Quantum Mechanics | Part IB, 2003

What is meant by the statement than an operator is hermitian?

A particle of mass mm moves in the real potential V(x)V(x) in one dimension. Show that the Hamiltonian of the system is hermitian.

Show that

ddtx=1mpddtp=V(x)\begin{aligned} \frac{d}{d t}\langle x\rangle &=\frac{1}{m}\langle p\rangle \\ \frac{d}{d t}\langle p\rangle &=\left\langle-V^{\prime}(x)\right\rangle \end{aligned}

where pp is the momentum operator and A\langle A\rangle denotes the expectation value of the operator AA.

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