Angular momentum by harmonic oscillator

I’ve been thinking about Sakurai problem 3.19 for the past few days.

The problem reads:

19.) What is the physical significance of the operators

K+ ≡ a+†a†        and       K ≡ a+a

in Schwinger’s Scheme for angular momentum? Give nonvanishing matrix elements of K±

I’m not sure I completely understand this problem yet. The ‘a’ operators are the creation (daggered operator) and annihilation operators (undaggered operator) for the simple Harmonic oscillator. Applying these operators to simple harmonic oscillator eigenstates increase or decrease, respectively, the energy quantum number of the state, allowing you to move up or down the energy spectrum. In Schwinger’s scheme, two harmonic oscillators are put together  to create an angular momentum eigenstate (thus proving that you really can create everything out of harmonic oscillators). These oscillators are represented by the ‘+’ and ‘-‘ subs on the ‘a’ operators. K+ and K- are almost but not quite the same as the ladder operators, defined in Schwinger’s Scheme as a+†a and a+a† which is definitely subtly different. In Schwinger’s scheme, the two harmonic oscillators are independent eigenspaces that are coupled together and applying a+†a increases the ‘+’ state while decreasing the ‘-‘ state. Eventually, the ‘-‘ state hits its ground state and a further application gives a zero. Going the other way with the second operator does the same thing, decreasing the ‘+’ state while increasing the ‘-‘ state until you hit the bottom of the ‘+’ state spectrum and annihilate the combination with a zero. The affect is like the Ladder operators walking across the ladder, either up or down ‘m’ values until you hit the end and kill the last state.

My conclusion about the K± operators is that they will essentially increment or decrement the total angular momentum (l-value) of the coupled state, which is actually really kind of wicked. I haven’t worked the math yet, but I think this is cool. This problem is also interesting to me because it gives a basic hit with a way that QFT deals with photon formalism. Photons are loaded into eigenstates using operators like these.

I’ve decided that Julian Schwinger was a very smart guy.

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