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In quantum mechanics, observables are quantities that can be measured, such as position and momentum. These are represented mathematically by operators that act on a quantum state. Observables can include any physical quantity that pertains to the state of a system. For instance, energy, position, and spin can all be considered observables.

Each observable is associated with a specific operator. The value of the measurement of an observable is determined by applying the corresponding operator to a particular quantum state, known as the wave function or simply state vector. Importantly, in the framework of quantum mechanics, the result of measuring an observable is not deterministic but probabilistic, often expressed as an eigenvalue when the measurement is performed.

Understanding observables is key to decoding the behavior of quantum systems, as they provide the link between the theoretical mathematical constructs and the actual physical measurements we can perform in a lab.

The commutator of two operators, say \( A \) and \( B \), is expressed as \([A, B] = AB - BA\). This mathematical entity is pivotal in quantum mechanics for determining relationships between different observables.

- **Zero Commutator**: If \([A, B] = 0\), the operators \( A \) and \( B \) commute, meaning their operations can be performed interchangeably without affecting the final outcome. This is a sign of compatibility between the observables they represent.
- **Non-Zero Commutator**: When \([A, B] eq 0\), the operators do not commute, implying a more complex relationship where the order of operations matters. This often indicates that the corresponding observables cannot be measured simultaneously with accuracy, introducing a level of uncertainty.

The commutator provides insight into the fundamental uncertainties and compatibilities of measurements in quantum mechanics, aligning with the Heisenberg uncertainty principle.

In quantum mechanics, the concept of simultaneous measurement pertains to the ability to observe two or more properties of a quantum system at the same time with exact precision. This is possible only if the operators associated with those observables commute.

When operators commute (\([A, B] = 0\)), it is mathematically feasible to define a mutual set of eigenstates where both observables can be sharply measured. This implies that you can accurately predict the outcomes of both measurements simultaneously without interference from the other. Conversely, if the operators do not commute (\([A, B] eq 0\)), any attempt to measure these observables simultaneously will result in an uncertainty, where the precise determination of one observable affects the certainty of the other.

Understanding whether observables can be measured simultaneously is crucial for conducting precise experiments and obtaining reliable data in quantum physics.

Eigenstates are fundamental concepts in quantum mechanics, representing specific states of a quantum system that have well-defined values, known as eigenvalues, for a particular observable. These are solutions to the eigenvalue equation involving an operator. For an operator \( A \), an eigenstate \(|abla\rangle\) satisfies the equation \( A|abla\rangle = a|abla\rangle\), where \( a \) is the eigenvalue.

Eigenstates are crucial because they reveal the possible outcomes of a measurement of an observable. These states retain clarity when the measurement is applied, meaning that if a system is in an eigenstate of an observable, measuring that observable will yield the corresponding eigenvalue with certainty.

When two operators commute, they share a common set of eigenstates. This shared eigenstate property allows for simultaneous certain measurements of the related observables. Hence, eigenstates play a critical role in determining the measurable properties of a system, linking the mathematical framework with the physical phenomena observed in experiments.