The particle in an infinite potential well in QM is usually solved by easily solving Schrodinger differential equation. On the other hand particle in the harmonic oscillator oscillator potential can be solved elegantly algebraically using the creation and annihilation operators to find its spectrum.
Is it possible to do the particle in a box problem using creation and annihilation operator and how?
Best Answer
Probably not, but you could always define $a=\sum_n \sqrt{n}\left|n-1\right>\left<n\right|$ which acts on the eigenstates, and write $H=E(a^\dagger a)^2.$ This doesn't help you solve the problem to begin with, but it does let you write $H$ with $a,a^\dagger$ once you know the spectrum.