# About Schroedinger Equation in Josephson Junction

In Charles Kittel's Introduction to Solid State Phyiscs (page 367), it derives the equation for DC Josephson Effect. I understand this is not a rigorous derivation. But it has a step making me think I don't understand the meaning of time dependent S.E. well.

It says, assume $\psi_1$ and $\psi_2$ are the probability amplitude of electron pairs on the two sides of the junction respectively, then applied time-dependent S.E. $i\hbar \frac{\partial \psi}{\partial t}=H\psi$ to the two amplitudes gives,

$$i\hbar \frac{\partial \psi_1}{\partial t}=\hbar T\psi_2$$

$$i\hbar \frac{\partial \psi_2}{\partial t}=\hbar T\psi_1$$

where $\hbar T$ represents the transfer interaction across the insulator.

My question is, isn't that the RHS of SE is Hamiltonia (H). Why one can use T? And also why $H\psi_1 = \hbar T\psi_2$? What is the physical meaning of submit "substitution"? Thanks a lot!