If uncertainty principle is explained by wave function, then doesn't wave function collapse when we measure position or momentum?

Question

I have been using an app called Quantum on the play store. It says that the uncertainty principle can be explained by wave function, that is when we try to determine position , the wave function should be localised which means momentum is uncertain and vice versa.

My question is that if we measure anything of the particle, the wave function should collapse and any uncertainty should vanish.

Sorry if the question is dumb.

EDIT: My textbook (published by the GOVERNMENT) explains it like: When we measure the position of a particle, a photon hits it and changes its momentum. This has been refuted by the app & wikipedia (looking at your answers, even you guys) as a confusion between uncertainty principle and observer effect. Can you refer a good reputed book for me to complain?

Answer 1

My question is that if we measure anything of the particle, the wave function should collapse and any uncertainty should vanish.

If an ideal position measurement is made and the particle if found to be at $\mathbf{x}$, and then another ideal position measurement is made immediately after the first, the result of the second measurement is certain to be $\mathbf{x}$. So uncertainty in the result of this second position measurement does indeed vanish.

However, if the second measurement is instead an ideal momentum measurement, any result is equally probable, i.e., there is 'infinite' uncertainty in momentum.

So it isn't true that any uncertainty should vanish.

Answer 2

The question isn't dumb at all, it's great to think about these things.

Yes, when you measure a wavefunction, you collapse it. But it's what you collapse it into that's interesting. You collapse it into a special wavefunction called an "eigenfunction" of the operator.

In general, we can build any wavefunction by adding up these different eigenstates. The eigenstates are building blocks for our wavefunction.

Let's talk about the classic position and momentum uncertainty principle. If you measure the position of a particle, you collapse it into a function which is just a spike at the location of the particle. This is called a delta function. Whereas if you measure the momentum of a particle, you collapse it into a momentum eigenfunction, which is basically an endlessly repeating wave.

These two functions are very different. They're about as different as you can imagine. One exists as just one point in space, the other is defined at all points in space. And here is where the uncertainty is. Although you have measured with precise certainty the position of a particle, as a result the wavefunction is now a spike. Now if we try to express that spike, in terms of possible momentum wavefunctions, we find we need to include all possible momentum wavefunctions. So by being in a position eigenstate, the spike function, it necessarily means the particle has all values of momentum.

The same applies the other way round. If we collapse it into a momentum eigenfunction, an endlessly repeating wave, then to build this wave from the spike functions, we find we need to add up an infinite amount of them, and so the particle could be anywhere!

Generally speaking, the more position eigenstates we need to describe a wavefunction, the fewer momentum eigenstates we'll need. Vice versa is also true. This is the uncertainty principle.

EDIT: It would be good if you could indicate your education background, e.g. knowledge of maths/physics, so we could direct you to further resources, as you clearly could go a step further than the app you're using now.

Answer 3

If uncertainty principle is explained by wave function, then doesn't wave function collapse when we measure position or momentum?

The uncertainty principle as the other answers state comes in pairs of variables, being measured whereas the (in)famous collapse of the wave function happens with every measurement.

The wavefunction squared with its complex conjugate gives a probability distribution for finding a particle ( for simplicity) at (x,y,z,t) or with a fourvector (p_x,p_y,p_z,E). A probability distribution is a statistical measure, it needs many instances.

In throwing a dice, if the 6 face comes up the probability that it is up from then on is 1, and one needs to throw again to build up a probability distribution. You can say that the probability distribution has collapsed, because there is no difference with the quantum mechanical probability distribution. One measurement gives a value and from then on it is fixed.

Look at this double slit experiment one electron at a time.

The accumulation is the probability distribution for the quantum mechanical problem "electron scattering through two slits". A single measurement is a point on the screen, and the probability of being at that point ,once it has been measured at that point is 1. That electron's wavefunction has "collapsed".

So any measurement will "collapse" the wavefunction for the individual particle detected. One has to accumulate measurements to register the behavior of the wavefunction.

The uncertainty principle is basic in quantum mechanics but is one step further than the wavefunction, because it separates variables that can be measured together with any accuracy, for each individual particle, with the ones that cannot be measured with great accuracy together.

For example in particle detectors

Incoming K- beam towards +y axis.

The V in the picture with no incoming track is a K0 decay to π+ π- , at a point in the bubble chamber, a "collapse" of its wavefunction.

The connection with the HUP: We measure the momentum from the curvature of the tracks of the particles with an error Δ(p) for each. The Heisenberg uncertainty tells us that we cannot localize the decay point better than the limit given by Δ(p)Δ(x)>h. The greater the accuracy in momentum the more the uncertainty in position. The experimental errors are such that the constraint is fulfilled within experimental errors in the existing measurements.