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I'm confused about how energy and time are linked. On the one hand, the Hamiltonian seems to describe the time evolution of the system because in the time dependent Schrodinger equation, $$\hat H ... 4answers 252 views Does Heisenberg equation of motion imply the Schrodinger equation for evolution operator? Let us choose to postulate (e.g. considering the analogy of the Hamiltonian being a generator of time evolution in classical mechanics)$$ i\hbar \frac{d\hat{U}}{dt}=\hat{H}\hat{U}\tag{1} $$where ... 3answers 104 views Solving the Schrödinger equation where the initial wave function is an energy eigenfunction I was watching Allan Adams' lecture on energy eigenfunctions, and there's one part (around 43 minutes into the lecture) that confuses me. Suppose we have the initial wave function \Psi (x,0) such ... 1answer 45 views Time evolution of states - Is total energy constant or not? Suppose the state of the particle is given as follows:$$ |\psi_{(t)}\rangle = \frac{1}{\sqrt2} \left( e^{-\frac{i\omega t}{2}} |0\rangle + e^{-\frac{3i\omega t}{2}} |1\rangle \right) $$Where the ... 2answers 252 views The formal solution of the Schrodinger equation Let's have Schrodinger equation (or some equation in Schrodinger form)$$ \tag 1 i \partial_{0} \Psi ~=~ \hat{H} \Psi . $$One likes to write that it has formal solution$$ \tag 2 \Psi (t) ~=~ ...
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 ...
Why do we consider evolution of a wave function and why is the evolution parameter taken as time, in QM. If we look at a simple wave function $\psi(x,t) = e^{kx - \omega t}$, $x$ is a point in ...