(Example question QM) Raising and lowering operators

Why are they important? Why some people call them creation and annihilation operators? Who are they trying to annihilate?

Wikipedia has a very cool article about them: Creation and annihilation operators - Wikipedia

They are important because they can help you find many solutions (eigenstates) to sometimes difficult differential (or operator) equations. Once you find one solution to the problem, applying these operators to this solution can give you other solutions for free.

Imagine a building that is hard to access (who knows, it might only have one door that is blocked half the time), but once inside you can use the stairs to explore different floors easily. That is why they are called “ladder” operators.

The creation and annihilation names are related to the application of similar operators in the second quantization formalism (Second quantization - Wikipedia). In that case, they truly “create” and “annihilate” particles (or things that behave like particles, such as states or pseudo particles) in a quantum system. You could think that even for the quantum harmonic oscillator, the creation operator is “creating” a quantum of energy on your state, and the annihilation operator is removing one.