What is interaction and measurement in QM?

Question

My understanding of "measurement in quantum mechanics" was that any interaction causes the probability "fuzziness" to get "resolved" to a particular value for an observable. However, unlike that, a non-observed, non-interacting particle has some probability to be "anywhere" until "interaction" with some other particle.

If an unobserved particle has a probability to be anywhere, then doesn't it sort-of "exist" everywhere? Wouldn't it interact with something, somewhere? Therefore, a non-interacting particle cannot exist by definition unless it is the only particle in the universe. If this conclusion actually correct in quantum mechanics, or my idea of interacting probability distributions just plain wrong?

Answer 1

The Definition of an interacting particle is a particle that CAN interact with other particles. An interacting particle is able to interact with other particles or even with itself (self-interactions are predicted by Quantum field theory).

Quantum mechanics/ Quantum field theory introduces randomness in the Dynamics of particles; this Quantum-mechanical random behavior of a particle gets less random if there are other particles, where the particle does interact with. It is often called "decoherence" or "collapse of wavefunction".

Formally, we can express the loss of random behavior of the particle due to interaction with its surroundings as follows:

Suppose the particle is in state $|\psi> = \sum_nc_n|n>$ where $n$ is a Quantum number like (angular) momentum (orthonormal basis) and $c_n$ the Expansion coefficients at the beginning. The surrounding system is in state $|\phi>$ at this time. Now let the particle evolve in time. It may interact e.g. by self-interactions, vacuum fluctuations. Now we wait until the first interaction with the System took place. At this time, the System changes to $|\phi'(m)>$ induced by the transferring of the Quantum number $m$ of the particle to the system. Moreover, the particle changes to the state $|\psi'(m)>$. Thus, we have the process

$|\psi> \otimes | \phi> \rightarrow |\psi'(m)> \otimes | \phi'(m)>$.

Now, let again be all possible outcomes a Superposition of above outcomes, i.e.

$|outcome> = \sum_m d_m |\psi'(m)> \otimes | \phi'(m)>$

with probability coefficients $d_m$. If we ask for the probability that the outcome will be the transferring of Quantum number $l$, we compute

$|<\psi'(l)|outcome>|^2 = |\sum_m d_m <\psi'(l)|\psi'(m)> \otimes | \phi'(m)>|^2 =$

$|d_l|\phi'(l)>|^2$

$= d_ld_l^*$

where orthonormality was used in the second and third line. This is simply a classical probability without any overlap contributions.

Quantum-mechanically, wave functions can also overlap, and due to interaction with surroundings, this overlap is lost; a decoherence took place.

Answer 2

My understanding of "measurement in quantum mechanics" was that any interaction causes the probability "fuzziness" to get "resolved" to a particular value for an observable

The best way to understand interactions is the iconal representation of Feynman diagrams. Let us take the example of electron electron scattering , for simplicity:

The interaction is the calculated cross section for an electron to be repulsed from another electron as a function of (x,y,z,t) , given by doing the integration that the diagram prescribes. The two electrons "interact" with each other. A measurement means real numbers, which will define a single (x,y,z,t). (or momentum and energy) One has to accumulate measurements to get the measured interaction crossection ( qm probability dependent) and check it against the calculation.

Here is an electron in a bubble chamber in a magnetic field.

The curly line was produced by an electron that was struck by one of twelve $K^-$ passing beam particles in a liquid hydrogen bubble chamber. It curves in an applied magnetic field and loses energy rapidly, spiralling inwards.

It is one point in measuring the $K^-$ e- scattering, electron at rest, as from the magnetic field one can get the momentum and thus the energy, one measurement of the interaction that can be calculated using the feynman diagrams for $K^-$ e- to $K^-$ e- scattering.

The electron itself is losing energy because it is kicking off electrons from the hydrogen atoms of the bubble chamber in an interaction of e-e- scattering, as in the diagram above. Each dot of ionisation is a measurement ( though a lot of assumptions have to be made in order to get the functional form of the interaction crossection).

Rule of thumb: measurements imply interactions, interactions are a much larger set than measurements.