Significance of annihilator and creation operators in the harmonic oscillator

[JohnPrior3]

We have been doing the harmonic oscillator in lecture and I am very confused with the annihilator operator and creation operator. Could someone explain their significance? I am taking linear algebra right now too, but we haven't gotten into operators yet. I've been very discouraged lately with Quantum.

 

[sweet springs]

Hamiltonian
$$H=\frac{k}{2}x^2+\frac{1}{2m}p^2$$ is sum of two terms. This can be written as
$$H=\hbar \omega (a^\dagger a + \frac{1}{2})$$ product of operators plus constant

We can interpret this product as successive operation, FIRST DESTROY THEN PRODUCE.
This is the stimulating point of view.