he operators - we get (9.11) If we just rearrange some...

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he operators and . In a previous chapter I have discussed a solution by a power series expansion. Here I shall look at a different technique, and define two operators and , (9.4) Since (9.5) or in operator notation (9.6) (the last term is usually written as just 1) we find (9.7) If we define the commutator (9.8) we have (9.9) Now we see that we can replace the eigenvalue problem for the scaled Hamiltonian by either of (9.10) By multiplying the first of these equations by
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Unformatted text preview: we get (9.11) If we just rearrange some brackets, we find (9.12) If we now use (9.13) we see that (9.14) how that (9.15) We thus conclude that (we use the notation for the eigenfunction corresponding to the eigenvalue ) (9.16) So using we can go down in eigenvalues, using we can go up. This leads to the name lowering and raising operators (guess which is which?). We also see from ( 9.15 ) that the eigenvalues differ by integers only!...
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he operators - we get (9.11) If we just rearrange some...

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