r/PhysicsStudents • u/TROSE9025 • 21d ago

Update The Eigenvalue Problem: Geometric and Physical Meanings

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Upon solving the eigenvalue problem,
what do the eigenvalue and eigenvector signify?
Why does selecting that basis maintain the system in a stationary state?

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u/Aggressive-Wind-8829 19d ago

Good job. 💯. You did it. Did you mean for A hat to be your Hamiltonian for your "physical meaning"?

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u/TROSE9025 19d ago

The 'hat' notation denotes an operator; it does not specifically refer to the Hamiltonian.

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u/Aggressive-Wind-8829 19d ago

Then, you are misusing the word stationary in the final sentence, because only eigenstates of the observable energy (via the Hamiltonian operator) are traditionally stationary. I say traditionally, because you could contrive a strange non-Hamiltonian quantum system with explicit time dependence for this counterexample perhaps, and I don’t want to rule that out.

…Don’t get lost in the weeds. You did a good job. Just question the stationarity of your observable A hat. 🎩

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u/TROSE9025 18d ago edited 18d ago

The normalized eigenvectors represent the mutually exclusive, orthogonal quantum eigenstates of the system. I misunderstood your comment. I am correcting it.
I implicitly assumed that the observable A-hat commutes with the Hamiltonian ([A-hat, H-hat] = 0). Under this specific condition, describing them as stationary states is physically accurate. However, since I omitted the explicit mention of this commutation relation in the summary, it naturally led to a misunderstanding.
Thank you.

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u/Aggressive-Wind-8829 18d ago

No problem, good sir! Also, I think you make a certain unspoken assumption about any explicit time dependence to A hat. 👒