Assume that H is an operator valid on some hilbert space, N, with nondegenerate eigenvalues, E_n, whose corresponding eigenbasis is {g_n}. If f is a vector that lies on N, then f can be expanded as a linear combination of the eigenbasis of H. Now mathematically, we can say that

H|f> = sum(n, E_n*c_n*|g_n>😉.

This implies that H changes the the vector, but it is still expanded in the entire eigenbasis. However, if we consider this problem from a physical perspective, |f> collapses into one of the eigenstates it is expressed in, or

H|f> ---> E_n*c_n|g_n>

where there is no summation. Which view is correct?

H doesn't necessarily have to collapse f into one of the eigenstates, it can be a linear combination of eigenstates.

In fact for most "measurement" processes you will not be able to find a well-defined H that does what you just described.

I am under the impression that an observation on a vector returns a single value. If f is expressed in the eigenbasis of H, and the eigenvalues of H are non-degenerate, wouldn't this force f to return an eigenvalue of H, and therefore collapse it into an eigenstate of H.

For instance, assume |f> = c_1|g_1> + c_2|g_2>.

Also assume P is an operator such that

P|g_1> = 1
P|g_2> = -1

Then P|f> must return either 1 or -1, correct? Then for instance, if P|f> = 1 wouldn't |f> have collapsed into |g_1>?

H acting on f is not an observation. H describes how f evolves in time.

This is where it gets confusing for me. I understand That H|f>=E|f> describes the evolution of the wave-function. However, I was also under the impression that a hermitian operator acting on a state-vector can be seen as an observation. I am having a hard time making these two views coincide. I am very confused by all this, sorry...

Originally posted by amolv06
This is where it gets confusing for me. I understand That H|f>=E|f> describes the evolution of the wave-function. However, I was also under the impression that a hermitian operator acting on a state-vector can be seen as an observation. I am having a hard time making these two views coincide. I am very confused by all this, sorry...