, I asked about what freq in the quantum SHO means physically as you cannot say the particle is oscillating since you cannot pinpoint exactly the position and momentum of a particle at any given time. He explains that the best way to understand it is that while freq measures the rate of oscillation in the classical pendulum, in the quantum case it only measures the energy of the particle (using de brogile, which I am fine with the explanation because of how I understand form the course that it is a hypothesis from something we observed experimentally that is later supported by more experiments, thus it is "easy to follow"). After discussing with him more about the quantum SHO, I came up with a picture which the pendulum is like a fuzzy cloud where particles can be found in some places more readily than another

    PS to elaborate 2. my situation is like
    If I try to explain to a friend how orbital angular momentum in atoms give rise to a magnetic moment as
    1. Something that pops out of an equation known as scrodinger equation which then have an associated magnetic field
    2. Something that is quantized (i.e. in bits, rather as something continuous) and interacts under a magnetic field and photons in some ways <insert the description of the ways> because of how it possess a magnetic field

    He/she will look at me wide eyed with confusion and will next ask, So how EXACTLY is the magnetic field arises from the orbital angular momentum?

    If I then explain it using analogy (assuming he/he knows a little bit of classical physics, (umm I suppose moving charges generate magnetic field is a daily knowledge, right?)
    3. Classical orbital angular momentum is what you get when you have something orbiting another thing, which is proportional to the distance between the objects and how fast it is orbiting. A moving charge will generate a magnetic field
    Quantum orbital angular momentum is similar, except that you cannot think of an electron as orbiting a nucleus like a planet is orbiting, this then give rise to a magnetic field

    Then he/she will question me with a barrage of question (as expected when one thinks classically), and also ask then how EXACTLY it orbits?

    If I then explain it using the rotating clouds suggest by Cptbork (or similar explanation that are easy to follow), then he/she will nail the concept, and then when he/she actually formally did a quantum course, it will be easier for him/her to relate the maths with the concepts

    ::Safety measure statement::
    I am not saying that you can throw away all the maths with these easy to understand yet highly accurate explanations, since you still need the maths to compare and investigate the physics in detail

    :

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    ossible diversion that will be made into a thread later::
    The discussion in this thread have caused me to wonder about the analogy of wavefunctions. More details on that query later
     
  17. exchemist Valued Senior Member

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    Yes KE and electrostatic PE interconvert, due to the (undefined) motion of the electron in all cases. In l=0 orbitals however, there is no net rotational component of the motion, hence no angular momentum and hence no orbital magnetic moment.

    I've always found helpful the fact that in orbitals where l>0 there is a node at the nucleus, meaning the electron cannot go right up to it, which makes intuitive sense if it is in pseudo-orbit, whereas by contrast, in s orbitals the electron density persists right up to the nucleus.

    (This "penetration" by s orbitals into the space adjacent to the nucleus exposes electrons in s orbital to more of the nuclear charge than p d or f electrons in the same shell experience. This has major consequences in chemistry, due to the effect on the relative energy levels of subshells as one goes up the Periodic Table, and thus the sequence in which they are filled (AufbauPrinzip).

    One of the objections to the Bohr model of the atom was that without angular momentum an electron should by rights fall into the nucleus. It seems to me one way to think of s orbitals is that these represent a quantum version of exactly that condition, and the reason they don't fall in - and stay in - is that in QM there is zero point energy, i.e. there is a bottom energy state that the system cannot go below, due to the wave nature of matter (the fundamental mode of resonant vibration).

    Also, regarding the orbitals with angular momentum, if one associates the number of nodes with successively higher harmonics, there is intuitive sense in the idea that with l=1 there is one nodal plane, with l=2 two nodal planes and with l=3 three, i.e. the more "units" of circulatory motion, the higher the "rotational" harmonic you have.

    Anyway, these are just some pictorial ways that I as a mere chemist have found useful to visualise what is going on.
     
  18. Secret Registered Senior Member

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    299
    1. Ok, except that the undefined explanation need to be swallowed

    2. Umm, can you elaborate more about pseudo-orbits, cause I don't know how to picture that?
    As for the rest, yes, one of these consequence is 4s filled before the 3d when you build up the ground state electronic configuration of an atom, which affects the transition metal chemistry

    3. I found the "quatized energy level" explanation more complete, as that will also explain why it does not even spontaneously fall in the first place to some other lower levels actually scrap that, it's wrong, as we are dealing with the ground state

    4. Bolded: That's a really good way to interpret the azimuthial quantum number, and it ties well with the figure 8 knot analogy. Now I have more ways to recall the sequence of l just in case my memory failed me in exam

    5. I am doing both chemistry and physics in uni, thus I have foundations in both subjects, and know the way a chemist and physicist think (although I am weaker in the physicist's mind since in that respect I am more mathematician like, often questioning every math detail that got left off in the physics proofs and also concern about infinities and zeros and how these quantities in the equation manifest themselves physically (i.e. unlike physicists, I don't hate infinities, as long they are meaningful in the equation in describing something real). I also read about and discuss with others about maths during my free time, so I am alright with the maths (save for careless mistakes in calculations).

    I can do long algebra if that is necessary to get a concept, but don't tell me to do pages and pages of calculations that make use of huge amount of data inputs (that is, instead of symbols, a very long number in scientific notation with n decimal places, and they are m of these!), as that is too dry and I will pass out

    I also have some foundations in evolutionary biology and biotechnology,molecular biology thanks to the biology course I took in 1st year and I have been trying my best to keep in touch with my biological side of the learning by reading science magazine, chatting with biology friends and sometimes read some biology journal articles, though I must say it is always less developed compared to chemistry and physics. I cannot keep in touch with biology formally because it is impossible to do a triple major, an in fact, I want to do a quadruple major (phys, chem, bio (aka live sciences), maths) as this will give me a very solid foundation that can tackle any science and engineering subjects, but that's impossible at least in the confines of a degree

    The way the courses in my uni interrelates in an interesting way (though not the intended aim). When came across a difficult physics course, it is often in the chemistry course that present the same thing that I manage to understand it more easily, and for other courses, vise versa. The exception is electromagnetism, a course that I am fundamentally weak at, don't understand why I am so bad at optics...

    PS I number people's quotes, that is how I deal with long posts, by responding to every relevant sections of it and ensure I did not miss out any questions in the response
     
  19. exchemist Valued Senior Member

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    Well pseudo-orbit is just a term I made up, to express the QM paradox that we have angular momentum (as experimentally verified by orbital magnetic moment), so there is a some kind of net circulation abut the nucleus, BUT, due to quantum indeterminacy we cannot determine any specific path that the electron takes.

    Mathematically, we have this wavefunction and we can operate on it to find out a limited number of properties, one of which is angular momentum. But we can't define position and momentum simultaneously, because they are non-commuting operators in QM.
     
  20. Secret Registered Senior Member

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    Thanks for reminding and clarifying

    How could I have forgot these important concepts (noncommutativity of momentum and position operators, Heisenberg uncertainty principle implies you cannot determine a path a particle takes (even just a rotating one)!

    So to summarize I guess the best way to think of it is that

    {Cptbork} the electron is a probability cloud where each individual bits is rotating (about the nucleus) (and this explains how the cloud as a whole can rotate){/Cptbork}, {exchemist} although we cannot say for certain how fast or in what orientation each piece rotates {/exchemist}. (and in fact the magnitude and direction details does not interested me) {Physics} these then give rise to a magnetic moment we observe {/Physics} and {Billy T} according to quantum mechanics, this and the orbital angular momentum are quantized {/Billy T}. <Some bits from other explanations might be missing that keep this explanation from being complete>

    {exchemist}And we came up with the above because from experiments we detected a magnetic moment associated with the atom, and then we build a model to describe and explain it{/exchemist}
     
  21. exchemist Valued Senior Member

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    Yes I think one needs a collage of concepts to get a physical picture. One qualification, though, re your comment about the "orientation" of rotation.

    We CAN know what proportion of the orbital angular momentum is aligned along a defined axis. This too is quantised: you can have a series of allowed projections of the total angular momentum vector along the z axis, but the orientation of the remaining component (in the x-y plane) is then undefined. It is the orientation of this projection that distinguishes the pz, py and px orbitals from one another, for example.
     
  22. CptBork Valued Senior Member

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    Right, and my point was just to illustrate that there were many pieces of evidence and a long line of reasoning before Stern and Gerlach conducted their spin experiment (they're also famous for directly measuring the Maxwell-Boltzmann distribution for particle energies in a hot gas, and probably some other things too). University undergrad courses and even the grad courses skip over such a huge range of historical details which really detracts from the power of the arguments in favour of the theories they're teaching. There's a reason Einstein wasn't confined to rambling in public layman forums, desperately searching for any sympathetic ear who would listen to his outlandish theories- there was already a lot of evidence to back him up going back a full century, he'd already begun establishing his prowess in classical and statistical mechanics, and several other prominent physicists had already been moving toward similar conclusions for more than a decade prior. Universities focus on teaching a functional knowledge of the material and leave many of the historical and derivational details to the individual, of whom few ever make or find the time to go seeking these things out, so I like to help fill in some of the gaps from time to time and try pointing to whatever else is out there to be known on the subject.

    You make a good point as to why Pauli had to postulate a new quantum number in order to explain multiplicities in the various electron orbits permitted by the Sommerfeld model. It wouldn't have been enough to simply argue that there could be two electrons sharing each orbit, because that would have violated the exclusion principle he conjectured as an explanation for why electrons don't all decay towards the lowest ground state orbit. I looked it up on Wikipedia, and it said that the concept of electron spin was actually first introduced to explain spectral line splitting in alkali metals. That would make sense, because the surface electrons in alkali metal lattices are only very loosely bonded to their atoms, so they mostly sit there without orbiting or drifting around, and it would be easy to study the effects of magnetic fields on those spectral lines without having to account for spin-orbit interactions and Thomas precession.

    From what I've read, Pauli initially just wanted a new quantum degree of freedom to explain how pairs of electrons could share nearly identical orbits, and then it was sunsequently proposed soon afterwards that this degree of freedom could be intrinsic electron spin with an associated magnetic moment, to explain the magnetic effects described above (and subsequently apply it to hydrogen once Thomas precession had been properly accounted for). I remember in my intro quantum textbook (anyone else ever use Townsend?), it started off discussing the Stern-Gerlach experiment for electron spins and using that as the basis to motivate the rules of quantum mechanics, but I knew that couldn't have been the true historical approach, because the book showed the postcard they sent to Niels Bohr with a labelled photographic exposure showing their experimental results, along with comments along the lines of "Congratulations on the confirmation of your theory."
     
  23. Farsight

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    See The discovery of the electron spin by S A Goudsmit and note this:

    "When the day came I had to tell Uhlenbeck about the Pauli principle - of course using my own quantum numbers - then he said to me: "But don't you see what this implies? It means that there is a fourth degree of freedom for the electron. It means that the electron has a spin, that it rotates".

    It's a modern myth that the electron intrinsic spin is nothing to do with rotation. See the Einstein-de Haas effect which "demonstrates that spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies as conceived in classical mechanics". I don't know where this myth comes from, but it seems to be connect to the non-sequitur you can find in older versions of the Wikipedia Stern-Gerlach article:

    "If this value arises as a result of the particles rotating the way a planet rotates, then the individual particles would have to be spinning impossibly fast. Even if the electron radius were as large as 2.8 fm (the classical electron radius), its surface would have to be rotating at 2.3×10^11 m/s. The speed of rotation at the surface would be in excess of the speed of light, 2.998×10^8 m/s, and is thus impossible.[2] Instead, the spin angular momentum is a purely quantum mechanical phenomenon."

    It's cargo-cult science. The electron is a spin half particle, a bispinor. It doesn't spin like a planet. It has a wave nature, it isn't some billiard-ball thing, it's more like an optical vortex, like a tornado of light with two orthogonal rotations. It spins thisaway AND it spins thataway, and the AND acts as a multiplier.
     

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