# The concept of Rotation in 3-d in QM

• toffee
In summary, the concept of "Rotation in 3-d" in QM is explained. The commutation relations between the operators in 3 dimensions are explained. The importance of phase factors is explained.
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?

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).

dextercioby said:
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 probablity distribution, but surely the expression is not mathematical.

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

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.

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).

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.

toffee said:
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.

Last edited by a moderator:

## 1. What is the concept of rotation in 3-d in quantum mechanics?

The concept of rotation in 3-d in quantum mechanics involves the mathematical description of the rotation of particles in three-dimensional space. In quantum mechanics, this rotation is described using the principles of quantum theory, which allows for the description of particles as both waves and particles.

## 2. How is rotation described in quantum mechanics?

In quantum mechanics, rotation is described using operators known as rotation operators. These operators act on the wave function of a particle and describe how the wave function changes under rotation in three-dimensional space. The rotation operators are based on the principles of symmetry and are used to calculate the probabilities of different states of a particle after rotation.

## 3. What is the significance of rotation in 3-d in quantum mechanics?

Rotation in 3-d in quantum mechanics is significant because it allows for the description of the behavior of particles in three-dimensional space. It is also essential for understanding the properties of quantum systems, such as spin, angular momentum, and energy states. The concept of rotation in 3-d is also crucial for developing theories and models in quantum mechanics, such as the quantum mechanical model of the atom.

## 4. Can the concept of rotation in 3-d be observed in experiments?

Yes, the concept of rotation in 3-d can be observed in experiments. For example, the rotation of particles can be observed in experiments involving quantum systems and their behavior under different external forces. The effects of rotation can also be observed in experiments involving the measurement of spin and angular momentum of particles.

## 5. How does the concept of rotation in 3-d relate to other concepts in quantum mechanics?

The concept of rotation in 3-d is closely related to other concepts in quantum mechanics, such as angular momentum, spin, and symmetry. These concepts are all integral to the understanding of the quantum world and play a crucial role in the development of theories and models in quantum mechanics. Additionally, the concept of rotation in 3-d is also closely related to other mathematical concepts, such as matrices and tensors, which are used to describe the rotation of particles in three-dimensional space.

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