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Hi, I am a string theorist and a publicist.


Jun
19
comment Which derivative with respect to time is which in the Heisenberg picture of quantum mechanics?
The latter two derivatives are equal to $v_{Heis} = p_{Heis}/m$ and $-\nabla \cdot V_{pot.\,energy} (\vec x_{Heis})$ in the simplest examples of quantum mechanics. These Heisenberg equations for the $d/dt$ derivatives mimic the classical equations of motion. That's of course no coincidence. All the time derivatives come from the $d/dt$ of the operators in the Heisenberg picture while $|\psi \rangle$ states are fixed. So the expectation values have to evolve, in the classical limit, just like the classical quantities: the operator equations have to agree in the classical limit. And they do.
Jun
19
comment Which derivative with respect to time is which in the Heisenberg picture of quantum mechanics?
Dear Qiaochu, even if $A_{Heis}$ meant nothing else, we still wrote its "definition" in terms of $A_{Schr}$, by the conjugation, so it is definitely not an undefined object. It is totally well-defined - if you decide that the conjugation is its definition - and all questions about it may be answered by pure thought. Yes, it is true that $\partial x_{Heis} / \partial t = 0$ and similarly for $p_{Heis}$. However, $dx_{Heis}/dt$ is not zero, and neither is $dp_{Heis}/dt$.
Jun
19
comment Which derivative with respect to time is which in the Heisenberg picture of quantum mechanics?
One more general comment about $d/dt$ vs $\partial / \partial t$. The distinction between these two occurs not only in quantum physics but also e.g. in classical hydrodynamics. When the velocity $v(x,y,z,t)$ is a function of the spacetime coordinates, it's clear what $\partial v/\partial t$ means - here $x,y,z$ are kept fixed. However, we may also study $dv/dt$ of a water molecule which is $\partial v / \partial t + (v\cdot \nabla) v$.
Jun
19
revised Which derivative with respect to time is which in the Heisenberg picture of quantum mechanics?
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Jun
19
revised Which derivative with respect to time is which in the Heisenberg picture of quantum mechanics?
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Jun
19
comment Which derivative with respect to time is which in the Heisenberg picture of quantum mechanics?
Just to be sure, in the equation $dA/dt = i/\hbar [H,A] +\partial A/\partial t$, all the symbols $A$ are obviously still $A_{Heis}$, including one in the last term. This whole equation is in Heisenberg picture - it is the dynamical equation determining the time evolution of all of physics in the Heisenberg picture so of course that it cannot rely on other pictures.
Jun
19
comment Which derivative with respect to time is which in the Heisenberg picture of quantum mechanics?
You make it sound contrived but there is nothing contrived about it. If $A_{Heis}(t) = e.A_{Schr}.e$, then obviously $d A_{Heis}(t)/dt = d(e.A_{Schr}.e)/dt$. This is called substitution and I am sure that most mathematicians would agree that this is true. Obviously, when we use symbols, we must know what they mean. The Wikipedia page assumes that the reader knows which $A(t)$ is the Schr. picture and which one is the Heis. picture. I just made this distinction more explicit. But when people know what the symbols mean, the notation of derivatives is unambiguous and standard.
Jun
19
answered Electric potential energy in curved space-time
Jun
19
answered Which derivative with respect to time is which in the Heisenberg picture of quantum mechanics?
Jun
15
awarded  Nice Answer
Jun
14
reviewed Approve Two spheres (A physics olympiad problem)
Jun
13
comment Can quantum annealing be used for factorization?
possible duplicate of What can the D-Wave quantum computer do?
Jun
13
answered infinite grid of planets with newtonian gravity
Jun
13
answered Concrete example of divergence of a vector field
Jun
12
answered Do color-neutral gluons exist?
Jun
12
reviewed Approve Can the effects of gravity be broken by jumping?
Jun
12
answered Is this algorithm for simulating a quantum computer accurate?
Jun
11
comment Commutators and Hermiticity - Exam question
You would need to know that $B$ has no zero eigenvalues, or something like that, because you may always find solutions to the conditions by creating $A,B$ as block-diagonal combinations of simpler solutions.
Jun
11
comment Two spheres (A physics olympiad problem)
Sorry, (b) is zero, isn't it? An inelastic collision is one in which the spheres merge into one object moving by one velocity, and by momentum conservation, its momentum is zero, isn't it? Am I missing something?
Jun
11
answered What causes the Pauli exclusion principle (and why does spin 1/2 = fermion)?