The concept of Rotation in 3-d in QM

[toffee]

The concept of "Rotation in 3-d" in QM

I'm having difficulty with two ideas and wud appreciate some help:

1)commutation relations:

$$[L_i, L_j]=i\hbar\epsilon_i_j_kL_k$$

reflect the law of combination of rotation in 3 dimensions.

What is the this law and why must in be satisfied whatever be the nature of the wave’functions’ they rotate?

2)Since rotation in 3 dimensions always obey composition rules the spin operators $$S_i$$ obey the same commutation relations:

$$[S_x,S_y]=i\hbar S_z$$ and cyclic

what are compostion rules and why does the above hold true?

thanks in advance

 

[dextercioby]

I can tell u that the 3-rd chapter of J.J.Sakurai's masterpiece is illuminating.So how about reading from there first and then come to us with things that u didn't find clear.

Daniel.

P.S.I hope u've done your HW with classical mechanics (Goldstein,actually).

 

[toffee]

I can tell u that the 3-rd chapter of J.J.Sakurai's masterpiece is illuminating.So how about reading from there first and then come to us with things that u didn't find clear.

Daniel.

P.S.I hope u've done your HW with classical mechanics (Goldstein,actually).

Unfortunaltly, I can’t get those books at my library (and I certainly can't afford them!). I tried googling for information but to no avail. It was basically a side comment on my lecture notes which i didn't really undestand, so even a one line explanation would be useful.

Well, if it isn’t possible to answer those questions (well the first one anyway), I’d like to ask another one:

Why is it that one can multiply a state vector by an arbitary phase factor? I can understand that it would not alter the probability distribution, but surely the expression is not mathematical.

ie $$\mid\psi >= e^i^\varphi\mid\psi >$$

 

[dextercioby]

It's because physicists do not measure vectors of abstract separable Hilbert spaces,but objects with physical significance.Probability & probability densities can be computed and tested with experiments...The III-rd principle contains the mathematical expressions that u need.And in those,an arbitrary phase factor is irrelevant.

The quantum theory of chemical bond is an example of domain in which phase factors matter.But it's only theory.

Daniel.

 

[physicsboy]

The mathematical content is contained in the ket itself. The phase factor multiplied infront does not change the "direction" of the ket but only scales it (makes it shotrter or bigger).

 

[dextercioby]

That "phase factor" must be a complex # of unit modulus,however...The "shorter"/"bigger" part is pointless,as the ket would have the same norm...(unit,usually).

Daniel.

 

[QMrocks]

I'm having difficulty with two ideas and wud appreciate some help:

1)commutation relations:

$$[L_i, L_j]=i\hbar\epsilon_i_j_kL_k$$

reflect the law of combination of rotation in 3 dimensions.

What is the this law and why must in be satisfied whatever be the nature of the wave’functions’ they rotate?

These commutation relations will always hold true independent of the wavefunction it acts on. This is because these commutation rules are identities abt the operators. The LHS operators are the same as RHS operators, literally the same if you write them down in the set of complementarity operators $$\vec{R}$$ and $$\vec{P}$$.

i got some good info from http://farside.ph.utexas.edu/teaching/qm/rotation/nodel.html

hope this helps.