Thread: Raising/lowering operators

Also known as creation/annihilation operators or ladder operators.

It seems these were introduced by Paul Dirac to simplify the solution to certain eigenvalue-eigenvector equations in quantum physics, especially QFT

Anyway, essentially these operators allow one to "climb up and down the ladder" of eigenvalues for any Hermitian operator on an Hilbert space with very much simplified calculations.

These operators are, as far as I can see, usually handed to us on a plate, and take on slightly different forms, depending on who you read and more particularly, what operator for which we are trying to find the eigenfunction solutions.

Consider the quantum harmonic oscillator in one dimension whose Hamiltonian may be written as , where the are respectively momentum and position operators, and is a constant defined by, say, (Note I have not included a mass term; this doesn't matter as it drops out in all subsequent calculations).

So let me attempt a "factorization" of . That is, quite illegally, treat as though they were simple vector-valued variables, not quantum operators. I find that factorize up to a multiplicative constant. (Notice these two are Hermitian conjugates).

Now I said (or implied) that I had seen these operators written in different forms. Since they all factorize (I have checked) they must be equal which I can assert without putting pen to paper ( This is not entirely true as written. I can explain if needed, but requires a little more work, and involves Cartesian products, equivalence relations and other mathy stuff.)

In other words, by defining,say, and I easily find, by simple rearrangement, that .

Now this is a cheat, of course, since the mixed terms in my expansion should not cancel (the momentum and position operators don't commute!). But setting and and

Then so that .

This immediately tells me how to express the operators and in terms of these creation and annihilation operators, using the commutator

So by being honest about my non-commuting operators, all I have done is added a constant as compared to when I was "cheating". I find this a little surprising. Should I? Well perhaps not; I have done the calculations and they are good. But I am still a little surprised.

Wow, that Dirac! Clever or what?