# A Question on SU(N)

1. Apr 28, 2013

### Clandestine M

In quantum mechanics, ladder operators could be constructed within SU(2). The examples should be ladder operators in Quantum Harmonic Oscillator and ladder operators in angular part of Hydrogen Atom (Lx + i Ly, Lx - i Ly).

In Field Theory, QED SU(2) and QCD SU(3), the creation and annihilation operators (an extended version of simple ladder operator) could also be constructed.

And finally in SU(N), the infinite dimensional Quantum Harmonic Oscillator also allows for the construction of ladder operator.

My question is:

is that because SU(2) is the subgroup of all SU(N) group? in this way the ladder operator in SU(2) could be extended to SU(N)?

Is there any good reference helps me understanding ladder operator? And the Factorial Methods solving differential equations? Is there any relation between ladder operator methods solving differential equation and Symmetry Methods solving differential equation (relying on Killing vector Field)?

Thanks a lot!

2. Apr 29, 2013

### fzero

Not quite. It turns out that any semisimple Lie algebra admits a ladder-operator basis for the generators, called the Cartan-Weyl basis. It's obvious that the reduction to an su(2) subalgebra of su(N) can be done in terms of the elements of the C-W basis, preserving the ladder-operator structure. But not all of the properties of a subalgebra have to extend to an entire algebra. For instance, every Lie algebra has a u(1) subalgebra (actually many of them), but that doesn't mean that every Lie algebra is abelian.

For the ladder operators, any decent text on Lie algebras, like the one I linked to above should discuss the Cartan-Weyl basis and the root systems. I'm not so familiar with the general theory of Lie groups and differential equations, the references on this page may or may not be useful.

3. May 1, 2013

### Clandestine M

Thanks for your reply.

Now I begin to understand that Cartan-Weyl Basis is specially for semi-simple Lie Algebra, and Killing vector field is another kind of basis (generators).

I have checked amazon.com and listened to advice from my peers.

In the book, "Factorization Method in Quantum Mechanics" by Shi-Hai Dong Springer (2007) Springer
the applications of ladder operator are well developed.

(I heard that) Ladder operators are called "coherent states" in Quantum Control, Quantum Optics, and some Mathematical Physics literatures.

Newly published book
"Coherent States and Applications in Mathematical Physics" by Monique Combescure, Didier Robert (2012) Springer
I think would be very useful to me.

:)

4. May 1, 2013

### fzero

There are a few different objects called coherent states in the literature. In terms of some ladder operators $\hat{A},\hat{A}^\dagger$, the canonical coherent state $| \alpha \rangle$ is defined as the eigenstate of the annihilation operator

$$\hat{A} |\alpha\rangle = \alpha | \alpha \rangle.$$

Some additional discussion can be found at http://en.wikipedia.org/wiki/Coherent_states#Quantum_mechanical_definition.

5. May 1, 2013

### Clandestine M

Thanks a lot!

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