Exploring Quantum Spin: Visualizing Spin Observables in Real and C-2 Space

[TimH]

I'm reading Hughes book The Structure and Interpretation of Quantum Mechanics. I was wondering if people knew of any good web resources that graphically (maybe with Java applets, etc.) illustrate how the spin observables Sx Sy and Sz are related to each other, either in real space or in their representation in C-2. Basically I'm just looking for any kind of illustrations involving quantum spin that will help me get a better feel for it. Thank you.

 

[vincentm]

http://www.falstad.com/qmatom/

 

[Gokul43201]

http://www.falstad.com/qmatom/

Am I missing something on that page? All I see are orbital representations. How does this help with spin and its components?

 

[TimH]

Yes I'm looking for something showing the mathematical connectedness of the incompatible observables in spin. But thank you very much for the link which has many other cool applets.

 

[Gokul43201]

Yes I'm looking for something showing the mathematical connectedness of the incompatible observables in spin.

That's not really clear to me. Are you asking for a visualization of the commutation relations for the spin operators?

 

[TimH]

I'm trying to understand the spin observables of the electron at a beginner level. I understand (I think!) that the three spin observables can be simultaneously represented in C-2 (i.e. a 2-dimensional complex Hilbert space). Since you can't visualize a complex 2-dimensional space I was wondering if anybody had taken a subset of the whole space and could display it, or somehow use a gimmick or shortcut to help show how the spin x,y, and z observables are interrelated in an applet.

 

[vincentm]

Am I missing something on that page? All I see are orbital representations. How does this help with spin and its components?

My bad

 

[Sojourner01]

I understand (I think!) that the three spin observables can be simultaneously represented in C-2 (i.e. a 2-dimensional complex Hilbert space)

Why is it that I've just finished a masters-level quantum mechanics course and don't understand this sentence?

 

[genneth]

Perhaps you should look for visualisations of the Bloch sphere. Remember that rays in C^2 has 3 degrees of freedom -- 2x2 from the components, -1 for the normalisation constraint. As it happens, this gives a very nice geometrical representation. It's useful for visualising the sometime obtuse algebra, but should not be afforded too much physical meaning. As usual, start with the wiki: http://en.wikipedia.org/wiki/Bloch_sphere

 

[Hurkyl]

It's useful for visualising the sometime obtuse algebra, but should not be afforded too much physical meaning.

I don't understand your warning; the Bloch sphere is (equivalent to) the space of pure states of such a qubit, and the corresponding ball is (equivalent to) the entire state space, is it not?

 

[genneth]

I don't understand your warning; the Bloch sphere is (equivalent to) the space of pure states of such a qubit, and the corresponding ball is (equivalent to) the entire state space, is it not?

Yes. But a 2-component system doesn't exhaust the physics of spin-1/2 particles and the like. The key point is that a direction on the sphere isn't a direction in "real life". Usually, the point is moot, but when you have something like a uniform magnetic field that changes the symmetries it's not quite as useful. But yes -- the surface of the sphere is exactly equivalent to the states of a 2-component system; in fact, the interior of the sphere is the space of density matrices over the system.

 

[quantumfireball]

i don't think its a good idea to imagine electron spin just as you would imagine a spinning ball.
An electron has spin even though its not spinning in the literal sense.
That the electron possesses spin has been proved experimnetantally in stern -gerlachs experiment.
One more thing is that only one component of spin can be determined ,
its impossible to dettermine the componenet of spin is say both x and z directions simultaneously.
Since an electron is a pont particle ,it make absolutely no sense whatsoever in imagining electron spin ,like say a spinning tennis ball

 

[ozymandias]

It might not make sense but it sure does get you a long way in obtaining the right orders of magnitude

-----
Assaf
http://www.physicallyincorrect.com"

 

[pellis]

Take a look at Greg Egan's applets, especially:

http://gregegan.customer.netspace.net.au/APPLETS/23/23.html

but also

Permanent link: http://www.gregegan.net/APPLETS/Applets.html
SO(3) |... | Dirac | SO(4) | ...| Spin Networks (Applet) |

The rest of his site is a blast too

Paul

 

[TimH]

Thanks for the link. This is the kind of visualization I was looking for, though its a little beyond my level.