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For quantum mechanics to be internally consistent, the following must be true:
In other words, it is not alright for the solution space (to the Schrodinger equation) to span only a subspace of the Hilbert space. It has to span the entire Hilbert space. But why? What's inconsistent about it?
I find this quite unbelievable, because it's saying any function in Hilbert space is a solution to Schrondinger equation. But when we solve an equation, we don't expect everything to turn out to be a solution.
Reference: Introduction to Quantum Mechanics, 2nd ed., David J. Griffiths, p 102.
The eigenfunctions of an observable operator are complete: Any function (in Hilbert space) can be expressed as a linear combination of them.
In other words, it is not alright for the solution space (to the Schrodinger equation) to span only a subspace of the Hilbert space. It has to span the entire Hilbert space. But why? What's inconsistent about it?
I find this quite unbelievable, because it's saying any function in Hilbert space is a solution to Schrondinger equation. But when we solve an equation, we don't expect everything to turn out to be a solution.
Reference: Introduction to Quantum Mechanics, 2nd ed., David J. Griffiths, p 102.