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Simple harmonic oscillatorsQuantum mechanics 
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#1
Nov1110, 04:01 PM

P: 59

1. The problem statement, all variables and given/known data
An ion in a harmonic ion trap sees a potential which is effectively that of a simple harmonic oscillator. It has a natural oscillation frequency given by v = 1 MHz. Ignoring any internal excitations, it is known to be in a superposition of the n = 0, 1 and 2 SHO energy states. A measurement is then made and it is found to be in the n = 2 level. a)What is the energy of the ion after the measurement has been made? 3. The attempt at a solution Why is the answer E_n = (2n+1)/2 [tex]\hbar\omega[/tex] I do not understand the (2n+1) / 2 Thanks! 


#2
Nov1210, 11:53 AM

P: 599

Hi,
The average energy in the n^{th} state (or in the phonon picture: number of phonons in a mode associated with frequency [tex]\omega[/tex]) for a single harmonic oscillator is given by: [tex]E_n=\frac{2n+1}{2}\hbar\omega=(n+\frac{1}{2})\hbar\omega.[/tex] where [tex]h\nu=\hbar\omega.[/tex] 


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