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Unformatted text preview: Physics 6210/Spring 2007/Lecture 17 Lecture 17 Relevant sections in text: § 2.3 Spectrum of the SHO Hamiltonian (cont.) It is shown in detail in the text that the eigenvalues of N are nondegenerate and are nonnegative integers, λ ≡ n = 0 , 1 , 2 , . . . . The essence of the proof is to note that the number operator must have a nonnegative expectation value: h ψ  N  ψ i = h ψ  a † a  ψ i = ( h ψ  a † )( a  ψ i ) ≥ , where h ψ  N  ψ i = 0 ⇐⇒ a  ψ i = 0 . On the other hand, since a lowers the eigenvalue of N by one unit, there must be a unique (up to normalization) lowesteigenvalue eigenvector  i such that a  i = 0 . One then easily infers that the spectrum of N is nondegenerate, consisting of the non negative integers: N  n i = n  n i , n = 0 , 1 , 2 , . . . . Therefore, the energy spectrum is purely discrete: E n = ( n + 1 2 )¯ hω, n = 0 , 1 , 2 , . . . and the energy eigenvectors are labeled  n i . These results arise from the assumption that X and P are selfadjoint operators on a Hilbert space (or, equivalently, that a and a † are adjoints of each other with respect to the Hilbert space scalar product). Finally, it is shown in the text that we have a  n i = √ n  n 1 i , a †  n i = √ n + 1  n + 1 i , h n  m i = δ nm ....
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This note was uploaded on 02/18/2012 for the course PHYSICS 6210 taught by Professor M during the Spring '07 term at AIU Online.
 Spring '07
 M
 Physics, mechanics

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