Harmonic Oscillator: Energy Levels Explained

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why is the energy levels of the harmonic oscillator En=(N+1/2)hf?
 
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Is it derived in your textbook? It's not a particulary short derivation regardless of the method you use do I'd rather not type it out here. As for why the energy levels are like that because that's what you get when you solve Schrödinger's equation for a potential of the harmonic oscillator form.
 
no it's not derived in my textbook... thank you~
 
That expression for E arises when solving the Schroedinger Equation analytically using the Power Series Technique (I don't know of any other analytical technique - if anyone else does let me know).
The solution is not hard but it is long.

An easier way to derive E is to use raising and lowering operators.
 
thank you very much! :)
 

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