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Consider the energy conservation principle, than the total amount of energy in the universe is a fixed value $E$.
Let us denote with $| \psi \rangle$ the wave function of the entire universe.
Is it right to write the Schödinger equation for the entire universe as $ i \hbar \frac{d |\psi \rangle}{dt}=E |\psi \rangle$ ?

If this is right, it means that the universe's wave function is fundamentally stable?

ScottyW
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  • I think that (and I mean this in the most literal sense possible) everything that ever happened would be experimental evidence that the universe as a whole is not in an energy eigenstate. – By Symmetry Feb 17 '16 at 19:02
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    I think these consideration go beyond the scope of Quantum Mechanics. They are purely academic considerations. Having said so, why would the wavefunction of the universe (or any isolated system) be an eigenfunction? –  Feb 17 '16 at 20:00
  • @AccidentalFourierTransform 1)I never pondered very deeply on the energy conservation principle. Now I see that universe's energy is not supposed to have a fixed value in time. (link). So... Thanks for the question! However let suppose it is constant. 2)Maybe stable is not the right word. I mean that this Schrödinger equation imply only a phase rotation for the time evolution of the wave function. – ScottyW Feb 17 '16 at 20:26
  • The only way that a quantum system can get into an energy eigenstate is by contact with an external system, so that it can shed or gain the right amount of energy. Just by that one can virtually guarantee that the definitions of universe and energy eigenstate are incompatible. – CuriousOne Feb 17 '16 at 21:26
  • There is no way to tell from the inside whether the universe is in an energy eigenstate. The t in the Schrodinger equation is just a parameter that doesn't correspond to time as measured on actual clocks. For more detail see Evolution without evolution: Dynamics described by stationary observables Don N. Page and William K. Wootters Phys. Rev. D 27, 2885 – alanf Feb 17 '16 at 23:16

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Well, why don't we consider a smaller problem before we tackle the whole universe. For instance, a particle with a wavefunction $\psi(t)$ in an infinite potential well that has eigenstates with energies $E_0$, $E_1$, $E_2$, etc...

This particle can always be written as a superposition of energy states.

$$ |\psi(t) \rangle = \sum_{i=0}^\infty c_i e^{iE_it} |E_i \rangle $$

Now, what is the energy of the particle? From this equation, the particle can exist in many possible different total energy states based on some probability. Thus, even if you have steady state solutions, that does not guarantee you know the exact total energy of the system. One can instead look at the average total energy

$$ \langle E \rangle = \langle \psi(t) | H | \psi(t) \rangle $$

but there is no guarantee that $\langle E \rangle$ is equal to any eigenenergy $E_i$. So, if we consider your E the average energy of the universe, this does not mean we are in a steady state.

Greg Petersen
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