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In quantum mechanics, it's common to encounter Hamiltonians that are formulated in terms of complex vectors, typically wave functions or state vectors. However, the expectation value of the Hamiltonian is always real, as it corresponds to the energy of the system, a physically measurable quantity.

Suppose you have a complex state vector $\Psi$, which belongs to a complex Hilbert space. The Hamiltonian $\hat{H}$ is an operator acting on this Hilbert space. The expectation value of the Hamiltonian is given by:

$$\langle H \rangle = \langle \Psi | \hat{H} | \Psi \rangle$$

This expression involves a bra-vector $\langle \Psi |$ and a ket-vector $| \Psi \rangle$. The bra-vector is the complex conjugate transpose of the ket-vector.

The Hamiltonian operator $\hat{H}$ is typically Hermitian, meaning its eigenvalues are real and it is equal to its own Hermitian conjugate:

$$\hat{H} = \hat{H}^\dagger$$

Due to the Hermitian nature of $\hat{H}$, the expectation value $\langle H \rangle$ will always be real, even though $| \Psi \rangle$ is complex.

In this formulation, you can think of the Hamiltonian as a function that takes a complex vector $| \Psi \rangle$ and returns a real value $\langle H \rangle$, consistent with physical interpretations.