Where Can I Find Comprehensive Notes on Quantum Mechanics Interpretations?

[vanhees71]

@Eloheim, there are two versions of the Born Rule.

In the first version, stated by Weinberg, if we make a measurement of O, the system that is in state ψ will collapse into an eigenstate |o_i> with eigenvalue o_i, with probability |<o_i|ψ>|².

In the second version, if we make a measurement of O, the system that is in state ψ will give the result o_i with probability |<o_i|ψ>|². So there is no collapse in the second version. If this version is used, most textbooks add collapse as a distinct postulate.

As I understand it, "collapse" and the "projection postulate" are the same, but vanhees71 is using the second version of the Born rule without collapse. Both versions of the Born rule cannot be derived from the Schroedinger equation.

I do accept that there are correct interpretations without collapse, such as Bohmian mechanics and probably also many-worlds. What I am skeptical about is whether the ensemble interpretation without collapse is correct, if one allows filtering type measurements as a means of state preparation.

I never understood, why the state must collapse to the eigenstate, while when we apply QT to real-lab experiments we use the second version only. QT doesn't say, what happens to the measured system but that's given by the measurement apparatus we use to measure the observable. The Born postulate just states that the probability to find a value as you said above (tacitly assuming that the eigenvalue is non-degenerate, i.e., the eigenspace of this eigenvalue is one-dimensional).

There are, of course, special cases, where you (can) do an ideal von Neumann filter measurement. The most famous example is the Stern-Gerlach experiment, which nowadays can be done with practically arbitrary precision using neutrons.

The good old original setup by Stern and Gerlach is, however, better to discuss this in principle: You use an oven with a little opening to get a particle beam that can be described by a mixed state (thermal in the restframe of the gas, making up the particle beam). The particles then go through an inhomogeneous magnetic field. One solve the corresponding dynamical problem in very good approximation analytically and even exactly to arbitrary precision numerically, see e.g.,

G. Potel et al, PRA 71, 052106 (2005).
http://arxiv.org/abs/quant-ph/0409206

You end up with well-separated partial beams that are "sorted" (with high precision) after their spin components given by the direction of the magnetic field, usually chosen as the z direction. In other words, after running trough the magnet you have a quantum state, where the position and spin-z component are entangled, and you can get a particle beam in a (nearly) pure spin state by just forgetting all unwanted partial beams, blocking them by some absorber material. There is no collapse necessary but just to put some absorber material in the way of the "unwanted" partial beams. Of course, you may now ask, how the absorption process happens microscopically in terms of quantum theory, and this might not be a simple issue, but experiment clearly shows that you can block particles, and nothing hints at some "collapse mechanism" that may ly outside of the quantum dynamics of a particle interacting with the particles in the absorber.

 

[atyy]

I never understood, why the state must collapse to the eigenstate, while when we apply QT to real-lab experiments we use the second version only. QT doesn't say, what happens to the measured system but that's given by the measurement apparatus we use to measure the observable. The Born postulate just states that the probability to find a value as you said above (tacitly assuming that the eigenvalue is non-degenerate, i.e., the eigenspace of this eigenvalue is one-dimensional).

There are, of course, special cases, where you (can) do an ideal von Neumann filter measurement. The most famous example is the Stern-Gerlach experiment, which nowadays can be done with practically arbitrary precision using neutrons.

The good old original setup by Stern and Gerlach is, however, better to discuss this in principle: You use an oven with a little opening to get a particle beam that can be described by a mixed state (thermal in the restframe of the gas, making up the particle beam). The particles then go through an inhomogeneous magnetic field. One solve the corresponding dynamical problem in very good approximation analytically and even exactly to arbitrary precision numerically, see e.g.,

G. Potel et al, PRA 71, 052106 (2005).
http://arxiv.org/abs/quant-ph/0409206

You end up with well-separated partial beams that are "sorted" (with high precision) after their spin components given by the direction of the magnetic field, usually chosen as the z direction. In other words, after running trough the magnet you have a quantum state, where the position and spin-z component are entangled, and you can get a particle beam in a (nearly) pure spin state by just forgetting all unwanted partial beams, blocking them by some absorber material. There is no collapse necessary but just to put some absorber material in the way of the "unwanted" partial beams. Of course, you may now ask, how the absorption process happens microscopically in terms of quantum theory, and this might not be a simple issue, but experiment clearly shows that you can block particles, and nothing hints at some "collapse mechanism" that may ly outside of the quantum dynamics of a particle interacting with the particles in the absorber.

It would be nice to have an example of how this is done for successive measurements. The reason I am skeptical is that by including the apparatus and environment in the unitary evolution, to get the state for a subsystem, one has to trace out degrees of freedom, which in general will result in an improper mixed state, not a pure state. If we interpret the improper mixed state as a proper mixed state, we can get a pure state for a sub-ensemble. So I think the assumption still has to be made that an improper mixed state can be treated as a proper mixed state.