In quantum mechanics, while studying the harmonic oscillator, I learnt about the ladder operator. And I realised that if you are able to find or determine any energy state of the quantum harmonic oscillator then, using the ladder operator, you can determine the other energy states as well. However, in none of the texts that I was reading, did I find mention of the following fact:

The energy states determined by the above procedure, that is, using the ladder operator are the only possible energy states of the harmonic oscillator. There exists no energy state that is not given by the ladder operator.

So this is my question:

Are the energy states determined by the ladder operator in case of a harmonic oscillator, the only possible energy states? Is any other energy state possible? And what is the proof?

It is well known that the above-mentioned states are the only possible energy States but I want a rigorous proof that no other state is possible. However the analytical procedure mentioned in different books show an approximate solution of the Schrodinger equation which can be considered to be rigorous in the sense that it solves the equation to derive the solutions although it considers certain approximations.

I have searched for this answer in books related to quantum mechanics written by DJ Griffiths, Gasiorowicz, Dirac and online resources like OCW, University of Columbia's courses, caltech.edu. But I couldn't find a proper answer.

Thank you for your help.

In quantum mechanics, while studying the harmonic oscillator, I learnt about ladder operators. And I realised that if you are able to find or determine any energy state of the quantum harmonic oscillator then, using the ladder operators, you can determine the other energy states as well. However in none of the texts I read I found the following fact:

The energy states determined by the above procedure, that is, using the ladder operator are the only possible energy states of the harmonic oscillator. There exists no energy state that is not given by the ladder operator.

So this is my question:

Are the energy states determined by the ladder operator in case of a harmonic oscillator, the only possible energy states? Is any other energy state possible? And what is the proof?

It is well known that the above-mentioned states are the only possible energy states but I want a rigorous proof that no other state is possible. However the analytical procedure mentioned in different books show an approximate solution of the Schrodinger equation which can be considered to be rigorous in the sense that it solves the equation to derive the solutions although it considers certain approximations.

I have searched for this answer in books related to quantum mechanics written by DJ Griffiths, Gasiorowicz, Dirac and online resources like OCW, University of Columbia's courses, caltech.edu, but I couldn't find a proper answer.