On Tuesday, July 29, 2025 at 6:01:15 PM UTC-6 Bruce Kellett wrote:
On Wed, Jul 30, 2025 at 9:42 AM Alan Grayson <[email protected]> wrote: On Tuesday, July 29, 2025 at 5:33:40 PM UTC-6 Brent Meeker wrote: On 7/29/2025 1:12 PM, Alan Grayson wrote: On Tuesday, July 29, 2025 at 2:04:31 PM UTC-6 Brent Meeker wrote: On 7/29/2025 7:18 AM, Alan Grayson wrote: Assuming we know all possible results of the measurements of a quantum system, that is, the set of possible eigenvalues, and suppose we also know the associated eigenfunctions, and we write the wf of the system as a linear sum of eigenfunctions each multiplied by a complex constant, is it mathematically assumed, or proven somewhere (perhaps by Von Neumann), that these eigenfunctions are orthogonal and form a basis for the Hilbert space in which they reside? TY, AG -- Yes, that's pretty much it. The physical system, including the ideal measurement, is modeled by a certain Hilbert space in which the basis states are the eigenfunctions the measurement. This is implicit in the concept of an ideal measurement as one, which if immediately repeated on the same system, returns the same value again. Brent But is it proven or assumed the eigenfunctions in the sum are basis states which span the space? If proven, where, by whom; if not, then the construct lacks rigor. AG It's true by construction that the eigenstates span the Hilbert space. "The Hilbert space" is the space whose bases are the eigenstates. What does "true by construction" mean? Does that include orthogonality of the basis eigenstates? AG A lot of these things are proved in Dirac's book "The Principles of Quantum Mechanics". For example, the orthogonality of the eigenfunctions of a single operator is proved on page 32 (of my edition). "Two eigenvectors of a real dynamical variable belonging to different eigenvalues are orthogonal". Thanks for the reference. I recently bought that book! AG Any set of linearly independent vectors forms a possible basis of a vector space if the number of linearly independent vectors equals the dimension of the space (The linearly independent vectors need not form a mutually orthogonal set, as long as they are linearly independent.) Bruce -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion visit https://groups.google.com/d/msgid/everything-list/5ae241ef-9976-4fd8-bce3-a1b78c84e53bn%40googlegroups.com.
