Multitude of Quantum Mechanics questions

[Master J]

I am studying Quantum mechanics at the moment, and so naturally a few questions arise!
So I will post them in this thread.

Cheers for any input, I really appreciate all the help that I get!

First off:

1. My picture of the Hamiltonian is that it is the total spectrum of possible energy values for a system, but to know what energy state it is actually in, one must perform a measurement on the system.This measured valued would then be an eigenvalue of the Hamiltonian. Is this correct?

2. For the time independent Schrödinger equation in one - dimension, can one find a normalizable solution for any x in the case the E<V (V is the potential)?? I know the graph of such a solution would look like a parabola ( and a parabola heading towards negative infinity also), and so I would presume no, but why not exactly?

3. The solutions to the time indep Schrödinger equation in an infinite well form a complete set. Does this mean ANY function in mathematics? This would in fact be the definition of a Fourier Series in this case...I find that odd and somewhat hard to believe!

 

[Dale]

1. My picture of the Hamiltonian is that it is the total spectrum of possible energy values for a system, but to know what energy state it is actually in, one must perform a measurement on the system.This measured valued would then be an eigenvalue of the Hamiltonian. Is this correct?

I would say that the set of all eigenvalues of the Hamiltonian are the total spectrum of possible energy values for a system.

2. For the time independent Schrödinger equation in one - dimension, can one find a normalizable solution for any x in the case the E<V (V is the potential)?? I know the graph of such a solution would look like a parabola ( and a parabola heading towards negative infinity also), and so I would presume no, but why not exactly?

What kind of potential are you thinking of here? Are you thinking of a finite well a harmonic oscillator or what?

3. The solutions to the time indep Schrödinger equation in an infinite well form a complete set. Does this mean ANY function in mathematics? This would in fact be the definition of a Fourier Series in this case...I find that odd and somewhat hard to believe!

In general it does not mean any function in mathematics. It means any vector in some space of functions, for example L2(0,1). Typically this could be described as any "well behaved" function in the well.