In quantum mechanics, both the expectation value and the value obtained using quantum operators provide information about physical observables of a quantum system, but they represent different quantities and serve different purposes.
- Expectation Value: The expectation value of an observable in quantum mechanics represents the average value one would expect to measure if the same observable were measured repeatedly on an ensemble of identically prepared quantum systems. Mathematically, the expectation value of an observable A is given by the expression ⟨A⟩ = ⟨ψ|A|ψ⟩, where |ψ⟩ represents the state of the quantum system and A is the corresponding quantum operator.
The expectation value provides a probabilistic prediction of the measurement outcome and takes into account the underlying quantum state. It gives the average value of a measurement that can be obtained when measuring the observable multiple times on identically prepared systems.
- Value using Quantum Operators: On the other hand, the value obtained using quantum operators represents the result of applying the corresponding operator directly to a quantum state. If |ψ⟩ is the state of the system and A is an observable, then applying the quantum operator A to the state yields a new state given by A|ψ⟩.
The value obtained using quantum operators provides information about the state resulting from the application of the operator to the given quantum state. It describes a particular outcome corresponding to the application of the operator to the state, rather than an average or probabilistic prediction.
In summary, the expectation value gives the average value of a measurement obtained by repeatedly measuring the observable on identically prepared systems, taking into account the quantum state. On the other hand, the value obtained using quantum operators represents the specific outcome resulting from the application of the operator to a given quantum state, without considering statistical averages.