Matrix Version

We can use matrix notation to re-do the problem above. The time-dependent Schrödinger equation in the original $\left\{ \Phi_1({\bf r}), \Phi_2({\bf r}) \right\}$ basis becomes

$$i \hbar \left[ \begin{array}{c} {\dot c}_1 \\ {\dot c}_2 \end{ar... ...2} \end{array} \right] \left[ \begin{array}{c} c_1 \\ c_2 \end{array} \right].$$ (42)

Now, just as in eq. 3, we can see that if H was a diagonal matrix, then the equations for c₁ and c₂ would become decoupled. Again, we can make H diagonal by going to the special basis made of the eigenvectors of H. In this new basis, we will denote vector coefficients and matrix elements with tildes as a reminder that the basis set has changed, and we obtain:

 

$$i \hbar \left[ \begin{array}{c} {\dot {\tilde c}}_1 \\ {\dot {\t... ...ght] \left[ \begin{array}{c} {\tilde c}_1 \\ {\tilde c}_2 \end{array} \right].$$ (43)

Now remember that we've gone to the eigenvector basis (which we'll also denote with tildes to distinguish it from the original basis), so we know that

$${\hat H} {\tilde \Phi}_1({\bf r}) E_1 {\tilde \Phi}_1({\bf r})$$ = (44)

$${\hat H} {\tilde \Phi}_2({\bf r}) E_2 {\tilde \Phi}_2({\bf r}).$$ = (45)

Thus we can further simplify the diagonal elements as

$${\tilde H}_{11} \int {\tilde \Phi}_1^*({\bf r}) {\hat H} {\tilde \Phi}_1({\bf r}) d{\bf r}$$ = (46)

$$E_1 \int {\tilde \Phi}_1^*({\bf r}) {\tilde \Phi}_1({\bf r}) d{\bf r}$$ =

= E₁

for normalized basis functions. Likewise of course ${\tilde H}_{22} = E_2$. Hence, we can expand our matrix equation 43 as

$$i \hbar {\dot {\tilde c}}_1 {\tilde c}_1 E_1$$ = (47)

$$i \hbar {\dot {\tilde c}}_2 {\tilde c}_2 E_2$$ = (48)

which is the same thing we got before when we assumed the given functions were orthonormal. The only difference is that here we emphasized the diagonalization of H rather than getting the eigenvectors, but of course it is the same process. These decoupled equations can be solved the same way as before to give

$${\tilde c}_1(t) {\tilde c}_1(0) e^{- i E_1 t / \hbar}$$ = (49)

$${\tilde c}_2(t) {\tilde c}_2(0) e^{- i E_2 t / \hbar},$$ = (50)

which we could write back in matrix notation (still in the eigenvector or tilde basis) as

$$\left[ \begin{array}{c} {\tilde c}_1(t) \\ {\tilde c}_2(t) \end{array} \right] \left[ \begin{array}{cc} e^{- i E_1 t / \hbar} & 0 \\ 0 & e^{- i... ...left[ \begin{array}{c} {\tilde c}_1(0) \\ {\tilde c}_2(0) \end{array}\right] .$$ = (51)

We can identify the matrix as the propagator,

 

$${\mathbf {\tilde G}}(t) \left[ \begin{array}{cc} e^{- i E_1 t / \hbar} & 0 \\ 0 & e^{- i E_2 t / \hbar} \end{array} \right]$$ = (52)

in the eigenvector basis.

In the vector/matrix representation, we can go from our original to our tilde coefficients and back as

$$\left[ \begin{array}{c} c_1(t) \\ c_2(t) \end{array} \right] {\bf U} \left[ \begin{array}{c} {\tilde c}_1(t) \\ {\tilde c}_2(t) \end{array} \right]$$ = (53)

$$\left[ \begin{array}{c} {\tilde c}_1(t) \\ {\tilde c}_2(t) \end{array} \right] {\bf U}^{\dagger} \left[ \begin{array}{c} c_1(t) \\ c_2(t) \end{array} \right],$$ = (54)

where ${\bf U}$ is the matrix made by making each column an eigenvector of ${\hat H}$ in the original basis. We could transform our propagator ${\mathbf {\tilde G}}(t)$ from the eigenvector basis to the original basis by

$${\mathbf G}(t) = {\mathbf U} {\mathbf {\tilde G}}(t) {\mathbf U}^{\dagger}.$$ (55)