# Simultaneous measurement of two observables

In quantum physics the configuration of a particle is fully defined by it's wave function. When a measurement of a particular observable ( eg. position, angular momentum etc.) is made on the particle , it's wave function collapses to one of it's eigen states (of the operator used to represent the observable) whose corresponding eigen value gives the result of the measurement. What happens if the measurement of two observables are made simultaneously? Does the wave function collapse into a linear combination of two eigen states ( one for each of the two operators corresponding to the observable)? How do we get the result of the measurements then?

Note: I know that if the operators do not have common eigen function, the corresponding observables cannot be measured simultaneously with accuracy ( Uncertainty principle) but I would like to know the scenario in terms of collapsing wave functions.

• To be clear, are you asking about the simultaneous measurement of compatible observables or of incompatible ones? If the latter, then you're asking, to the letter, "I know this is a mathematical impossibility, but how do you describe it mathematically?". – Emilio Pisanty Mar 14 '18 at 12:51
• I am asking about both the cases along with calculations for each – D.K. Mar 14 '18 at 13:52

## 1 Answer

• If the observables are compatible then you just project on their shared eigenfunctions.

• If the observables are incompatible then they are incompatible, period. It's not a question of whether you can "observe them simultaneously with accuracy" or not: if the observables are incompatible then there isn't a shared eigenprojector and the very notion of simultaneous measurement is meaningless. You can't say what "happens" to the wavefunction because the scenario is nonsensical to begin with.

• But we can measure two incompatible quantities with some uncertainty, right? How is measurement possible if the the wave function does not collapse? – D.K. Mar 14 '18 at 15:11
• @D.K. No, you can't perform simultaneous measurements of incompatible quantities. You can perform partial measurements of a given quantity and then subsequently perform a measurement of other (possibly incompatible) quantities, but that is not what your question's phrasing entails. – Emilio Pisanty Mar 14 '18 at 16:06
• can you elaborate on what you mean by partial measurement? – D.K. Mar 14 '18 at 17:21