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I had a math question about the following steps.
The Shroedinger's equation can be written as follows.
\LARGE i\hbar \frac{d}{dt}U(t) |\psi(0)> = HU(t)|\psi(0)>
Where H is the hamiltonian and U is the time evolution operator.
So U satisfies the Schrödinger equation.
\LARGE i\hbar\frac{d}{dt}U(t) = HU(t)
This is the part that I don't quite understand.
If the hamiltonian H is independent of time, the solution to the above equation is
\LARGE U(t) = e^{-iHt/\hbar}
This is a solution of integration with respect to U to solve for the differential equation.
My problem is that i can't seem to justify the expression:
\LARGE \int i\hbar \frac{dU(t)}{U(t)} = \int Hdt
Am I allowed to divide operators and shuffle them around? Furthermore, when I think of operators in their matrix representation, I am even more confused, since I do not know of matrix division, only inverse operations on matrix.
Any help would be appreciated. Thanks.
The Shroedinger's equation can be written as follows.
\LARGE i\hbar \frac{d}{dt}U(t) |\psi(0)> = HU(t)|\psi(0)>
Where H is the hamiltonian and U is the time evolution operator.
So U satisfies the Schrödinger equation.
\LARGE i\hbar\frac{d}{dt}U(t) = HU(t)
This is the part that I don't quite understand.
If the hamiltonian H is independent of time, the solution to the above equation is
\LARGE U(t) = e^{-iHt/\hbar}
This is a solution of integration with respect to U to solve for the differential equation.
My problem is that i can't seem to justify the expression:
\LARGE \int i\hbar \frac{dU(t)}{U(t)} = \int Hdt
Am I allowed to divide operators and shuffle them around? Furthermore, when I think of operators in their matrix representation, I am even more confused, since I do not know of matrix division, only inverse operations on matrix.
Any help would be appreciated. Thanks.