- #1

dark_matter_is_neat

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- Homework Statement
- An electron in the hydrogen atom in the ground state is described by the wavefunction: ##\Psi(x,y,z) = Ae^{-\frac{r}{r_{1}}}##

where ##r = \sqrt{x^{2}+y^{2}+z^{2}}## and A and ##r_{1}## are constants.

Use the Schrodinger equation to find ##r_{1}## and the energy eigenvalue E in terms of the electron mass and charge.

- Relevant Equations
- ##-\frac{\hbar^{2}}{2m} \nabla^{2} \Psi + V \Psi = E \Psi##

In this case, ignoring derivatives that go to zero, (denoting the charge of the electron as q to avoid confusion) ##-\frac{\hbar^{2}}{2m} \frac{1}{r} \frac{\partial^{2}}{\partial r^{2}} (rAe^{-\frac{r}{r_{1}}}) - \frac{q^{2}}{4 \pi \epsilon_{0} r} Ae^{-\frac{r}{r_{1}}} = E A e^{-\frac{r}{r_{1}}}##.

So going through the derivatives:

##(\frac{\hbar^{2}}{mrr_{1}} - \frac{\hbar^{2}}{2mr_{1}^{2}} - \frac{q^{2}}{4 \pi \epsilon_{0} r}) A e^{-\frac{r}{r_{1}}} = E A e^{-\frac{r}{r_{1}}}##.

I can cancel ##A e^{-\frac{r}{r_{1}}}## on each side to get ##E = \frac{\hbar^{2}}{mrr_{1}} - \frac{\hbar^{2}}{2mr_{1}^{2}} - \frac{q^{2}}{4 \pi \epsilon_{0} r}##, which isn't good since it contains two unknowns ##r_{1}## and E, and it contains r. I'm not sure how to get two separate equations for ##r_{1}## and E from just the Schrodinger equation and I'm not sure how to get rid of r, since neither expression should depend r.

So going through the derivatives:

##(\frac{\hbar^{2}}{mrr_{1}} - \frac{\hbar^{2}}{2mr_{1}^{2}} - \frac{q^{2}}{4 \pi \epsilon_{0} r}) A e^{-\frac{r}{r_{1}}} = E A e^{-\frac{r}{r_{1}}}##.

I can cancel ##A e^{-\frac{r}{r_{1}}}## on each side to get ##E = \frac{\hbar^{2}}{mrr_{1}} - \frac{\hbar^{2}}{2mr_{1}^{2}} - \frac{q^{2}}{4 \pi \epsilon_{0} r}##, which isn't good since it contains two unknowns ##r_{1}## and E, and it contains r. I'm not sure how to get two separate equations for ##r_{1}## and E from just the Schrodinger equation and I'm not sure how to get rid of r, since neither expression should depend r.

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