Ladder operators are found in various contexts (such as calculating the spectra of the harmonic oscillator and angular momentum) in almost all introductory Quantum Mechanics textbooks. And every book I have consulted starts by defining the ladder operators. This makes me wonder why do these operators have their respective forms? I.e. why is the ladder operator for the harmonic oscillator
$$\hat{a}=\sqrt{\frac{m\omega}{2\hbar}} \left( \hat{x} + \frac{i}{m\omega}\hat{p} \right) $$
and not something else?
On a similar note, does anyone know the physicist/paper who/which proposed this method? Wikipedia mentions Dirac, but I have been unable to find any leads.
Answer
You may recall from high school algebra that $x^2 + y^2 = (x + iy)(x - iy)$. Because the way the adjoint operator works, you could define an operator $\hat a = x + iy$, and its adjoint becomes $\hat a^\dagger = x - iy$. The hamiltonian for the quantum oscillator is just this relation with some constants. You have to be careful because the ladder operators don't commute; that causes the constant $\frac{1}{2}\hbar\omega$ to show up. Of all the sources that I've seen discuss the oscillator with the ladder operators Griffiths (section 2.3.1) is the only one who actually explains the problem this way. The others just pull the ladder operators seemingly out of nowhere, then demonstrate that they work.
The ladder operators date at least to Dirac's Principles of Quantum Mechanics, first published in 1930. That's a really good example of Dirac just inventing the ladder operators and then showing that they solve the problem. Dirac had a tendency to bring in math that physicists at the time weren't familiar with. So it's possible that he saw the ladder operators in math, realized they could solve physics problems, and introduced them to physics. He doesn't provide a citation in Principles, so it's also possible that he invented them. The best citation for where Dirac got the ladder operators should be in one of his original papers.
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