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I recently started learning quantum mechanics and when I solved the Schrödinger equation for the Hydrogen atom, in particular the Radial equation, I found that I had normalized it but a term in the normalization constant was the Energy itself. I feel like there is something wrong with this, but I can't seem to find any literature to back it up. Is having an energy term in the normalization okay? I guess the more general question is that is there some kind of restriction on the kinds of terms that may appear in the normalization constant (besides the fact that they must be constant)?

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  • $\begingroup$ The normalisation constant N is often expressed as the equation $N^2\int _0 ^\infty \psi^* \psi d\tau = 1$ where $\tau $ represents the coordinates, $\psi$ the wavefunction and * the complex conjugate. (Note not all wavefuntions are complex quantities) $\endgroup$ – porphyrin Jul 4 '16 at 8:37
  • $\begingroup$ and 'no' the normalisation is a constant i.e. a number. In this case you have an extra twist in the normalisation. Because the real wavefunction also has angular parts there is an extra r in the normalisation due to the spherical volume element being $r^2sin(\theta )dr d\theta d\phi $. This means that the normalisation equation becomes $\int _0 ^\infty R^*_{nl}(r)R_{nl}(r)r^2 dr=1$ where $R_{nl}(r)$ is the radial wavefunction with quantum numbers n and l. Hope this helps. $\endgroup$ – porphyrin Jul 4 '16 at 8:51
  • $\begingroup$ Your feeling about something wrong is "wrong". If you find non-normalized eigenfunctions of the radial equation, say $\:R_n(r)\:$, the eigenvalues are the eigenenergies $$ E_{n}=\dfrac{E_{0}}{n^{2}}, \quad E_{0}=\text{ground state energy}=-13.6eV \tag{01} $$ where $n=0,1,2,\cdots.$ the principal quantum number. So , it's reasonable the normalization factor to be a function of $n$. The term "constant" concerns its independence of the radius variable $r$ or other variables of the hydrogen equation as $\phi,\theta$. $\endgroup$ – Frobenius Jul 4 '16 at 10:46

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