64 votes
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Why do excited states decay if they are eigenstates of Hamiltonian and should not change in time?

The atomic orbitals are eigenstates of the Hamiltonian $$ H_0(\boldsymbol P,\boldsymbol R)=\frac{\boldsymbol P^2}{2m}+\frac{e}{R} $$ On the other hand, the Hamiltonian of Nature is not $H_0$: there ...
AccidentalFourierTransform's user avatar
49 votes
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How does a particle know how to behave?

I think this question makes hidden, inarticulated assumptions about reality. In physics, we make observations and then try to find models that match them. The models, though, belong only to us and ...
innisfree's user avatar
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31 votes

How can the universe evolve unitarily if there's no clock outside it?

You are assuming that time does not exist without clocks. That is analogous to assuming that space does not exist without rulers, and both assumptions are unjustified. As far as we know, we live in a ...
Marco Ocram's user avatar
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28 votes
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Is there a fundamental reason for the exponential dependence of the evolution of the temperature in an electronic deviced that is powered on?

This behavior is basically described by Newton cooling with heat generation, using the equation: $$MC\frac{dT}{dt}=G-k(T-T_{\infty})$$where T is the temperature, t is time, M is the mass, C is the ...
Chet Miller's user avatar
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28 votes
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Why is the second law of thermodynamics not symmetric with respect to time reversal?

The arrow of time in thermodynamics is statistical. Suppose you have a deterministic system that maps from states that can have character $X$ or character $Y$, to other states that can have character $...
g s's user avatar
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27 votes

Why do excited states decay if they are eigenstates of Hamiltonian and should not change in time?

The hydrogen atom in an excited state is not really in an energy eigenstate. There are two ways of looking at it. One way is to recognize that the atom is not isolated. It is always coupled to ...
garyp's user avatar
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25 votes

Is there an equivalent of computation of physical processes in nature?

Nature doesn’t need to find out where the droplets need to go. They just go. This is the kind of question that has slowed down science from the Greek time to the Middle Ages, after which people slowly ...
Oбжорoв's user avatar
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21 votes

Why can't two different quantum states evolve into the same final state?

Is it true that two different states cannot evolve into the same final state? That depends on exactly what you mean. If we consider the total state of a closed system, then two different states will ...
DanielSank's user avatar
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20 votes

Is there an equivalent of computation of physical processes in nature?

That is, in my opinion, a good question. Re-phrased, "Does a physical process constitute a computation?" The answer depends on the meaning of "computation". Most often, ...
S. McGrew's user avatar
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19 votes
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How is Liouville's theorem important to statistical mechanics?

Because you do equilibrium statistical mechanics. In the usual ensemble theory we associate to a system (a macrostate) a big number of corresponding microstates, each microstate is a point in phase ...
RenatoRenatoRenato's user avatar
18 votes
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Treating the Schrödinger equation as an ordinary differential equation

$\psi$ is a curve through $\mathcal H:=L^2(\mathbb R^{Nd})$, not through $\mathbb R^{Nd}$ itself. That is, for each $t\in \mathbb R$ we have that $\psi(t)\in L^2(\mathbb R^{Nd})$ is loosely$^\ddagger$...
J. Murray's user avatar
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16 votes
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What's the reasons to use time-ordering operator?

The time ordering enters as a consequence of the definition of the Hamiltonian as the generator of time translations. In the Schödinger picture: $$|\psi(t)\rangle \approx \left(1 - \frac{i}{\hbar} H(t'...
Sean E. Lake's user avatar
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16 votes

What is meant by unitary time evolution?

Yes, there is a difference. Unitary time evolution is the specific type of time evolution where probability is conserved. In quantum mechanics, one typically deals with unitary time evolution. ...
Avantgarde's user avatar
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16 votes

Do non-commuting Hamiltonians have non-commuting time evolution operator?

Here is a counterexample to the first part of OP's question: Imagine $K$ is diagonalizable with eigenvalue spectrum $\subseteq \mathbb{Z}$ within the integers. Then $e^{i2\pi K}={\bf 1}$ is the ...
Qmechanic's user avatar
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15 votes
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Why I cannot write the time evolution operator $e^{-i(T+V)t}$ as the product of operators $e^{-iTt}e^{-iVt}$

This is because of the BCH formula $$\begin{align}e^Z~=~&e^Xe^Y \cr\Downarrow~&\cr Z~=~&X+Y+\frac{1}{2}[X,Y]+{\cal O}(X^2Y,XY^2),\end{align}$$ or equivalently, the Zassenhaus formula. But ...
Qmechanic's user avatar
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15 votes
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How can the universe evolve unitarily if there's no clock outside it?

I'm not sure an answer exists to your question because it is not stated sufficiently precisely. However I think it is worth pointing out that we need to distinguish between the time coordinate and the ...
John Rennie's user avatar
14 votes
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How to avoid paradoxes about time-ordering operation?

Time ordering $T$ (like any other operator ordering, such as, normal ordering $: ~:$, radial ordering ${\cal R}$, etc.) is technically speaking a linear map from symbols to operators, not a linear map ...
Qmechanic's user avatar
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14 votes
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Generality of the Schrödinger equation

There are two theorems relevant here. The former proves that actually, under natural hypotheses, the multipliers $\omega$ can be removed and one can safely apply the Stone theorem. This theorem is an ...
Valter Moretti's user avatar
13 votes

The formal solution of the time-dependent Schrödinger equation

The existing answer by Qmechanic is entirely correct and extremely thorough. But it is very long and technical, and there's a danger that the core of the answer can get buried under all of that. The ...
Emilio Pisanty's user avatar
13 votes

How does a particle know how to behave?

One way to think about it is that a particle "sniffs out" its immediate surroundings and reacts to gradient: a trend like a declining potential in one direction. Single-celled organisms do this. ...
Job Stancil's user avatar
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12 votes
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Do atomic orbitals "pulse" in time?

Short answer: yes, but only the phase factor has the time dependence. The spatial profile is constant in time because the eigenstates of the Hamiltonian are Stationary states. Maths: The time ...
SuperCiocia's user avatar
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12 votes
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Time evolution in quantum mechanics of states not contained in the Hilbert space

Generalized eigenfunctions are most naturally formalized as tempered distributions - linear maps from $\mathcal S\subset L^2(\mathbb R)$ to $\mathbb C$, where $\mathcal S$ is the Schwartz space of ...
J. Murray's user avatar
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12 votes

Why is the second law of thermodynamics not symmetric with respect to time reversal?

A long comment. Thermodynamics can be shown mathematically to be an emergent theory from statistical mechanics. Its laws are observational laws, deduced from variables and their measurements, that are ...
anna v's user avatar
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11 votes

Time-evolution with a time-dependent Hamiltonian

Actually in this specific case you are helped a bit because the time dependence is contained in a term proportional to the unit matrix $$ \hat H = E_0e^{t/\omega_0}\hat I + E_1\left(\begin{array}{cc} ...
ZeroTheHero's user avatar
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11 votes
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Why is the momentum-space wavefunction for a free particle not a function of time?

The Fourier transform $\phi(k)$ is a function only of $k$ and not of time because it indicates the amplitude of each plane wave that compose the wave function. The amplitudes are conserved in time, ...
Annibale's user avatar
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11 votes

Logarithm of Operators in Quantum Mechanics

The Stone's theorem proves the following. Consider a group of unitary operators $(U(t))_{t\in\mathbb{R}}$ acting on a Hilbert space $\mathscr{H}$ (i.e. satisfying $U(t+s)=U(t)U(s)$, in more ...
yuggib's user avatar
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11 votes

Time evolution operator in classical mechanics?

The formula is formal, and isn’t very useful for actual computation. You have to view $\{\cdot,H\}$ as a linear operator acting on the vector space of observables, i.e. real functions defined on phase ...
LPZ's user avatar
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11 votes
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Is unitary time evolution the same as obeying the Schrödinger equation?

As other answers have pointed out, assuming unitary time evolution does get you part of the way there — but not all the way. In particular, you could call it a postulate that the Hermitian operator ...
Jbag1212's user avatar
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11 votes

What physical processes other than scattering are accounted for by QFT? How do they fit into the general formalism?

We have a model (the standard model) that is formulated as a particular QFT, that seems to describe every experiment we've ever done on Earth (excluding anything gravitational). My point being that ...
QCD_IS_GOOD's user avatar
  • 6,785
10 votes

Is it obvious that the Hamiltonian observable in Quantum Mechanics should also be the Energy observable?

The fact that the energy should act as the generator of time translations is a fundamental postulate of the theory. For starters, you couldn't even define $H$ to be $i\hbar \partial_t$ because $H$ ...
Gold's user avatar
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