Evalutaion of Schrodinger's equation

[student354]

hi!
i asked to evaluate the schrodinger 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.