Word Count : 754

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To solve the problem, we need to determine the normalization constant \( N \) and the expectation values of the energy, \( L^2 \), and \( L_z \) for the given wavefunction: \[ \psi(\mathbf{r}) = N \left( \psi_{100} + 2\sqrt{2}i \psi_{210} + 4 \psi_{21-1} \right) \] Here, \( \psi_{nlm} \) are the Hydrogen atom eigenfunctions, which are orthonormal. This means: \[ \int \psi_{nlm}^* \psi_{n'l'm'} \, d^3r = \delta_{nn'} \delta_{ll'} \delta_{mm'} \] --- ### Step 1: Normalization constant \( N \) The wavefunction must be normalized, so: \[ \int |\psi(\mathbf{r})|^2 \, d^3r = 1 \] Substitute \( \psi(\mathbf{r}) \): \[ |N|^2 \int \left| \psi_{100} + 2\sqrt{2}i \psi_{210} + 4 \psi_{21-1} \right|^2 \, d^3r = 1 \] Expand the integrand: \[ \left| \psi_{100} + 2\sqrt{2}i \psi_{210} + 4 \psi_{21-1} \right|^2 = |\psi_{100}|^2 + |2\sqrt{2}i \psi_{210}|^2 + |4 \psi_{21-1}|^2 \] Because the \( \psi_{nlm} \) are orthonormal, cross-terms vanish, and we get: \[ _____ ___ ________ _________ _____.
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