- #1
TFM
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Homework Statement
The groundstate energy of the hydrogen atom can be calculated using the variational
principle. The normalised groundstate wavefunction is:
[tex] \psi_{100} = R(r)_{10} \cdot Y_{00} [/tex]
with [tex] R_{10} = 2Ae^{-3/2}e^(r/a) [/tex] and [tex] Y_{00} = \frac{1}{\sqrt{4z\pi}} [/tex]
A is the so called variational parameter.
a)
Calculate the potential energy of the groundstate as a function of A.
b)
Calculate the kinetic energy of the groundstate as a function of A.
Homework Equations
Schrödinger Equation:
[tex] -\frac{\hbar^2}{2m_e}\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\partial R}{\partial r}\right) +\frac{\hbar^2l(l+1)}{2m_er^2}R - \frac{Zr^2}{(4\pi\epsilon_0)r}R = ER [/tex]
The Attempt at a Solution
I am not quite sure, but would I be ruight in saying that if a solve the LHS, and then divide through by R, this would give:[tex] \frac{1}{R}\left( -\frac{\hbar^2}{2m_e}\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\partial R}{\partial r}\right) +\frac{\hbar^2l(l+1)}{2m_er^2}R - \frac{Zr^2}{(4\pi\epsilon_0)r}R\right) = E [/tex]
Would this be the potential energy, or is it the total energy?
Many thanks,
TFM