- #1
Emspak
- 243
- 1
Homework Statement
The Hamiltonian of helium can be expressed as the sum of two hydrogen Hamiltonians and that of the Coulomb interaction of two electrons.
[itex]\hat H = \hat H_1 + \hat H_2 + \hat H_{1,2}[/itex]
The wave function for parahelium (spin = 0) is
[itex]\psi(1,2) = \psi_S(r_1, r_2)\dot \xi_A(s_1, s_2)[/itex] with the first being a symmetric spatial function and the second being an antisymmetric one.
We can separate this into the normalized function
[itex]\psi_S(r_1,r_2) = \frac{1}{\sqrt{2}}[\psi_1(r_1)(\psi_2(r_2)+ \psi_1(r_2)(\psi_2(r_1)]=\psi_S(r_2,r_1)[/itex]
For orthohelium the functions look like this:
[itex]\psi(1,2) = \psi_A(r_1, r_2)\dot \xi_S(s_1, s_2)[/itex]
[itex]\psi_A(r_1,r_2) = \frac{1}{\sqrt{2}}[\psi_1(r_1)(\psi_2(r_2) - \psi_1(r_2)(\psi_2(r_1)]=\psi_A(r_2,r_1)[/itex]
Show the ground state of helium is parahelium. The hint is what happens to the wavefunction.
Homework Equations
OK, so I start with that the Hamiltonian of the given wave function(s)
$$H_1 = \frac{\hbar^2}{2m}\frac{\partial \psi}{\partial^2 r_1^2} = E_1 \psi, H_2 = \frac{\hbar^2}{2m}\frac{\partial \psi}{\partial^2 r_1^2} = E_2 \psi,
H_{1,2} = -\frac{e^2}{4\pi\epsilon_0 r_{1,2}}$$
The Attempt at a Solution
OK, I was trying to get a handle on how to get started with this. My first thought was to treat each Hamiltonian separately and just get a wavefunction the way I would solving any Schrodinger, though I am not sure of the boundary conditions. Then I can get a value for energy. But my sense is there is a simpler way to do it. I don't need a whole walk-through I don't think, I am just trying to understand some of the slutions I do see out there.