Can Multiple Competitive Operators Transform a Quantum State Simultaneously?

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In a quantum state, if you use the position operator, it gives you position, momentum operator, momentum, Hamiltonian, energy.. can you give an example or all experiments done where a quantum state has 3 competitive operators acting on it.. I want to see the quantum state changing in between position, momentum and Hamiltonian.. and what is the biggest system that has been experimented on?
 

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Probably the experiment most students are familiar with, which "switches operators" (i..e changes what is being measured) at the quantum scale would be the Stern Gerlach experiment, where more than one S-G apparatus is used. Rotating the apparatus 90deg changes the operator from z-spin to y-spin.

Pretty much every experiment involves changing operators.
i.e. to do experiments on a particle in a box, first you have to get the particle in the box. The act of putting it in the box is a measurement of it's position - to "someplace inside the box" ...

In an experment, you don't think "I'll switch operators from momentum to hamiltonian", you think "I'll measure momentum then energy". You are just changing what you want to measure. Physically it involves changing the apparatus used to do the measurement.

Atoms have had something like 4 measurements - to prepare them in a definite quantum state: this is why you have 4 quantum numbers.
The experiments would involve hyperfine splitting for eg.
 
Simon Bridge said:
Probably the experiment most students are familiar with, which "switches operators" (i..e changes what is being measured) at the quantum scale would be the Stern Gerlach experiment, where more than one S-G apparatus is used. Rotating the apparatus 90deg changes the operator from z-spin to y-spin.

Pretty much every experiment involves changing operators.
i.e. to do experiments on a particle in a box, first you have to get the particle in the box. The act of putting it in the box is a measurement of it's position - to "someplace inside the box" ...

In an experment, you don't think "I'll switch operators from momentum to hamiltonian", you think "I'll measure momentum then energy". You are just changing what you want to measure. Physically it involves changing the apparatus used to do the measurement.

Atoms have had something like 4 measurements - to prepare them in a definite quantum state: this is why you have 4 quantum numbers.
The experiments would involve hyperfine splitting for eg.

And it's possible for two observers to have different operators and different results of the same quantum system at the same time?
Remember Zurek propose quantum Darwinism where the quantum system would produce only one outcome.. why does he need to propose this if experiment can show 2 results at the same time?