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I am reading Griffiths Quantum Mechanics textbook, and am having some difficulty with a derivation on page 56. To me, there seems to be something logically wrong with his arguments, but I can not pin-point precisely what it is.

To provide you with a little background, Griffiths is trying to solve the Schrodinger equation for a quantum harmonic oscillator potential, employing what he refers to as the "algebraic method." He already has defined the raising and lower operators, and has already shown us how to factor the Hamiltonian operator. Here is where I am having trouble:

"Now, here comes the crucial step: I claim that if [itex]\psi[/itex] satisfies the Schrodinger equation with energy E (that is:[itex]H \psi = E \psi[/itex]), then [itex]a_+ \psi[/itex] satisfies the Schrodinger equation with energy [itex]E + \hbar \omega[/itex]..."

The rest involves him going through the proof. As I mentioned above, there seems to be something wrong with the author's reasoning, but I can't quite pin-point it. To me, it seems that I could have chosen absolutely operator I wish, call it [itex]*_1[/itex], and say that the energy is [itex]E+*_2[/itex], and just find a relationship that looks like the Schrodinger equation, which appears to be what he did.

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# Finding A Solution Using the Ladder Operators

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