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Jul
20
comment Does “dark matter” explain how I can have -1 apples?
No, that apple would obviously be made of antimatter.
Jul
20
comment Interpretation of a density matrix as an observable
Why is the identity not an observable? I can even give you a "measurement device" for it: Just write an "1" on the display. And while I'm not familiar with the nLab approach, I'd be very surprised if the identity would not correspond to the constant function with value 1, which clearly is a smooth function.
Jul
13
comment could it be possible to create a pentane steam engine?
The basic principle of a steam engine (convert heat to mechanical motion through boiling water) is used in many types of power plants. Even a nuclear power plant in the end does nothing but drive a modern version of the steam engine with the heat of the nuclear reaction. However, a low-heat steam engine would be very inefficient, no matter how it is realized, due to the fundamental laws of thermodynamics.
Jun
22
comment Classical and quantum probabilities in density matrices
A third way to prepare the same $\rho$ is to generate a polarization-entangled two-photon state and send one of the photons. If you retain the other photon, you can turn the sent-out photon into either Alice's or Bob's ensemble by measuring the other photon in either the horizontal/vertical or circular polarization basis. Note that the actual measurement statistics of the sent-out photon won't depend on which way (or even whether) you measured the second photon.
Jun
15
awarded  Nice Answer
Jun
14
revised How can I find the motion equations of the 2-dim harmonic oscillator?
Fixed sign error
Jun
14
comment How can I find the motion equations of the 2-dim harmonic oscillator?
It's the Hamiltonian of a harmonic oscillator; the Hamiltonian is used in the Hamilton formalism (another way to derive the equations of motion). Basically, $H=T+V$, except that $T$ is described with the momentum. Otherwise, that's a special case of the potential I've given (with $a=\tfrac12w_x^2$, $b=0$, $c=\tfrac12w_y^2$), achieved by using the main axes of the potential as coordinates. BTW, I just notice that I've got a sign error in the equations of motions; I'll correct it in a moment.
Jun
14
comment How can I find the motion equations of the 2-dim harmonic oscillator?
@math12: "Two-dimensional harmonic oscillator" means, by definition, "two-dimensional system with attractive quadratic potential". "Quadratic potential" means "potential that is a polynomial of degree 2 in the position", and "attractive" means "when you go away from the equilibrium position, the potential gets greater".
Jun
14
comment How can I find the motion equations of the 2-dim harmonic oscillator?
@math12: No. A specific implementation (not the most general one) would be a mass that can move on a plane (horizontally, so no gravitation, and frictionless) which is connected through a (frictionless, massless, arbitrarily bendable) fiber through a small hole in that plane to a spring which is in its rest position when the mass is exactly above the hole.
Jun
14
comment How can I find the motion equations of the 2-dim harmonic oscillator?
$x$ and $y$ are just two directions in which the oscillator can be displaced. They don't even need to be orthogonal. The potential energy $ax^2+bxy+cy^2$ is just the most general possible quadratic form without linear or constant term. It has that form because that's what makes the system a harmonic oscillator (you could plug another potential in here, but then you'd no longer have a harmonic oscillator — well, unless your coordinates are something else than displacements, then it may be a harmonic oscillator in different coordinates).
Jun
14
comment How can I find the motion equations of the 2-dim harmonic oscillator?
See the edit of my answer.
Jun
14
revised How can I find the motion equations of the 2-dim harmonic oscillator?
Added an explicit calculation with Lagrange formalism
Jun
14
comment How can I find the motion equations of the 2-dim harmonic oscillator?
The harmonic oscillator is where the force is proportional to the displacement. A physical implementation of the one-dimensional h.o. is the spring pendulum. The normal pendulum isn't a harmonic oscillator (but can be approximated as one if the amplitude is sufficiently low). Solving a spherical pendulum is much harder than solving a harmonic oscillator, so you'd not normally substitute a spherical pendulum for a harmonic oscillator (rather, the other way round).
Jun
14
answered How can I find the motion equations of the 2-dim harmonic oscillator?
Jun
14
comment Is the graviton hypothetical?
@Anixx: In a theory where there are no gravitons, gravitational waves will certainly not be composed of them. If there could also be theories where there are gravitons but gravitational waves are not composed of them, I don't know.
Jun
14
answered Is the graviton hypothetical?
May
17
revised Quantum Mechanics - Observable
terminology correction: average value → expectation value
May
17
comment Atomic nucleus consisting of only neutrons?
@ashpool: See my edit
May
17
revised Atomic nucleus consisting of only neutrons?
added information about higher mass number
May
17
answered Atomic nucleus consisting of only neutrons?