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What do you get when you Taylor expand a Magnus expansion?

OP's series (3) is expanding in powers of the Magnus Hamiltonian $\bar{H}$, which by itself contains infinitely many terms (and infinitely high order of $\hbar^{-1}$). This makes OP's series (3) ...
Qmechanic's user avatar
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1 vote
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How does $[J^2, J_+] = 0$, but $ | l, m \rangle $ is not an eigenstate of $J_+$?

It all comes down to the theorem that states that when two (finite-dimensional) linear functions commute and are both diagonalizable then they are simultaneously diagonalizable. The operators $J^2$ ...
nico617's user avatar
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Constant of Motion in Quantum Mechanics for explicit time-dependent Operators

Take any time-independent operator $X$ which commutes with the Hamiltonian and multiply it by an arbitrary real-valued function of time : $$Y(t) = f(t)X$$ Then $Y(t)$ commutes with the Hamiltonian, ...
SolubleFish's user avatar
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2 votes
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Constant of Motion in Quantum Mechanics for explicit time-dependent Operators

If I understand you correctly, you want to compute $$\frac{d}{dt}\underbrace{\left(X(t)-\frac{P(t)}{m} t\right)}_{X^\prime(t)}$$ using the Heisenberg equation of motion $$\frac{d X^\prime(t)}{dt} =\...
Hyperon's user avatar
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Why does $\hbar$ appear twice in the axioms of QM?

The Schrodinger equation is not at all a postulate of quantum mechanics, it is derived directly by the other axioms, so it inherits the constant from the Commutation relations. See Sakurai for a full ...
LolloBoldo's user avatar
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Why does $\hbar$ appear twice in the axioms of QM?

$\hbar$ (or $h$) is scattered in many places over the whole QM subject, including well-known Planck law : $$\large {\displaystyle B_{\nu }(\nu ,T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }...
Agnius Vasiliauskas's user avatar
3 votes
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Does the Hamiltonian always commute with the Time Evolution Operator?

What can be said about the general case, in which $H$ depends on time explicitly? Specifically: Do $U(t, t_0)$ and $H$ still commute? No, not in general. In general, you can write $U$ as a time-...
hft's user avatar
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