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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 non-degenerate and are non-negative integers, λ ≡ n = 0 , 1 , 2 , . . . . The essence of the proof is to note that the number operator must have a non-negative 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) lowest-eigenvalue eigenvector | i such that a | i = 0 . One then easily infers that the spectrum of N is non-degenerate, 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 self-adjoint 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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