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Author  Topic: Introduction to Quantum Mechanics.  
Son Goku Inactive Member

I've been inspired by Dr. Adequate's series on Geology (an amazing thread), so I would just like to propose a similar introduction to quantum mechanics. I would plan to begin around May this year. I know it is pretty early to be proposing this, but I plan to treat it like a proper course, similar enough to those I actually lecture, so I would need some time to prepare.
The only mathematics involved will be matrices and vectors. EDIT: Vectors and matrices form the subject known as linear algebra, the amount of linear algebra necessary for the course will be explained in a separate pdf/post. To learn linear algebra you only need to know what a number is! Edited by Son Goku, : Little details on the mathematics. Edited by Son Goku, : I can't spell.


Son Goku Inactive Member 
I guess it could be put in one of the science forums, Miscellany or Big Bang and Cosmology would be fine.
Mainly to gauge interest and see what topics people might like to see discussed.


Son Goku Inactive Member 
Thanks for the offer, seems like a good idea. Hopefully I'll be able to do it myself, as I wouldn't like to bother you, but there's probably subtleties to the process I don't know about, so I might need your help!
I'm preparing it in advance, since every time I write a university course, only at the end do I realise exactly when certain ideas should be introduced. My "lesson by lesson" version wouldn't be as good.


Son Goku Inactive Member

I think you should. I showed my Dad your old 2.1 version a while back and he really enjoyed it, it's a really nice "hammering home" of the evidence for an old earth.
Edited by Son Goku, : No reason given.


Son Goku Inactive Member

So the idea is that the topics would be something like:
1. Derive quantum mechanics from looking at the SternGerlach experiment. Basically that all the features of Quantum Mechanics (e.g. Complex numbers, probabilities) naturally follow from taking a serious look at how the spin of electrons is affected by a magnetic field. 2. Then, look into more depth at the framework we have derived. Deriving the uncertainty principle and entanglement for example. 3. Next, focus on the probabilistic aspects of Quantum Mechanics and their meaning. Proof of the KochenSpecker theorem and other such results, which serve to prove how probability in quantum mechanics differs from normal statistics and probability. 4. Interpretations and Decoherence. 5. Quantum computing, as an application of the preceding four sections. Edited by Son Goku, : Forgot to mention Decoherence.


Son Goku Inactive Member

If you listen to your heart, you already know he's right about everything. Except you can't listen with ears of flesh, but with the ears of love, the cochlea of understanding and the external auditory meatus of compassion.


Son Goku Inactive Member 
Well, talking about spin will be a big part of the notes, since it's the property of matter that displays all the features of quantum mechanics while still only involving simple mathematics.
I'll try an explanation now though, since I don't need the full notes to discuss it. As you are aware particles have mass and charge, these are two numbers, one which determines how difficult it is to accelerate the particle and the other determines the strength of the electric field it produces. Angular momentum, in quantum mechanics, in simply another number like this. You can think of it as determining the strength of the magnetic field an object generates. It is simply another "charge" like electric charge. Some particles come with a certain amount of angular momentum, but you can also generate angular momentum from rotating around a fixed point. Angular momentum is actually just a charge. You don't have to rotate to possess angular momentum, that's just one of the ways of obtaining it. Think of energy. There is kinetic energy, mass energy, potential energy. They're all just different ways of possessing the same quantity, namely: Energy. However in classical mechanics the only way to possess angular momentum is via the second method of rotating about a fixed point. Hence the conflation of angular momentum and rotations. Even the name is unfortunate, reflecting its historical origin. Angular momentum, implies that it is simply the analogue of momentum for rotations. It should really be called "external SU(2) charge", for reasons I'll explain in the notes. So an electron isn't really spinning around or anything, it just possesses two charges:Electric charge. Angular momentum charge. Since, unfortunately, in classical mechanics the second charge only comes from rotations, this leads to the mental picture of an electron rotating, however this is false. If you analyse the other effects of rotation (rotation causes things besides angular momentum) an electron doesn't display them, hence it is definitely not rotating. In quantum mechanics, the angular momentum coming from rotations is called orbital angular momentum and angular momentum that the particle just naturally possesses is called spin angular momentum (again an unfortunate name as the particle doesn't really spin).


Son Goku Inactive Member

The quantity was called spin because initially it was thought that this new angular momentum was coming from the electron spinning on its own axis.
The idea was that the electron was like the Earth, it had orbital angular momentum (motion around the Sun) and spin angular momentum (rotation about its own axis). However when you really think about it (as people did almost immediately) this doesn't really make any sense, as the Earth's spin, is really just another form of orbital motion, simply the material of the Earth swinging around the main axis. Since the electron is fundamental, this doesn't really make any sense as the explanation of its "spin" value. However, yes, assuming that the electron spun on its own axis with only certain fixed values possible was consistent with the SternGerlach experiment. Or more accurately a quantum "indefinite" version of the Earth's spin was consistent with the SternGerlach experiment. If this was true, spin would be just like momentum, i.e. it has a classical analogue and all quantum mechanics would do is to make it probabilistic. However there are a few facts that show you that "spin" is not a quantum version of the Earth's axial rotation, as was initial thought (and which is the origin of the name). It is actually a quantum quantity that is not a random/indefinite version of any classical quantity. The clearest example I can think of is that there is a simple expression relating angular momentum to the strength of a magnetic field and a second expression relating angular momentum to rotational velocity. It turns out that the measured strength of an electron's field would imply a rotational speed exceeding the speed of light. This would be problematic enough, however if this were true you could scatter other particles off an electron and in the scattering those particles would steal some of the electron's rotational energy and rotate themselves. Ultimately you could make the electron stop spinning, reduce the magnetic field to zero. However in actual scattering experiments, nothing like this happens. The electron's magnetic field stays constant. So therefore there must simply be no rational energy to steal from. The "spin" angular momentum must in fact have nothing to do with spinning. As I said, we know now that it is simply a charge, like electric charge.


Son Goku Inactive Member 
Sorry for the incredibly late reply!!
Yes, spin is more like a "potential aspect", in the sense you have defined it. I'm much busier in work than I thought, but things a clearing up now and I hope to start this course soon.


Son Goku Inactive Member 
It was only discovered in Quantum Mechanics. It's not possible for classical particles, they can only generate angular momentum via orbital motion.


Son Goku Inactive Member 


Son Goku Inactive Member 
It's like the quantum number for spinup/down was reset during the spinleft/right test. (reset from "up" to "?", please excuse my layman's language)
Yes. The notes have not explicitly reached this point, but that is what will be developed next. Testing xspin resets zspin to "?". We say they are incompatible observables, they cannot both possess definite values. (This will turn out to be the origin of the uncertainty principle.) Most of the mathematics really boils down to precisely quantifying the many forms "?" can take and getting quantitative predictions from it.
In the "B" experiment this detection of spinleft/right is not made and the result of the second up/down test yields 100% "up". Even though the beam in the spinleft/right apparatus is split into separate spinleft and spinright paths (since the equipment is exactly the same, functions just the same (minus the detector)) the quantum number ("up") from the first up/down test block was not reset by the left/right test block.
Correct.
The only difference between the setup of the original and "A" experiments and the setup of the "B" experiment is that the original and "A" detect the left/right split while "B" does not.
Exactly. Essentially eliminating anything but the detection process as the source of the difference.
This is a "Double Slit" experiment for electron spin, yes?
Yes, it is the direct analogue of the Double slit experiment for spin. The unfortunate thing about the double slit is that to describe it you need advanced mathematics (Hilbert spaces, complexvalued partial differential equations). However for spin, the mathematics is far simpler, vectors and matrices.
So in a layman's analogy (not totally accurate but close enough for some understanding) is saying ...
Actually that's totally accurate. We only need to develop an understanding of the "?" state, i.e. are there different types of "?", how do we describe them mathematically.
"When the spinleft/right paths are detected the quantum number for spinup/down is reset to "?" thus allowing the second up/down test to spread the results among the quantum probabilities" ... acceptable?



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