The distracted experimentalist's problem

[Heidi]

TL;DR Summary

the experimentalist has forgotten the result of the quantum measurement. what can he do?

we have quantum particle which evolves according to an Hamiltonian H (it does not depend on t)
at t = 0 the experimentalist makes a measurement but he often forgets to read the result which is shortly shown on a screen. If he immediately makes the same measurement on the particle (it was not destroyed) he will see the same result on the screen.
suppose that time has passed when he realizes that he is not sure of the result.
the system has evolved, can he make another experiment which will confirm the first result?
Maybe by using the operator evolution in the Heisenberg picture?

 

[Nugatory]

the system has evolved, can he make another experiment which will confirm the first result?

It depends on what is being measured. Does it commute with the Hamiltonian? If not, the probability of getting a different result the second time will increase over time as the state evolves.

The state after the first measurement and its subsequent evolution are are the same whether the experimenter “forgets” the first result or not.

 

[Heidi]

Hi Nugatory,
It seems that you think that the experimentalist cannot use a different apparatus. suppose that he measured a spin along the vertical direction at t = 0.
at t = 1 he can measure what he wants , maybe the same thing along another direction a t = 1.
The aim is to find second device which will confirm the first result he forgot to read the first time in the general case.
I thought that if the operator was s1 v1><v1 + s2 v2><v2 + ...
(orhthormal eigenstates) the operator s1 v1(t)><v1(t) + ...
would be useful to find a new device. Is it so?

 

[Dale]

Summary:: the experimentalist has forgotten the result of the quantum measurement. what can he do?

the system has evolved, can he make another experiment which will confirm the first result?

Not in general. Since the system has evolved it is no longer in the original state. Furthermore, even making another measurement of the original state would not confirm the original measurement.

The scientist should hire a lab assistant.

 

[Heidi]

in the first measurement a t = t0 the operator O will give v one its eigenstates: v1 v2 v3 ...
a t = t1 this basis is rotated by the hamiltonian giving
v1(t1) v2(t1) v3(t1) ...
v has been rotated into v(t)
the trick would be to find another measurement on v(t) giving the same eigen value.
in the spin measurement it would be a second measurement in anoter direction?

 

[Dale]

the trick would be to find another measurement on v(t) giving the same eigen value.

That is not possible in general. It may be possible in some specific circumstances, but not usually.

 

[Heidi]

I found why it is not possible in general and it was a surprise for me.
in the Hilbert space H all the states can exist in the nature. we can produce superposition of two existing states. but it is not the same for the hermitian operators on H.
i wrote a hermitian opetator with v1(t)><v1(t) and v2(t)><v2(t) inside and i hoped to find a correspondinf device. but such a device does not exist. No device can tell you in a single shot that the particle is in a superposition of position or superposition of up and down. it is only possible with many measurements on many copies of the states.
all the hermition opetators do not correspond to really observable devices. there are those coming from classical physics: position, momentum ... MQ add projection of spins along a spatial direction but nothing else for superposition of things like that.

 

[PeterDonis]

such a device does not exist.

A more accurate statement would be that we do not currently know how to build such a device (at least for superpositions of position; see below).

No device can tell you in a single shot that the particle is in a superposition of position

No device that we currently know how to build. But that might be a limitation of our current knowledge, not a fundamental limitation of physics.

or superposition of up and down

We do know how to build a device that measures a superposition of spin up and spin down: just orient the spin measuring device horizontally instead of vertically. Any spin state is a superposition of other spin states.

I found why it is not possible in general

Nothing in this post of yours has anything to do with the problem you describe in your OP. The reason why you can't, in general, solve the problem you describe in your OP is that most measurement operators do not commute with the Hamiltonian, as @Nugatory pointed out in post #2.

 

[PeterDonis]

The aim is to find second device which will confirm the first result he forgot to read the first time in the general case.

You can't do this in general because, as @Nugatory pointed out in post #2 and as I pointed out in post #8, most measurement operators do not commute with the Hamiltonian. That means that, if you wait some time before trying to "confirm" the first measurement, time evolution has destroyed the information you are trying to obtain. There is no way to get the information even if you change what measurement you make on the second measurement; the information is simply no longer there.

 

[Heidi]

I understand but the projection of spin along a given direction seems to be different. if state rotates in the up, down basis in the Hilbert space , the experimentalist may wait during the time between 0 of his firs measurement and the moment when there was a pi/2 rotation in this basis. Then he reverses the direction of his axis measurement and he finds its first unread result. is it correct?

 

[PeterDonis]

the experimentalist may wait during the time between 0 of his firs measurement and the moment when there was a pi/2 rotation in this basis.

What is he conducting the second measurement on? Remember that, in a spin measurement such as Stern-Gerlach, the two possible output states come out in two different beams. If the experimentalists is only measuring one beam, then he does know what the result of the first measurement was, because he knows which beam he's looking at. If, on the other hand, he is measuring both beams, then they must have been recombined somehow, which destroys the information about the first measurement result, and his second measurement will have a 50-50 chance of either outcome.

 

[Heidi]

We do know how to build a device that measures a superposition of spin up and spin down: just orient the spin measuring device horizontally instead of vertically. Any spin state is a superposition of other spin states.

Are there other examples of such a possibility? is it the case with the total angular momentum?

 

[PeterDonis]

Are there other examples of such a possibility?

The case of spin is particularly simple because the spin of a single spin-1/2 particle has the simplest possible Hilbert space. One should not expect that level of simplicity in other examples that have more complicated Hilbert spaces.

However, one should also not conclude from that that measuring states that are "superpositions" of familiar observables like position is impossible.