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I've been studying the finite-dimensional representations of the lie algebra sl(2,C). I've read that these representations are related to the harmonic oscillator and the associated raising and lowering operators, but I'm not really sure how they are related. I've read you can generate a new energy state from an old one, but I'm not really sure how it works, and the energy states do not differ by 2 like the eigenspaces do for sl(2,C). I'm having trouble finding references about this, especially ones that are accessible. If anyone knows where I can find more info, or has some knowledge of the material, I would greatly appreciate their help.

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My understanding is that the eigenvalues of the representation of $\text{sl}(2,\mathbb{C})$ are integer valued but the standard physics convention has an additional factor of $\hbar/2$ which may fix your issue. – Alex Troesch Dec 14 '10 at 3:42
I'm guessing you're using the basis $X_+,X_-,H$ with $[H,X_+] = 2X_+, [H,X_-] = -2X_-, [X_+,X_-] = H$? Then I think in physics they use $\frac{1}{2}H$. So if $v$ is an eigenvector of $\frac{1}{2}H$ with value $r$ then $X_+ v$ is an eigenvector of $\frac{1}{2}H$ with value $r+1$. – Eric O. Korman Dec 14 '10 at 3:44
2 looks like the commutation relations in physics involve $[a_+, a_-] = 1$, so it's not quite the same since $H$ doesn't act as the identity in the rep space. – Eric O. Korman Dec 14 '10 at 3:51
Ryder section 2.3. Hall section 1.7. – Matt Calhoun Dec 14 '10 at 4:48

The Quantum Mechanical Harmonic Oscillator

If you look at the Wikipedia page on the quantum mechanical harmonic oscillator, you will see that it is not the Lie algebra $\mathfrak{sl}_2$ that enters the picture, but rather an algebra generated by two operators $a$ and $a^{\dagger}$ satisfying $[a,a^{\dagger}] = 1$ (see also Eric's comment above).

This is a familiar algebra in disguise, the so called Weyl algebra of differential operators with polynomial coefficients $\mathbb C[x,\partial_x]$. (Note that $[\partial_x,x] = 1$, by the Leibniz rule.) The product $x\partial x$ is the so-called Euler operator (it acts on $x^n$ as multiplication by $n$); you can see that in the harmonic oscillator interpretation, it acts as the Hamiltonian (up to a shift of $1/2$, to add non-zero groundstate energy, and a rescaling by $\hbar \omega$).

This algebra has a natural representation, namely on the space of polynomials $\mathbb C[x]$, and we see that this is isomorphic to the representation on the span of the eigenstates that occurs in the harmonic oscillator picture. (The polynomial $x^n$ corresponds to the eigenstate with energy $\hbar \omega(n + 1/2).$)

The Lie algebra $\mathfrak sl_2$ in quantum mechanics

If my memory serves, the first time $\mathfrak sl_2$ entered the picture (in my undergrad QM class, at least) is in the analysis of the hydrogen atom. The point is that the Schrodinger equation in this case has a rotational symmetry, and differentiating this action, one gets an action of the Lie algebra of $SO(3)$ on the space of states. Complexifying, this gives an action of $\mathfrak sl_2$.

If you separate variables in the equation (using spherical coordinates, which are natural in view of the rotational symmetry), the radial part and the spherical part separate, the radial part is easily dealt with, and one is left studying a certain differential equation on the space of functions of the sphere $S^2$.

Mathematically, one is looking at $L^2(S^2)$, with its natural action of $\mathfrak sl_2$ (in the guise of the complexified Lie algebra of $SO(3)$). The Shrodinger equation involves the Laplacian, which is also the Casimir in the enveloping algebra of $\mathfrak sl_2$, and so if you fix the energy (eigenvalue of the Laplacian), one is trying to understand the corresponding eigenspace of the Casimir on $L^2(S^2)$. This is a special case of the Peter--Weyl theorem, which (in this case) also goes under the name of the theory of spherical harmonics.

The idea, roughly, is: $S^2 = SO(3)/SO(2)$. Now by Peter--Weyl, $L^2(SO(3))$ is the Hilbert space direct sum of $V\otimes V^*$, where $V$ runs over the irreps. of $SO(3)$. Thus $L^2(S^2)$ is the Hilbert space direct sum of $V\otimes (V^*)^{SO(2)}$. The space of $SO(2)$ invariants in each irrep. is one-dimensional, so in fact $L^2(S^2)$ is the Hilbert space direct sum of $V$, as $V$ runs over the irreps. of $SO(3)$.

Since $SO(3) = SU(2)/\{\pm 1\}$, these correspond on the Lie algebra level to the odd-dimensional irreps. of $\mathfrak sl_2$.

Each $V$ is an eigenspace for the Casimir (if $V$ has dimension $n$ then the Casimiar eigenvalue is some quadratic expression in $n$ that you can easily figure out, or look up), and distinct $V$ give distinct eigenvalues.

So, going back to the hydrogen atom, if you fix the energy of the electron, then the collection of states has dimension equal to the dimension of the corresponding $V$. (The fact that this is typically greater than one-dimensional is referred to in physics as degeneracy; the energy does not uniquely determine the state.) This has a natural basis corresponding to the $SO(2)$ eigenvectors. Physically, the derivative of this $SO(2)$ (the traditional element $H$ in the Lie algebra $\mathfrak sl_2$, perhaps up to some scaling, and often denoted $L_z$ in physics literature) is the operator corresponding to angular momentum around the axis in $S^2$ which is fixed by $SO(2)$ (often taken to be the $z$-axis, by convention, in the physics literature), and so the different eigenvectors for $SO(2)$ in the given $V$ are states with the same energy, but with different angular momenta around the $z$-axis.

Physically, one can apply a magnetic field along this axis which will then affect these different $L_z$-eigenstates differently (because its effect on the electron depends on the angular momentum of the electron around the field lines), and so one can split the degeneracy. This is the Zeeman effect. One sees the different energy lines in the spectrum of hydrogen split into a number of lines ($n$ lines when the energy is proportional to $n^2$).

I haven't looked, but this must be discussed in many, many places online, and is also in any standard intro. to quantum text. (But probably with less appearance of the symbols $L^2(S^2)$ and $\mathfrak sl_2$ then a mathematician would use!)

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A nice reference for some of this are Peter Woit's notes: – BBischof Dec 15 '10 at 7:20
Weyl's The theory of groups and quantum mechanics is a charming place where to read this :) – Mariano Suárez-Alvarez Jan 19 '12 at 3:31

First, I think there is potentially more than one question here. The finite-dimensional version, addressed by Matt E, is about the hydrogen atom. The "harmonic oscillator" sometimes means a different thing, not about finite-dimensional repns of the Lie algebra $\mathfrak{sl}_2(\mathbb C)$, but about infinite-dimensional ones. There there is interaction with repns of SO(n) for all n. (This is also an example of "Segal-Shale-Weil" repns, ... also existing over p-adic fields, and abstractly, after Weil.)

I'm sure there are other worked-out versions on-line, but at least I can offer my own worked-exercise, with (a mathematician's notion of) a bibliography:

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