Physical interpretation of hopping between orthogonal orbitals

[hokhani]

TL;DR

What causes hopping between orthogonal orbitals?

Suppose two orthogonal neighbouring orbitals $|\phi _1 \rangle$ and $|\phi _2 \rangle$ so that $\langle \phi_1|\phi _2 \rangle =0$. Applying an electric field adds a new term $u (c_1^{\dagger}c_1-c_2^{\dagger}c_2)$ to the Hamiltonian which u is a constant potential. Obviously, we still have the hopping as $\langle \phi_1|u (c_1^{\dagger}c_1-c_2^{\dagger}c_2)|\phi _2 \rangle =0$. I would like to know how the hopping arises between the two neighbouring orbitals.

 

[anuttarasammyak]

Cross terms including $c_2^{\dagger}c_1$, etc. seems required.

 

[hokhani]

Cross terms including $c_2^{\dagger}c_1$, etc. seems required.

Right, but my question is how this term may arise?

 

[anuttarasammyak]

A familiat perterbation Hamiltonian in applying electric field is
$$H'=eEx$$
where
$$x=\sqrt{\frac{\hbar}{2m\omega}}(c+c^\dagger)$$
I do not have enough knowledge of your system $|\phi_1>,|\phi_2>$.

 

[pines-demon]

TL;DR Summary: What causes hopping between orthogonal orbitals?

Suppose two orthogonal neighbouring orbitals $|\phi _1 \rangle$ and $|\phi _2 \rangle$ so that $\langle \phi_1|\phi _2 \rangle =0$. Applying an electric field adds a new term $u (c_1^{\dagger}c_1-c_2^{\dagger}c_2)$ to the Hamiltonian which u is a constant potential. Obviously, we still have the hopping as $\langle \phi_1|u (c_1^{\dagger}c_1-c_2^{\dagger}c_2)|\phi _2 \rangle =0$. I would like to know how the hopping arises between the two neighbouring orbitals.

The orbital are assumed to be orthogonal for convenience but in the true Hamiltonian they are not. There is some overlap, it is considered tiny but it is this overlap that allows hopping. Check the tight-binding theory.