Timeline for Can there be an infinite set of commuting quantum mechanical operators, and how are their common eigenfunctions to be determined?
Current License: CC BY-SA 4.0
9 events
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| Dec 2, 2020 at 16:35 | comment | added | Roger V. | @Doc I think you should reformulate your question, to make it more precise, or ask a new one. So that others can participate in the discussion. | |
| Dec 2, 2020 at 15:57 | comment | added | Doc | Let us continue this discussion in chat. | |
| Dec 2, 2020 at 15:40 | comment | added | Roger V. | It seems that you are confusing the operators and the wave functions (i.e., the functions on which the operators can act). | |
| Dec 2, 2020 at 15:33 | comment | added | Doc | The above hydrogenic wavefunctions are not eigenfunctions of the Kinetic Energy (Z^2/2n^2) and Momentum (Z/n) operators (their respective eigenvalues), and these operators do not commute with the above operators. Analogously, what are the atomic operators that with the Kinetic Energy and Momentum operators would form a complete set of commuting operators? | |
| Dec 2, 2020 at 15:33 | comment | added | Doc | The atomic wavefunctions with which I am most familiar are the hydrogenic wavefunctions in spherical polar coordinates that are the eigenfunctions of the atomic Total Energy (-Z^2/2n^2), Total Angular Momentum Squared l(l+1), and Azimuthal Angular Momentum (m) operators (their respective eigenvalues). | |
| Dec 2, 2020 at 13:03 | comment | added | Roger V. | Secondly, you can take the recipe proposed in my answer: any function of a momentum operator, $f(\hat{p})$, will commute with this operator. Note, however, that the functions are understood either as Taylor expansions or in terms of the spectral decomposition, as in my answer. But requiring analyticity is an additional constraint, which limits the number of possible operators. | |
| Dec 2, 2020 at 13:03 | comment | added | Roger V. | Firstly, it is not clear from your question what specific atomic operators you mean - how are they different from the standard kinetic energy and momentum operators? Do you simply mean that they should be considered in polar coordinates with atomic wave functions as the basis? - but then you are already talking about their matrix form. | |
| Dec 2, 2020 at 12:29 | comment | added | Doc | As was indicated, the operators of interest are the atomic KE and momentum operators with eigenvalues that are functions of hydrogenic quantum numbers with analytic eigenfunctions. Can you provide or suggest an answer that is explicitly analytic and not a solution in a general matrix and vector representation? | |
| Dec 2, 2020 at 11:44 | history | answered | Roger V. | CC BY-SA 4.0 |