An eigenstate, or determinate state, is a state where the measurement of some observable always yields the same result. This means that the standard deviation of the observable is zero. If a distribution has a standard deviation of zero, this means that every value is the same value. There is only one value. Does this mean that an eigenstate, represented by its eigenfunction, looks like a collapsed wavefunction?
This doesn't seem right to me because the eigenfunctions of the hamiltonian for the infinite square well are sine or cosine functions (they don't look like a spike).
Maybe I'm getting a little lost with what an eigenfunction means/represents. The state of a system in Quantum Mechanics is represented by a vector, usually an infinite dimensional vector. Wavefunctions give you the probability amplitude that a particle is at a specific location at a specific time. When I solve the time-independent Schrodinger equation, I'm solving for the eigenfunctions of the energy operator, the hamiltonian. So when the hamiltonian acts on a system in one of those eigenfunctions, that system always collapses to the same point in space?
If I was in a determinate state of position, then the wavefunction would actually look like a spike? If I was in a determinate state of momentum, the wavefunction would be a sinusoidal?