Archive for the Cute Problems Category

A Problem of Gravity

Posted in Cute Problems with tags , , on May 9, 2017 by telescoper

Here’s a nice one for the cute problems folder.

Two spherically symmetric stars A and B of equal mass M and radius r have centres separated by a distance 6r. Ignoring any effects due to the orbital motion of the stars, determine a formula (in terms of G, M and r) for the minimum velocity with which material can be ejected from the surface of A so as to be captured by B.

Answers through the comments box please. First correct answer receives 7 points.

 

A Question of Equilibrium

Posted in Cute Problems with tags , on March 30, 2017 by telescoper

It’s been quite a while since I’ve been able to find time to post any items in the cute problems folder, and I don’t have much time today either, but here’s a quickie. You may well find this a lot harder than it looks at first sight. At least I did!

An isolated system consists of two identical components, each of constant heat capacity C, initially held at temperatures T1 and T2 respectively. What is the maximum amount of work that can be extracted from the system by allowing the  two components to reach equilibrium with each other?

As usual, answers through the comments box please. There is no prize, even if you’re right.

 

 

The Neyman-Scott ‘Paradox’

Posted in Bad Statistics, Cute Problems with tags , , , , on November 25, 2016 by telescoper

I just came across this interesting little problem recently and thought I’d share it here. It’s usually called the ‘Neyman-Scott’ paradox. Before going on it’s worth mentioning that Elizabeth Scott (the second half of Neyman-Scott) was an astronomer by background. Her co-author was Jerzy Neyman. As has been the case for many astronomers, she contributed greatly to the development of the field of statistics. Anyway, I think this example provides another good illustration of the superiority of Bayesian methods for estimating parameters, but I’ll let you make your own mind up about what’s going on.

The problem is fairly technical so I’ve done done a quick version in latex that you can download

here, but I’ve also copied into this post so you can read it below:

 

neyman-scott1

neyman-scott2

I look forward to receiving Frequentist Flak or Bayesian Benevolence through the comments box below!

Reflections on Quantum Backflow

Posted in Cute Problems, The Universe and Stuff with tags , , , , on November 10, 2016 by telescoper

Yesterday afternoon I attended a very interesting physics seminar by the splendidly-named Gandalf Lechner of the School of Mathematics here at Cardiff University. The topic was one I’d never thought about before, called quantum backflow. I went to the talk because I was intrigued by the abstract which had been circulated previously by email, the first part of which reads:

Suppose you are standing at a bus stop in the hope of catching a bus, but are unsure if the bus has passed the stop already. In that situation, common sense tells you that the longer you have to wait, the more likely it is that the bus has not passed the stop already. While this common sense intuition is perfectly accurate if you are waiting for a classical bus, waiting for a quantum bus is quite different: For a quantum bus, the probability of finding it to your left on measuring its position may increase with time, although the bus is moving from left to right with certainty. This peculiar quantum effect is known as backflow.

To be a little more precise about this, imagine you are standing at the origin (x=0). In the classical version of the situation you know that the bus is moving with some constant definite (but unknown) positive velocity v. In other words you know that it is moving from left to right, but you don’t know with what speed v or at what time t0 or from what position (x0<0) it set out. A little thought, (perhaps with the aid of some toy examples where you assign a probability distribution to v, t0 and x0) will convince you that the resulting probability distribution for moves from left to right with time in such a way that the probability of the bus still being to the left of the observer, L(t), represented by the proportion of the overall distribution that lies at x<0 generally decreases with time. Note that this is not what it says in the second sentence of the abstract; no doubt a deliberate mistake was put in to test the reader!

If we then stretch our imagination and suppose that the bus is not described by classical mechanics but by quantum mechanics then things change a bit.  If we insist that it is travelling from left to right then that means that the momentum-space representation of the wave function must be cut off for p<0 (corresponding to negative velocities). Assume that the bus is  a “free particle” described by the relevant Schrödinger equation.One can then calculate the evolution of the position-space wave function. Remember that these two representations of the wave function are just related by a Fourier transform. Solving the Schrödinger equation for the time evolution of the spatial wave function (with appropriately-chosen initial conditions) allows one to calculate how the probability of finding the particle at a given value of evolves with time. In contrast to the classical case, it is possible for the corresponding L(t) does not always decrease with time.

To put all this another way, the probability current in the classical case is always directed from left to right, but in the quantum case that isn’t necessarily true. One can see how this happens by thinking about what the wave function actually looks like: an imposed cutoff in momentum can imply a spatial wave function that is rather wiggly which means the probability distribution is wiggly too, but the detailed shape changes with time. As these wiggles pass the origin the area under the probability distribution to the left of the observer can go up as well as down. The particle may be going from left to right, but the associated probability flux can behave in a more complicated fashion, sometimes going in the opposite direction.

Another other way of thinking about it is that the particle velocity corresponds to the phase velocity of the wave function but the probability flux is controlled by the group velocity

For a more technical discussion of this phenomenon see this review article. The exact nature of the effect is dependent on the precise form of the initial conditions chosen and there are some quantum systems for which no backflow happens at all. The effect has never been detected experimentally, but a recent paper has suggested that it might be measured. Here is the abstract:

Quantum backflow is a classically forbidden effect consisting in a negative flux for states with negligible negative momentum components. It has never been observed experimentally so far. We derive a general relation that connects backflow with a critical value of the particle density, paving the way for the detection of backflow by a density measurement. To this end, we propose an explicit scheme with Bose-Einstein condensates, at reach with current experimental technologies. Remarkably, the application of a positive momentum kick, via a Bragg pulse, to a condensate with a positive velocity may cause a current flow in the negative direction.

Fascinating!

 

 

 

 

A Cosmic Microwave Background Dipole Puzzle

Posted in Cute Problems, The Universe and Stuff with tags , , , , , on October 31, 2016 by telescoper

The following is tangentially related to a discussion I had during a PhD examination last week, and I thought it might be worth sharing here to stimulate some thought among people interested in cosmology.

First here’s a picture of the temperature fluctuations in the cosmic microwave background from Planck (just because it’s so pretty).

planck_cmb

The analysis of these fluctuations yields a huge amount of information about the universe, including its matter content and spatial geometry as well as the form of primordial fluctuations that gave rise to galaxies and large-scale structure. The variations in temperature that you see in this image are small – about one-part in a hundred thousand – and they show that the universe appears to be close to isotropic (at least around us).

I’ll blog later on (assuming I find time) on the latest constraints on this subject, but for the moment I’ll just point out something that has to be removed from the above map to make it look isotropic, and that is the Cosmic Microwave Background Dipole. Here is a picture (which I got from here):

dipole_map

This signal – called a dipole because it corresponds to a simple 180 degree variation across the sky – is about a hundred times larger than the “intrinsic” fluctuations which occur on smaller angular scales and are seen in the first map. According to the standard cosmological framework this dipole is caused by our peculiar motion through the frame in which microwave background photons are distributed homogeneously and isotropically. Had we no peculiar motion then we would be “at rest” with respect to this CMB reference frame so there would be no such dipole. In the standard cosmological framework this “peculiar motion” of ours is generated by the gravitational effect of local structures and is thus a manifestation of the fact that our universe is not homogeneous on small scales; by “small” I mean on the scales of a hundred Megaparsecs or so. Anyway, if you’re interested in goings-on in the very early universe or its properties on extremely large scales the dipole is thus of no interest and, being so large, it is quite easy to subtract. That’s why it isn’t there in maps such as the Planck map shown above. If it had been left in it would swamp the other variations.

Anyway, the interpretation of the CMB dipole in terms of our peculiar motion through the CMB frame leads to a simple connection between the pattern shown in the second figure and the velocity of the observational frame: it’s a Doppler Effect. We are moving towards the upper right of the figure (in which direction photons are blueshifted, so the CMB looks a bit hotter in that direction) and away from the bottom left (whence the CMB photons are redshifted so the CMB appears a bit cooler). The amplitude of the dipole implies that the Solar System is moving with a velocity of around 370 km/s with respect to the CMB frame.

Now 370 km/s is quite fast, but it’s much smaller than the speed of light – it’s only about 0.12%, in fact – which means that one can treat this is basically a non-relativistic Doppler Effect. That means that it’s all quite straightforward to understand with elementary physics. In the limit that v/c<<1 the Doppler Effect only produces a dipole pattern of the type we see in the Figure above, and the amplitude of the dipole is ΔT/T~v/c because all terms of higher order in v/c are negligibly smallFurthermore in this case the dipole is simply superimposed on the primordial fluctuations but otherwise does not affect them.

My question to the reader, i.e. you,  is the following. Suppose we weren’t travelling at a sedate 370 km/s through the CMB frame but instead enter the world of science fiction and take a trip on a spacecraft that can travel close to the speed of light. What would this do to the CMB? Would we still just see a dipole, or would we see additional (relativistic) effects? If there are other effects, what would they do to the pattern of “intrinsic” fluctuations?

Comments and answers through the box below, please!

 

A Problem in Lagrangian Mechanics

Posted in Cute Problems with tags , , on April 28, 2016 by telescoper

Today, as well as saying goodbye to Sally Church, I managed to finish my lecture course on Theoretical Physics. There’s still another week of teaching to go, but I have covered all the syllabus now and can use the remaining sessions for revision. The last bit of the course module concerned the calculus of variations and a brief introduction to Lagrangian mechanics so for a bit of fun I included this example.

Professor Percy Poindexter of the University of Neasden has invented a new theory of mechanics in which the one-dimensional motion of a particle in a potential V(x) is governed by a Lagrangian of the form

L=mx\ddot{x} +2V(x).

Use Hamilton’s Principle and an appropriate form of the Euler equation to derive the equation of motion for such a particle and comment on your answer.

UPDATE: Since nobody has commented I’ll just reveal the point of this question, which is that if you follow the instructions the equation of motion you should obtain is

m\ddot{x}= -\frac{\partial V}{\partial x},

which is exactly the same as you would have got using the usual Lagrangian

L= \frac{1}{2}m\dot{x}^{2} - V(x).

Anyone care to comment on that?

A Potential Problem with a Sphere

Posted in Cute Problems with tags , , on March 9, 2016 by telescoper

Busy busy busy again today so I thought I’d post a quick entry to the cute problems folder. I set this as a problem to my second-year Theoretical Physics students recently, which is appropriate because I encountered it when I was a second-year student at Cambridge many moons ago!

Sphere_problem

HINT: You can solve this by finding the general solution for the potential at any point inside the sphere, but that isn’t the smart way to do it!

FURTHER HINT: The question asks for the Electric Field at the origin. What terms in the solution for the potential can contribute to this?

Answers through the comments box please!

OUTLINE SOLUTION: A numerically correct answer has now been posted so I’ll give an outline solution. The potential V inside the sphere is governed by Laplace’s equation, the general solution of which is a series expansion in powers of r and Legendre polynomials, i.e. rn Pn(θ). The coefficients of this expansion can be determined for the given boundary conditions (V=V0 at r=a for θ = +1, V=0 for cos θ = -1). However this is a lot more work than necessary. The question asks for the electric field, i.e. the gradient of the potential, and if you look at the form of the potential there is only one term that can possibly contribute to the field at r=0, namely the one involving rP1(cosθ) =rcos θ (which is actually z). Any higher power of r would give a derivative that vanishes at the origin. Hence we just have to determine the coefficient of one term. Using the orthogonality properties of the Legendre polynomials this can easily be seen to be 3V0/4a. The electric field is thus -3V0/4a in the z-direction, i.e. vertically downwards from the top of the sphere.