For each of the following, use the completeness relation of the position eigenstates to show:
that the state of the system can be expressed as \(\left|{\psi}\right\rangle =\int \psi(x)\left|{x}\right\rangle dx\)
that the expansion coeffients for a state written in the energy basis, \(c_n = \left\langle {n}\middle|{\psi}\right\rangle \) are \(\int\varphi^*_n(x)\psi(x)dx\) in wavefunction notation.
that the norm of a quantum state, \(\left\langle {\psi}\middle|{\psi}\right\rangle \) is \(\int\psi^*(x)\psi(x)dx\) in wavefunction notation.
that the expectation value of position,
\[\langle \hat{X} \rangle = \left\langle {\psi}\right|\hat{X}\left|{\psi}\right\rangle =\int|\psi(x)|^2x~dx\] in wavefunction notation.