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Dark energy is not negative energy. It causes a repulsion because of its unusual equation of state, which causes it to behave as if it has a negative pressure. There is some discussion of this in the answers to Have negative pressures any physical meaning? and possibly also 'Negative pressure' counteracting gravity?. When general relativists talk ...

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Quoting Sean Carrol's article linked by Симон Тыран, which makes the case for energy not being conserved: Having said all that, it would be irresponsible of me not to mention that plenty of experts in cosmology or GR would not put it in these terms. We all agree on the science; there are just divergent views on what words to attach to the science. In ...

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If the particle is in an eigenstate of the Hamiltonian, you will get the same energy eigenvalue every time. We know that energy is conserved because the Hamiltonian obviously commutes with itself. The only time it is not conserved is if the Hamiltonian depends explicitly on time.

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The equation you wrote $$H|\psi\rangle=E|\psi\rangle$$ is the time-independent Schrödinger equation for an energy eigenstate. I.e., the state you are considering is already an eigenstate of the Hamiltonian with energy $E$. Therefore, as mentioned in the other answer, its time evolution is a simple phase factor, and you will always measure $E$ if you keep ...

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Law of conservation of energy states that the energy can neither be created nor destroyed but can be transformed from one form to another. Let us now prove that the above law holds good in the case of a freely falling body. Let a body of mass 'm' placed at a height 'h' above the ground, start falling down from rest. In this case we have to show that the ...

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When they say "Do not ignore electric force", they mean that there is both a magnetic and an electric force on the electron/positron, and you should not forget the electric force. In other words, you are asked to compute, for the $\vec v_+$, $q_+$ of the positron, the effect on the electron of its $\vec E$ and $\vec B$ field. Fortunately, 5 keV (kinetic ...

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Long story short: conservation of energy only holds locally where you can assume a static spacetime. On large scales the expansion of the universe gets relevant, so energy is said not to be conserved universally since the amount of dark energy per volume stays the same while the volume increases, see Sean Carroll's article, from which I quote: The famous ...

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Whether energy is or isn't conserved in an expanding universe is a somewhat vexed issue. On the one hand you have an experienced physicist claiming that energy is conserved, and on the other hand you have an experienced physicist claiming that energy is not conserved. The problem is that accounting for energy in general relativity is a complicated business. ...

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