# What makes expectation values real?

If you have some wave function of some particle, say...

|¥>

And you calculate the expectation value of momentum, say...

<¥|p|¥>

What ensures that that spatial integral is real valued?

Separately, all the components of the integral are complex valued

$$\langle \psi|\hat{p} \psi \rangle^*=\langle \psi|\hat{p}^{\dagger} \psi \rangle = \langle \psi|\hat{p} \psi \rangle,$$