Find the eigenvalues of the Hamiltonian - Harmonic Oscillator

1. The problem statement, all variables and given/known data



Find the eigenvalues of the following Hamiltonian.



[itex]Ĥ = ħwâ^{†}â + \alpha(â + â^{†}) , \alpha \in |R[/itex]





2. Relevant equations



[itex]â|\phi_{n}>=\sqrt{n}|\phi_{n-1}>[/itex]

[itex]â^{†}|\phi_{n}>=\sqrt{n+1}|\phi_{n+1}>[/itex]



3. The attempt at a solution



By applying the Hamiltonian to a random state n I get:



[itex]Ĥ |\phi_{n}> = E_{n}|\phi_{n}>[/itex]

[itex]Ĥ |\phi_{n}>= ħwâ^{†}â|\phi_{n}> + \alpha(â|\phi_{n}> + â^{†}|\phi_{n}>) [/itex]

[itex]Ĥ |\phi_{n}>= ħw\sqrt{n}\sqrt{n}|\phi_{n}> + \alpha(\sqrt{n}|\phi_{n-1}> + \sqrt{n+1}|\phi_{n+1}> )[/itex]

[itex]E_{n} |\phi_{n}> = ħwn + \alpha(\sqrt{n}|\phi_{n-1}> + \sqrt{n+1}|\phi_{n+1}>) [/itex]



This is where my problem arrives. I don't know how to prove that



[itex]\alpha(\sqrt{n}|\phi_{n-1}> + \sqrt{n+1}|\phi_{n+1}>) = 0[/itex]



Any help would be highly appreciated!

Thanks.

via Physics Forums RSS Feed http://www.physicsforums.com/showthread.php?t=720491&goto=newpost