No, in standard quantum mechanics, the expectation value of an observable cannot be negative. The expectation value is the average value of a measurement outcome that one would obtain when measuring a particular observable on a quantum system in a given quantum state.
Mathematically, the expectation value of an observable A is calculated using the wavefunction or density operator of the system and the corresponding operator for the observable. The expectation value is given by:
⟨A⟩ = ⟨ψ|A|ψ⟩
where |ψ⟩ represents the quantum state and A represents the operator associated with the observable.
Since the expectation value is essentially a weighted average of possible measurement outcomes, it is always a real number. In cases where the observable has a continuous spectrum, the expectation value can be any real number within the range of possible eigenvalues of the observable.
It's worth noting that individual measurement outcomes can be negative, positive, or zero, depending on the specific eigenvalues associated with the observable being measured. However, when taking the average over multiple measurements, the expectation value itself cannot be negative.