An operator is considered Hermitian if it is equal to its own conjugate transpose. In other words, the operator remains unchanged when its elements are switched from complex numbers to their complex conjugates.
The expectation of an Hermitian operator is determined by taking the inner product of the state vector with the operator applied to that vector. This can be represented mathematically as ⟨A⟩ = ⟨ψ|A|ψ⟩, where A is the Hermitian operator and ψ is the state vector.
This is because the inner product of a state vector with its complex conjugate is always a real number. Since the expectation of an Hermitian operator involves taking the inner product of the state vector with the operator applied to it, the result will always be a real number.
The expectation of an Hermitian operator is a key concept in quantum mechanics and is closely related to measurement outcomes. The squared magnitude of the expectation value represents the probability of obtaining a certain measurement outcome when the operator is measured in that state.
No, the expectation of an Hermitian operator cannot be negative. This is because the squared magnitude of the expectation value represents the probability of obtaining a certain measurement outcome, and probabilities cannot be negative. However, the expectation value can be zero, indicating a zero probability of obtaining a certain measurement outcome.