Evalutaion of Schrodinger's equation

[student354]

hi!
i asked to evaluate the Schrödinger equation using dirac notaion.
i saw some ways but didn't understand them.
is it true?
if it does, what are M and 1 represent?
thanks!

 

[Truecrimson]

The $|\psi (t+dt) \rangle - |\psi (dt) \rangle$ in the first line should be $|\psi (t+dt) \rangle - |\psi (t) \rangle$ (as in the second line). $M(dt)$ is an operator that takes $|\psi(t) \rangle$ to $|\psi (t+dt) \rangle$. 1 is the identity operator.

The operator that evolves a quantum state under Hamiltonian $H$ is $M(t) = e^{-iHt/\hbar}$. For small $dt$, keeping only the term linear in $dt$ of the the Taylor expansion of $M(dt)$ gives the right hand side of the second line.