Adding spin components on different axes x+x=Z?

[chopficaro]

sqrt(1/2)up = sqrt(1/2)right + sqrt(1/2)left

this is very counter intuitive for me, I am used to normal Cartesian coordinates where u can add and subtract magnitudes in the x-axis all day and get nothing on any other axis. furthermore, a spin to the right should be the opposite of a spin to the left, and so right should equal negative left, and adding the same magnitude of each should give u ZERO!

 

[SpectraCat]

sqrt(1/2)up = sqrt(1/2)right + sqrt(1/2)left

this is very counter intuitive for me, I am used to normal Cartesian coordinates where u can add and subtract magnitudes in the x-axis all day and get nothing on any other axis. furthermore, a spin to the right should be the opposite of a spin to the left, and so right should equal negative left, and adding the same magnitude of each should give u ZERO!

Spin is a fundamentally different entity from classical observables, and is treated in standard non-relativistic Q.M. as an intrinsic quantum mechanical quality of particles that cannot be represented in position space. So the "right" and "left" referred to in your equation above do not refer to opposite directions on some Cartesian axis. In fact, they are orthogonal basis vectors, so that means that the angle between them is actually 90 degrees.

I know that this is rather non-intuitive at first, since you probably started out talking about things as projections of the spin vector on various space-fixed Cartesian axes. However it is all consistent. You might want to google "Pauli spin matrices" if you are interested in the mathematical formalism behind this.

 

[PhilDSP]

Remember that there are 2 possible values to a square root: one positive and the other negative, with the same absolute value, though for simple calculations we often neglect the negative value. This gives us a really handy way to encode positions or times for instance involving symmetry.

A complex number is the standard way to take advantage of coding those types of symmetry in one or two dimensions. A spinor or a quaterion (which the Pauli spin matrices are associated with) does the same thing in additional dimensions.

 

[chopficaro]

ty i understand more, I am watching some tutorial videos on quantum mechanics but they aren't going over any of the math. i have a very robust history in math so I am a little disappointed. when i finish my tutorials i will look up those matrices