I Understanding the Irreducible Solution in Classical Harmonic Oscillators

[George444fg]

TL;DR

Harmonic Oscillator

Hellow. I am doing an introductory to Quantum Mechanics course, and the irreducible solution appeared in the harmonic oscillator. When we talk about the irreducible solution, this is the solution as a linear combination of the eigenbasis of the system. This is understandable, however, if I have a simple case of a harmonic oscillator, with solution sin(wt) then the irreducible solution would be ie^(-iwt)? Thank you in advance

 

[Haborix]

I'm a little confused by the terminology you use, particularly irreducible solution. But I read your post as defining the irreducible solution as the solution written as the sum of eigenvectors. Also, are you referring to the quantum harmonic oscillator or the classical harmonic oscillator in the last half of your post?

 

[George444fg]

I'm a little confused by the terminology you use, particularly irreducible solution. But I read your post as defining the irreducible solution as the solution written as the sum of eigenvectors. Also, are you referring to the quantum harmonic oscillator or the classical harmonic oscillator in the last half of your post?

It is the classic oscillator. But nonetheless, it can be derived from the quantum oscillator.