How much time do you have to make the same measurement?

[quantumfunction]

As I undersand it, if you make a measurement on particle and get some observable property like spin, you can quickly make a second measurement and get the same outcome. How long do you have to make that second measurement before it starts evolving according to Schrödinger's equation and you don't know if you will get the same outcome or not.

Thanks

 

[atyy]

It isn't true in general that an immediate repeat of a measurement will yield an identical result.

It is true for projective measurements with discrete outcomes, which is what is usually described in textbooks. For measurements with continuous outcomes, such as a measurement of position or momentum, it is not possible for immediate repetition to yield an identical result.

In the case where an identical result is possible, one must make the repeat immediately, otherwise the system will evolve under its Hamiltonian (Schroedinger's equation) to a state that is not an eigenstate of the observable that was measured.

In the special case in which the eigenstate of the measured observable is also an eigenstate of the Hamiltonian, then one can measure at any later time to get the identical result.

 

[quantumfunction]

Thanks for the answer @Science advisor, that clears things up. Ihave a question.

If this is the case, does this mean measurements with continuous outcomes are not quantized?

 

[atyy]

There are two definitions of "quantized".

In one, it means the classical observables are continuous, while the quantum observables are discrete. Energy is quantized in this sense, while position is not quantized in this sense.

However, another meaning of quantized means "described by the quantum mechanical formalism". In this sense, a quantum particle that can have position as a continuous observable is quantized.

 

[mfb]

before it starts evolving according to Schrödinger's equation

It always does that (even during your measurements, but it could get disturbed by this measurement). The question is "how strong is the influence of this evolution on the parameter you measure" and that depends on the system.

 

[kith]

How long do you have to make that second measurement before it starts evolving according to Schrödinger's equation and you don't know if you will get the same outcome or not.

The energy components of your state evolve in time with e^{\frac{-i E_n t}{\hbar}}
So if t \ll \frac{\hbar}{E_n} for all energy components of the state, the exponentials are approximately zero and your state stays approximately the same between time 0 and time t. That's what can be said in general. As mfb said, the rate of change of the parameters you measure in the second measurement are important.