Solving the Schrodinger Equation for a particle being measured

[physwiz222]

TL;DR Summary

Why doesnt anyone attempt to solve the Schrodinger Equation obviously numerically for the situation where a free electron is measured by simulating the interaction between a detector and an electron as a way to gain insight and possibly solve the measurement problem.

In Quantum Mechanics the measurement problem is that once a system is measured the wavefunction inexplicably collapses into an eigenstate we all know this. Many believe the localization is due to interacting with the detector. If thats so why doesnt anyone try and model this interaction numerically.

What I want to know is why has no one attempted to numerically solve obviously due to the large number of particles which make up a detector the Schrodinger Equation for a detector measuring an electron where you essentially model the detector quantum mechanically and the electron as usual and you essentially see if the localization due to detection emerges.

I want to know are there any issues with this preposition and why no one seems to have attempted this solution to the measurement problem. Is it because the number of particles in a detector is simply too big, do we just not have the computational power. If so why has no one discussed Numerical modelling of a detector measuring a particle.

 

[PeroK]

A computer simulation is only possible for simple quantum systems. Avogadros number is $6 \times 10^{23}$, so you could couldn't even have one bit per atom in a mole.

 

[Demystifier]

Detectors can be described by models with a small number of degrees of freedom. See e.g. Sec. 6.1 of https://arxiv.org/abs/1406.5535

 

[vanhees71]

You can, of course, treat this problem in quantum many-body theory. There's a vast literature about it. It solves the so-called measurement problem to a certain extent by the discovery of decoherence. A nice book about all this is

E. Joos, H.D. Zeh, C. Kiefer, D. Giulini, J. Kupsch I.-0. Stamatescu, Decoherence and the Appearance of a Classical World in Quantum Theory, 2nd edition, Springer (2003)

 

[LittleSchwinger]

The worked model in chapters 7 and 8 of Roland Omnes book on quantum mechanics is probably the easiest place to study decoherence in detail (do the exercises as well!)

 

[gentzen]

In Quantum Mechanics the measurement problem is that once a system is measured the wavefunction inexplicably collapses into an eigenstate we all know this.

Well, even if you reject the collapse postulate, the measurement problem still won't go away. Why are there definite measurement results that obey classical logic FAPP?

Many believe the localization is due to interacting with the detector.

Who believes this? And does he believe that the definite measurement result are caused by this, or does he believe that the collapse of the wavefunction into an eigenstate is caused by this?

If thats so why doesnt anyone try and model this interaction numerically.

Not sure what you mean by "numerically". You want to compute some high dimensional wavefunction on a computer? And what should the computer do with that high dimensional wavefunction? Check that large parts of the wavefunction associated with the system are close to being localized near an eigenstate? And repeat that simulation for many different (perhaps somehow special) initial quantum states of the detector?

What I want to know is why has no one attempted to numerically solve obviously due to the large number of particles which make up a detector the Schrodinger Equation for a detector measuring an electron where you essentially model the detector quantum mechanically and the electron as usual and you essentially see if the localization due to detection emerges.

You indeed seem to have something similar in mind to what I described above.

I want to know are there any issues with this preposition and why no one seems to have attempted this solution to the measurement problem. Is it because the number of particles in a detector is simply too big, do we just not have the computational power. If so why has no one discussed Numerical modelling of a detector measuring a particle.

The words "numerically" and "computational power" might be red herrings here. If you have some model, you can often analyse it by traditional means. But the question remains whether the "definite measurement results that obey classical logic FAPP" were not somehow put-in by hand into those models.

At least that is the question I ask myself when somebody suggests some paper(s) that at first glance seem to try to do something like you suggest (even so I guess that on closer study, I would learn that those papers have different goals and actually try to do something else):

I would suggest having a read of a full recent model of how macroscopic commutativity arises, such the Allahverdyan et al (2011). Environmental Decoherence is actually not the dominant reason for classicality. Equilibration processes, thermalisation and the contraction of the algebra of observables are stronger effects. See the following recent paper by Frohlich for a rigorous worked model of the latter.

Just to add decoherence isn't really the main factor responsible for the classical limit. Even before the investigation of decoherence in the 1970s there were detailed models of classicality being caused by ergodic effects or kinematic effects reducing the algebra of observables, such as in the WAY theorem.
... In fact if one thinks about it, ... can be easily shown to display quantum effects. ...
A good, and very long, guide to all this is the well known paper of Allahverdyan et al:
https://arxiv.org/abs/1107.2138

 

[gentzen]

The worked model in chapters 7 and 8 of Roland Omnes book on quantum mechanics is probably the easiest place to study decoherence in detail (do the exercises as well!)

This seems to refer to "The Interpretation of Quantum Mechanics" from 1994.

7. Decoherence
Orientation
1. An Intuitive Approach
Solvable Models
2. A Simple Model
3. Another Example: The Pendulum
More General Models
4. The General Theory
5. Decoherence by the External Environment
6. Back to Schrodinger's Cat
Can One Circumvent Decoherence?
7. A Criticism of Decoherence
8. One Cannot Circumvent Decoherence
9. Justifying the Assumptions*
10. The Direction of Time
Appendix: Decoherence from an External Environment
Problem

8. Measurement Theory
1. Reality and Theory. Facts and Phenomena
2. An Introduction to Measurement Theory
Measurement of a Single Observable
3. What Is a Measurement?
4. The Main Theorems
Wave Function Reduction
5. Two Successive Measurements
Actual Facts
6. Actual Facts and the Present Time
7. Everett's Answer
8. A Law of Physics Different from All Others
The Notion of Truth
9. The Criteria of Truth
10. Up to What Point Can One Know the State?
11. Explicit States
Appendix A: The Theorems of Measurement Theory
Appendix B: The Density Operator and Information Theory

The only excercise I could find in chapters 7 and 8 was the one at the end of chapter 7:

---

The decoherence factor (7A.3) can be written as $\exp(-t/T)$, where $T$ may be called the decoherence time. One considers as the object a sphere having a radius $R$, under the following conditions: (i) in air at normal temperature and pressure, (ii) in a perfect vacuum at the surface of the earth, in the full light of the sun, (iii) in an intergalactic vacuum, containing only the cosmological 3°K radiation, and (iv) in a laboratory vacuum with $10^6$ particles per $\text{cm}^3$.
Joos and Zeh give the value of the decoherence times (in seconds) for these various cases as the following:

Object	Dust	Aggregate	Big Molecule
R(cm)	10-3	10-5	10-6
(i)	10-36	10-32	10-30
(ii)	10-21	10-17	10-13
(iii)	10-6	10+6	10+12
(iv)	10-23	10-19	10-17

Comment on these values.