Time-evolution of an observer means how it changes over time; it's derivative over time. In quantum mechanics, it is described by the Hamiltonian. DO NOT use this tag for evolving laws of physics.

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What does a unitary transformation mean in the context of an evolution equation?

Let be the unitary evolution operator of a quantum system be $U(t)=\exp(itH)$ for $t >0$. Then what is the meaning of the equation $$\det\bigl(I-U(t)e^{itE}\bigr)=0$$ where $E$ is a real ...
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Is it possible to derive Schrodinger equation in this way?

Let's have wave-function $\lvert \psi \rangle$. The full probability is equal to one: $$\langle \Psi\lvert\Psi \rangle = 1.\tag{1}$$ We need to introduce time evolution of $\Psi $; we know it in ...
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Evolution operator for time-dependent Hamiltonian

When i studyed QM I'm only working with non time-dependent Hamiltonians. In this case unitary evolution operator has the form $$\hat{U}=e^{-\frac{i}{\hbar}Ht}$$ that follows from this equation $$ ...
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Does Heisenberg equation of motion imply the Schrodinger equation for evolution operator?

Let us choose to postulate (e.g. considering the analogy of the Hamiltonian being a generator of time evolution in classical mechanics) $$ i\hbar \frac{d\hat{U}}{dt}=\hat{H}\hat{U}\tag{1} $$ where ...