Measuring quantum mechanics changes the answer and energy?

Question

A thought occured to me thinking about the quantum slit experiment. Upon firing an electron or particle it would impart an opposing force on the machine shooting it. The direction if this force should give you the direction your electron should be going and allow you to predict if it went through a slit or not. When you do, does it count as a measurement and stop the interference pattern? Or does the particle need to be measured?

Imagine for a moment it does still create an interference pattern because you didnt measure the particle itself. If you then measure the electron if and when it goes through a slit AFTER you measured the force in the opposite direction, would the energy be at odds compared to a particle going the exact same direction but it wasn't measured (because it would go through both slits and none at the same time and thus have multiple potential directions)?

Answer 1

Even if you know the initial direction of the particle, this does not tell you which slit the particle will go through. The wavefunction (probability wave) of the particle will still follow the Schrodinger equation. For example, the initial condition in solving the Shrodinger equation could be a Gaussian wave packet in space centered at the measured starting position with some standard deviation determined by the device measuring the initial position. The Gaussian will also be determined by the measured momentum (with some magnitude and direction, each with their own associated uncertainty). This wavefunction will then evolve according to the Schrodinger equation.

Getting to the heart of your question: just because we know the initial position and direction of the particle does not mean it follows a well defined trajectory afterwards. This is just not the case according to basic interpretations of quantum mechanics.

Answer 2

There is only interference if the setup is such that it is not possible to determine through which slit the electron went.

Answer 3

Whenever you're analyzing such conceptual problems, it's very useful to try and make a concrete model of the system you're considering. Maybe even multiple models, to better see the common properties.

Suppose you have an apparatus, which fires an electron in one direction, and recoil (or whatever other mechanism) results in a second electron being ejected with the opposite momentum. To be able to use the momentum being opposite to predict where the first electron will be, you need the initial positions to also be accurately defined: otherwise even classically you'll get uncorrelated results.

Now, the two-electron system, being quantum mechanical, obeys Heisenberg uncertainty principle. Since both particles are in a precisely defined location, their total momentum is quite uncertain. This then means that, by measuring the momentum of the second electron, you'll obtain the momentum of the first electron... ± that additional uncertainty. The final result will be that you'll still get garbage results if you try to answer the which-way question for the first electron using this measurement of the second electron's momentum.

Answer 4

If you are able to determine which slit the particle went through, by any means, there is no interference pattern.

In practice, I think it would be difficult to set up the experiment as described. But in principle, yes, if you measured the recoil accurately enough to determine which slit the particle went through, the interference pattern would disappear.

(Although as Aaron's answer points out, knowing the momentum does not necessarily determine which slit the particle goes through. I'm assuming that the momentum is high enough, and the slits far enough apart, that the probability of a particle going through the "wrong" slit is minimal. The probability is never going to be zero, so there will always be some interference, but it could in principle be minimized - and that interference pattern would look different to the original one anyway.)

Answer 5

This (your first paragraph) is the same question/famous objection Einstein posed in the Bohr-Einstein debates: https://en.wikipedia.org/wiki/Bohr%E2%80%93Einstein_debates#Post-revolution:_First_stage

So basically Bohr reply was: You also need to apply QM to the double slit. So to measure the momentum transfer, you have to have a sharp momentum (of the measurement apparatus) first. But then your position is not determined leading to washing out of the interference pattern.