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- Homework Statement
- A certain Hamiltonian can be expressed as: ##\hat H = | \varphi_1 \rangle \langle \varphi_1| + 2 | \varphi_2 \rangle \langle \varphi_2| + 3 | \varphi_3 \rangle \langle \varphi_3| ##

where, ##| \varphi_1 \rangle## , ##| \varphi_2 \rangle## , ##| \varphi_3 \rangle## are normalized but not fully orthogonal.

##\langle \varphi_1 | \varphi_2 \rangle = \langle \varphi_2 | \varphi_1 \rangle = \langle \varphi_2 | \varphi_3 \rangle = \langle \varphi_3 | \varphi_2 \rangle = \frac {1} {2}##

##\langle \varphi_1 | \varphi_3 \rangle = \langle \varphi_3 | \varphi_1 \rangle = 0##

- Relevant Equations
- 1: ##E_\phi = \frac { \langle \phi | \hat H | \phi \rangle } { \langle \phi | \phi \rangle } ##

2: ##\sum_{j=1}^n (H_{ij} - ES_{ij} ) C_j = 0## , i = 1, 2,..., n

First I picked an arbitrary state ##|ϕ⟩=C_1|φ_1⟩+C_2|φ_2⟩+C_3|φ_3⟩## and went to use equation 1. Realizing my answer was a mess of constants and not getting me closer to a ground state energy, I abandoned that approach and went with equation two.

I proceeded to calculate the following matrix:

##\begin{pmatrix}

( \frac {3} {2} - E) & ( \frac {3} {2} - \frac {E} {2} ) & ( \frac {1} {2} ) \\

( \frac {3} {2} - \frac {E} {2} ) & ( 3 - E ) & ( \frac {5} {2} - \frac {E} {2} ) \\

( \frac {1} {2} ) & ( \frac {5} {2} - \frac {E} {2} ) & ( \frac {7} {2} - E )

\end{pmatrix}\vec C = 0##

Calculating the determinant, I end up with ##E^3 - 6E^2 + 9E - 3 = 0## which doesn't seem to have a rational solution. I must've gone through the calculation half a dozen times now, and confirmed with my professor that equation 2 was the one I should use, but I'm just not seeing how I'm supposed to find a solution.

I proceeded to calculate the following matrix:

##\begin{pmatrix}

( \frac {3} {2} - E) & ( \frac {3} {2} - \frac {E} {2} ) & ( \frac {1} {2} ) \\

( \frac {3} {2} - \frac {E} {2} ) & ( 3 - E ) & ( \frac {5} {2} - \frac {E} {2} ) \\

( \frac {1} {2} ) & ( \frac {5} {2} - \frac {E} {2} ) & ( \frac {7} {2} - E )

\end{pmatrix}\vec C = 0##

Calculating the determinant, I end up with ##E^3 - 6E^2 + 9E - 3 = 0## which doesn't seem to have a rational solution. I must've gone through the calculation half a dozen times now, and confirmed with my professor that equation 2 was the one I should use, but I'm just not seeing how I'm supposed to find a solution.