QM: translation and rotation operators : what's the point?

[koroljov]

Homework Statement

I understand, mathematically, that the translation operator (both for infinitesimal and finite translations) can be written as a function of the momentum operator. It is said then that momentum "generates" translation. Similiary, the rotation operator can be written as a function of the angular momentum operator.

I can't help but thinking there's some point here that I'm missing. What's the point here, beyond "you can write this in function of that?" Why would anyone even want to construct a translation operator? Why not just changing the coordinates x -> x+a ?

 

[malawi_glenn]

The formalism of QM deals with operators. And if you carefully see how you from the translation operator "derive" the momentum operator and its properties, you see how beautiful this works.

What will change the coordinates? An operator, hence the translation operator.

And the angular momentum operator is then derived from the rotation-operator.

And you "need" time evolution operator, to get quantum dynamics; in the same spirit of this formalism.

I think that you have "missed" (at least according to my knowledges of QM) that momentum and angular momenta operator is derived FROM the definitions of translation- and rotationoperator.

Hopefulle more guys will answer =)

 

[Dick]

It's because the invariance of systems under things like translation and rotation corresponds to the conservation of operators like linear momentum and angular momentum. To see this you need to convert the translation and rotation to operator form.

 

[meopemuk]

Homework Statement

I understand, mathematically, that the translation operator (both for infinitesimal and finite translations) can be written as a function of the momentum operator. It is said then that momentum "generates" translation. Similiary, the rotation operator can be written as a function of the angular momentum operator.

I can't help but thinking there's some point here that I'm missing. What's the point here, beyond "you can write this in function of that?" Why would anyone even want to construct a translation operator? Why not just changing the coordinates x -> x+a ?

One of the most important tasks of quantum mechanics (and physics, in general) is to find out how the same physical system looks from different inertial frames of reference. So, one needs to know how the wave functions and/or operators of observables change with respect to inertial transformations (space and time translations, rotations, and boosts). Mathematically this boils down to finding unitary operators which represent these inertial transformations in the Hilbert space of the system.

You are right that some of these operators are rather simple. I.e., space translations simply shift the arguments of wave functions. Space rotations are easy too. However, the action of time translations is very non-trivial in interacting systems. Another interesting and non-trivial question is how wave functions and observables transform with respect to boosts. All these questions can be answered only if we take into account that inertial transformations form a 10-parameter Poincare group. The theory of unitary representations of the Poincare group in relativistic quantum mechanics was developed by Wigner (for free particles) and Dirac (for interacting systems)

E. P. Wigner, "On unitary representations of the inhomogeneous Lorentz group", Ann. Math. 40 (1939), 149.

P. A. M. Dirac, "Forms of relativistic dynamics", Rev. Mod. Phys. 21 (1949), 392.

(See also S. Weinberg, "The quantum theory of fields", vol. 1 (1995).)

The identification of operators of (total) energy, momentum, and angular momentum with Hilbert space generators of time translations, space translations, and rotations, respectively, is an important part of this theory.

Eugene.

 

[koroljov]

Thank you all for your replies. I'll look further into it.