homework and exercises - Centrifugal force acting on ring

Take for example the case of a rod rotating about an axis passing through its centre of mass and perpendicular to it. It has a ring hung from one of its sides. The rotation of the rod causes the ring to move out wards and ultimately fall off the rod.

I asked my teacher what causes the ring to move out wards and he said the centrifugal force acting in the frame of reference of the ring causes motion. What exactly is the ring reference frame? Is it a rotating frame? How would u visualize motion in this frame of reference?

quantum mechanics - Local measurement of entangled particle pairs and interpretation of state

This just started to bother me after reading yet another entangled particle question, so I hate to ask one myself, but...

If we have two entangled particles and take a measurement of one, we know, with certainty, what the outcome of same measurement of the other particle will be when we measure it. But, as with all fundamental QM, we also say the second particle does not have a definite value for the measurement in advance of making it.

Of course, the particle pairs are always described by an appropriate wave function, and measurement of half the pair changes the probabilities associated with each term. But, and this is the part that bothers me, with knowledge of the first measurement, we know the outcome of the second measurement. There's nothing strange or forbidden or anything about this, no non-local affects, no FTL communication, etc.

But does it still make sense to talk about the second particle not being in a definite state prior to the measurement? Or can we consider a local measurement of one half of an entangled pair to be a measurement of the other half? The latter option feels distasteful because it implies communication between the particles when communication is possible, which is obviously wrong, but I find it hard to pretend the second particle in the pair, whose state I know by correlation, is somehow not in that state before I confirm it by measurement.

I should clarify that my uneasiness comes from a comparison with the usual "large separation" questions of entangled particles, where our ignorance of the first measurement leads us to write an "incorrect" wavefunction that includes a superposition of states. Although this doesn't affect the measurement outcome, it seems there is a definite state the second particle is in, regardless of our ignorance. But in this more commonly used scenario, it seems more fair to say the second particle really isn't in a definite state prior to measurement.

As a side note, I realize there's no contradiction in two different observers having different descriptions of the same thing from different reference frames, so I could accept that a local measurement of half the pair does count as a measurement of the other half, while a non-local measurement doesn't, but the mathematics of the situation clearly point to the existence of a definite state for the second particle, whereas we always imply it doesn't have a well-defined state (at least in the non-local case) before measurement.

At this point, my own thinking is that the second particle is in a definite state, even if that state could, in the non-local separation case, be unknown to the observer. But that answer feels too much like a hidden variables theory, which we know is wrong. Any thoughts?

Answer

According to quantum mechanics, the particles - whether it is the first particle or the second particle or any other particle in the Universe - refuse to have any well-defined state or property prior to the measurement.

The only "kind of" exception is the case - relevant for a maximally entangled scenario - in which we are interested in a property of the second particle - or any other particle - that is predicted to take a particular value with probability equal to 100 percent. Of course, if the probability is 100 percent, then you can be sure what the measured property will be, and you may assume that this value of the property exists even before the measurement.

However, for the very same particle that has some value of the quantity equal to something at 100 percent, there still inevitably exist other observables that are not known. (Just design a Hermitian observable with random off-diagonal matrix elements.)

In the basis of eigenstates of those other properties, the probability amplitudes are generic, and some of the options have probabilities that differ from 0 percent as well as 100 percent. For those quantities - and it is a majority of observables - the usual prescriptions of quantum mechanics hold: the value is not determined prior to the measurement. It is not just unknown to the physicists: it is unknown to Nature. A picture in which those observables are determined leads to wrong predictions and contradictions.

cosmology - Does gravity slow the expansion of the universe?

Does gravity slow the expansion of the universe?

I read through the thread http://www.physicsforums.com/showthread.php?t=322633 and I have the same question. I know that the universe is not being stopped by gravity, but is the force of gravity slowing it down in any way? Without the force of gravity, would space expand faster?

Help me formulate this question better if you know what I am asking.

astronomy - What parameters control the amount of thermal energy an object must possess for it to be detectable in space?

What parameters control the amount of thermal energy a space object must possess for it to be detectable by a sensor also in space (i.e. one that does not have to deal with interference from a planet's atmosphere)?

Answer

It's not the heat that matters, but the brightness. Sedna, discovered in 2003, has an apparent magnitude of 21.3. Wikipedia lists an apparent magnitude of 27 as the faintest object observable from an eight-meter ground-based telescope, and magnitude 36 as the faintest observable with a 40-meter telescope under construction.

These objects are not seen by their thermal radiation, like stars, but because they have nonzero albedo and reflect sunlight. Both incident and reflected sunlight fall off like the square of the distance. When Sedna moves from perihelion (now) to aphelion (in 11,000 years) its distance will increase by about a factor of ten; the sunlight incident on the surface will decrease by a factor of 100 (which is five magnitudes) and the brightness at Earth of the reflected light will decrease by another factor of 100. At aphelion, then, Sedna will have an apparent magnitude of around 37 and might or might not be visible from Earth's surface using a telescope the size of a baseball diamond. That's at a distance of 0.015 light-year.

So we don't have to worry about seeing Oort cloud objects by their reflected light at a light-year. For blackbodies (like stars) there is a straightforward relationship between absolute magnitude, distance, and size: a big, cold object can emit more light and be more luminous than a small, hot object. Absolute magnitudes are based on brightness at a distance of 10 parsecs or 32 light-years, so our barely-detectable magnitude-37 blackbody at one light-year would have an absolute magnitude of 42.5 or so. Such an object would either be a stray planet (of which, apparently, there are many) or a black dwarf. I'm not certain whether a brown dwarf could be that cool and still retain its atmosphere, and a brown dwarf that lost its atmosphere would just be a rocky stray planet. (I may come back later and edit in a calculation, hmmmm.)

A quantum particle moving from A to B will take every possible path from A to B at the same time

If a quantum particle can take an unlimited number of paths to get from point A to point B wouldn't a quantum particle never get from point A to point B?

A quantum particle takes every path at the same time to get from A to B? How is that even possible? Can anyone really explain what is going on? And maybe shed some light on the math?

quantum mechanics - Observing the exponential growth of Hilbert space?

One of the weirdest things about quantum mechanics (QM) is the exponential growth of the dimensions of Hilbert space with increasing number of particles. This was already discussed by Born and Schrodinger but here's a recent reference:

http://arxiv.org/PS_cache/arxiv/pdf/0711/0711.4770v2.pdf

and another one:

http://www.scottaaronson.com/papers/are.pdf

Another difference between quantum mechanics and classical physics is that QM is discrete.
A good example is the line spectrum of hydrogen.

Is there any good observational example which demonstrates the exponential growth of Hilbert space dimensions?
Even for low numbers?
(Does it perhaps show up in the line spectrum of a three or four valent atom?)

optics - Far-field intensity from scattering of small particles

Howdy, I'm building a simulation for looking at the light field underwater. In order to verify my simulation, I'm looking for some data showing the far-field intensity that comes from single scattering from many small particles in suspension. I suspect Mie theory plays a part here, but I'm having a hard time finding some results, rather than doing all the derivations myself.

In other words, I want to know the power distribution on a plane after a beam of light has been scattered by a bunch of small particles through a volume. I know Oregon Medical has a nice online simulation that produces scattering phase functions (http://omlc.ogi.edu/calc/mie_calc.html), but that doesn't give me the power on a plane - only the scattering profile from individual particles. I'm fine with only a single scattering result.

I want to do initial verification using a fixed particle size. Having a hard time finding a reference with this data. Help?

Answer

The main problem about a rigorous solution to such a scattering proplem is that computations are extremely demanding. Just imagine you have a wavelength $\lambda$ of some $400$nm to $700$nm for visible light (from here):

Now, to do physically meaningful simulations, you will need a sub-wavelength lattice which makes any computational cell above, say $10\,\mu m^3$ not accessible since you have in the order of one million grid points.

Approximative Approaches

But of course there can be ways out of it if you are willing to make some approximations which will largely depend on the characteristics of the particles you are looking at. It is best to assume that we only have spherical particles since we can apply Mie theory in this case.

Large Particles

First of all, let us consider particles which are much larger than the wavelength. Then, the radius $R$ times the wave vector $k=2\pi/\lambda$ is much bigger than one, $$kR\gg1$$ which basically means that one observes reflection at a plane interface. You can implement these particles using geometrical optics (mixed with Fresnel reflection if you like) since nothing really wave-like will happen as in this image (taken from here):

Small Particles

Second, the particles should be much smaller than the wavelength, $$kR\ll1\,.$$ Then, everything what is observed is a sum of dipolar responses of the particles in the so-called Rayleigh-scattering. Then,

the intensity of light scattered by a single small particle from a beam of unpolarized light of wavelength $\lambda$ and intensity $I_0$ is given by:

$$I=I_0(1+\cos^2\theta)\frac{(kR)^6}{2(kr)^2}\left(\frac{n_p^2-1}{n_p^2+2}\right)$$

where I have chosen the variables to be consistent with the used terminology and $r$ is the distance to the object, $\theta$ is the scattering angle and $n_p$ is the sphere's refractive index. Here is an image of such a situation with some metal particles also having quadrupolar excitation (from here):

A Mean Field Approach - Effective Permittivity

If you have a lot of these small objects, you may use the Clausius-Mossotti relation which gives you an effective permittivity $\epsilon_p=n_p^2$ depending on the concentration of the particle in some volume: $$\epsilon_{eff} = \epsilon_p + \frac{n\alpha}{1-\frac{n\alpha}{3\epsilon_p}}$$ where $\alpha$ is the polarizability of the sphere, for details see e.g. Electromagnetic mixing formulas and applications by Sihvola. This would be something like a mean-field approach. You can make some very neat effects using this effective approach since it allows you to calculate a continuous refraction around some particle streams under water.

However, if the particles size is in the order of the wavelength, $$kR\approx 1$$ then you may have to take higher multipole moments into account which may be a very demanding task.

For much more on the subject I would recommend Bohren & Huffmanns classic Absorption and Scattering of Light by Small Particles.

Sincerely