- #1
pellman
- 684
- 5
Theoretical problems often begin with "given a system in state ψ0" For example, the 2-slit experiment begins with the assumption of a plane wave incident on the slits. I had always understood to this mean some prior set of measurements had been made to determine the initial state. But how can this be done in principle?
For simplicity, suppose our system is such that a single observable O is sufficient for a complete description, and the eigenvalues of O have discrete spectrum. Let ϕk be the orthonormalized eigenstate corresponding to the kth eigenvalue of O. Then any initial state of the system has the form
ψ0 = ∑ ak ϕk
To know the initial state ψ0 is to know all the ak . But repeated measurements of O can only give us |ak|2. So how can we ever know the relative phases?
I expect there must be some way of making measurements so as to take advantage of "interference effects" to get the phases. Can someone else explain further?
In the case of spin 1/2, I derived a way of getting the relative phases of the up and down coefficients by making measurements of the spin along the other two axes, and using the resulting amplitudes to calculate the relative phase along the desired axis. This works. But what about the general case?
For simplicity, suppose our system is such that a single observable O is sufficient for a complete description, and the eigenvalues of O have discrete spectrum. Let ϕk be the orthonormalized eigenstate corresponding to the kth eigenvalue of O. Then any initial state of the system has the form
ψ0 = ∑ ak ϕk
To know the initial state ψ0 is to know all the ak . But repeated measurements of O can only give us |ak|2. So how can we ever know the relative phases?
I expect there must be some way of making measurements so as to take advantage of "interference effects" to get the phases. Can someone else explain further?
In the case of spin 1/2, I derived a way of getting the relative phases of the up and down coefficients by making measurements of the spin along the other two axes, and using the resulting amplitudes to calculate the relative phase along the desired axis. This works. But what about the general case?