Who needs doors, when I can tunnel?

Tunneling is one of the mysterious features of quantum mechanics, but there is a very nice way to visualize it. There is a simulation method in order to obtain the ground state of quantum systems, called path integral Monte Carlo. It is based, as its name suggests, on Feynman’s path integral approach to quantum mechanics, but I will not go deeper in that… The idea is the following: a particle becomes a set of many, many copies, beads or replicas. Which one is the real particle? All of them, and none. Then, link them all, each one to the next, with a spring whose natural length is zero, and with a spring constant which increases with the mass. Now, put all the system in a “fake temperature”, which depends on \hbar… and that’s all! Simulate that, just using Monte Carlo, and the equilibrium distribution that you obtain is the ground state of the quantum system.

In the simulation, the potential is represented with the background colors: blue is low, orange is high. So, you see the potential consists of many minima, the central one is deeper. In fact, the energy of the particle is not enough to jump over the barriers… but it does not matter now. The “ring polymer” can jump, even if a classical particle can’t. The height of the barrier is not a huge problem if it is thin, because in that case the “spring” can stretch, and make one of the beads jump over it! That’s the tunneling, indeed.

So, what you’re observing in that video is the quantum cloud. In fact, each ring polymer represents a possible history for the particle, returning to the initial point.  Each bead corresponds to the position of the particle at a certain instant, and the energy in the springs corresponds to… the kinetic energy, which will not be zero because of the uncertainty principle.

If you need more explanations (I would!), read qfluct

Quantum dreams

Quantum mechanics, the dreams stuff is made of… (David Moser)

A quantum particle, prisoner in a square box of infinite walls, starts out with minimal energy, which grows and grows, slowly… although, no matter how much energy it gathers, no matter it grows quadratically… it will never escape…

You can also see it as the vibrational modes of a square drum. It looks continuous because I interpolated between them for a smoother visualization…

Emmy Noether

March 8th is the international working woman’s day, so I guess it’s just fair to write a blog entry about my favourite woman physicist… which happens to be Amalie (Emmy) Noether. I will not focus so much on her life, but on the most wonderful theorem on mathematical physics imagined by human minds, which was her brain-child…

About her life, I will only remind you that she was the first woman teacher at the University of Göttingen, recruited by Hilbert and Klein, in 1915. Göttingen was the most important center for theoretical physics at that time. It took a lot of arguing… One faculty member said “What will our soldiers think, when they come back home and are asked to study at the feet of women?”, and Hilbert gave his famous response: “This is a university, not a bath house”… Being a jew and socialist, she had to flee from Germany when Hitler came to power, and escaped to Russia and then to the US… You can read Wikipedia and many other sources for more info.

About her work… well, for me, the most impressive result of mathematical physics is known as Noether’s theorem, I’ll try to explain it in simple terms: if your physical system has a symmetry, then it has a conserved quantity. Conservation of energy is due to the invariance under time translation: physics is the same today or tomorrow. Conservation of momentum, due to invariance under spatial translations: physics is the same here, in Vladivostok or in alpha-Centauri. And so on. How come? I’ll try to give a derivation that makes you feel the thrill, yet does not get stuck in technical details…

Let us consider the space of all possible physical configurations of a system. In classical mechanics of point particles, a configuration is specified when you give all the positions and momenta, so a point in it will be given by x=(q_1,q_2,\cdots,p_1,p_2). Time-evolution is a flow in this configuration space. A flow is just putting a vector at each point of space, indicating the direction and speed with which you should move if you’re there. But there are many other interesting flows in configuration space, which correspond to other operations different from time evolution. You might consider the flow induced by rotating the whole system, or translating it, or stretching it…

All of those flows can be expressed in terms of generating functions. Consider any scalar function defined on the configuration space,  f(x). Its flow is defined in the following way. Get the gradient, \nabla f, which is a vector field. You might consider it to be the flow, but it is not convenient. We apply on it a certain matrix, call it ω, the symplectic matrix. This way, the flow of a function f is given by u=\omega \nabla f.  The only thing that you need to know about ω is that ωu is always perpendicular to u. If you move along a direction which is perpendicular to the gradient of a function, you keep the value of that function constant, right? So, moving along the flow \omega\nabla f preserves the value of f. The flow of f preserves f.

Now, apply this story to time evolution. Its flow is induced by the hamiltonian: u_t=\omega\nabla H. Of course, this means that time evolution will preserve the value of H. OK, we knew that! The equations of motion are

{\dot x}={\partial x\over \partial t}=\omega \nabla H(x)

What about other flows? Since I’m trying to keep things non-technical, I won’t prove the following assertions. Spatial translations are generated by the momentum f(x)=p. Rotations are generated by the angular momentum (on the z-coordinate, say): f(x)=L_z=yp_x-xp_y… What does it mean? Let’s say that you’re rotating your system by an angle α around the z-axis. You want to know the position of all the particles after such a rotation. Then, you get the “equations of motion”:

{\partial x\over \partial \alpha} = \omega \nabla L_z(x)

Let’s say that we want to know how one of these functions f evolves with time. Then, we derivate that thing with respect to time:

{\partial f\over \partial t}= {\partial f \over \partial x} {\partial x\over\partial t} = \nabla f \omega \nabla H

This object is important, so we give a name to it, the Poisson bracket, {f,H}.

So,  {f,g} means “how evolves f under the flux induced by g. Its main property is that {f,g}=-{g,f}, because of the properties of ω.

Now, Emmy Noether’s magic in action. Let us say that f is a symmetry of the system. This means that the hamiltonian does not evolve under the flux induced by f. So, {H,f}=0. But then, {f,H}=0 also! And this means that f does not change under the flux induced by H, i.e: under time evolution. So, f is a conserved quantity!

And this is Noether’s theorem: for every continuous symmetry of a system, there is a conserved quantity. It is, of course, the generator of that symmetry. If you have translation symmetry, momentum is preserved. Rotation-symmetry: angular momentum is preserved. For more intrincate symmetries, there are more abstract conserved quantities. For example, the esoteric gauge symmetry explains, via Noether’s theorem, the conservation of charge! And the conservation of energy? That’s the easiest, it’s just the symmetry under time-evolution…

For more info, besides Wikipedia (not the best site…), check John Baez’s explanation, or this page, or any good book on classical mechanics.

OK, this was a tribute to my favourite woman physicist of all times… But,  as of today, I also want to pay tribute to the ones I’ve met in my life: Silvia, Pushpa, Mar, Lourdes, Carmen, Nuria, Lola, Nina, Sagra, Elena, Vanessa, Susana, Rosa, Arantxa, Diana and all the rest…

There’s music in the primes… (part I)

OK, now a new section at physicsnapkins, in which I will discuss a bit about my own research… Recently, Germán Sierra and I have submitted to the ArXiv a paper about the Riemann hypothesis, which you can see here. To be honest, the real expert in the field is Germán, my contribution is mostly technical. Anyway, I’ll try to convey here the basic ideas of the story… We’ll give a walk around the concepts, assuming only a freshman maths level.

It’s fairly well known that the sum of the reciprocal numbers diverges: \sum_{n=1}^\infty 1/n\to\infty. Euler found, using an amazing trick the sum of the inverse squares and, in fact, the sum of the inverse of any even power. This formula is simply amazing: \sum 1/n^2 = \pi^2/6, isn’t it? Now, Riemann defined the “zeta” function, for any possible exponent:

\displaystyle{\zeta(z)=\sum_{n=1}^\infty {1\over n^z}}

So, we know that \zeta(1)=\infty, \zeta(2)=\pi^2/6, and many other values. Riemann asked: what happens when z is complex? Complex function theory is funny… We know that z=1 is a singularity.  If you do a Taylor series around, say, around z=2, the radius of convergence is the distance to the nearest singularity, so R=1. But now, your function is well defined in a circle of center z=2 and radius R=1. This means that you can expand again the function in a Taylor series from any point within that circle. And, again, the radius of convergence will be the distance to the closest singularity. This procedure is called analytical continuation.

 

The series of circles show how to compute the analytical continuation of a complex function...

Well… in the case of the Riemann \zeta function, the only singularity is z=1. Therefore, I can do the previous trick and… bypass it! Circle after circle, I can reach z=0 and get a value, which happens to be… -1/2. So, somehow, we can say that, if we had to give a value to the sum 1+1+1+1+…, it should be -1/2. Also, and even more amazing, \zeta(-1)=\sum_{n=1}^\infty n=1+2+3+4+\cdots=-1/12. Hey, that’s really pervert maths! Can this be useful for real life, i.e.: for physics. Well, it is used in string theory, to prove that you need (in the simplest bosonic case) dimension D=26… but that’s not true physics. Indeed, it’s needed for the computation of the Casimir effect. Maybe, I’ll devote a post to that someday. Anyway, this is the look of the Riemann zeta function in the complex plane:

 

The Riemann zeta function. Color hue denotes phase, and intensity denotes modulus. The white point at z=1 is the singularity.

Even more surprises… 1^2+2^2+3^2+\cdots=0. In fact, it’s easy (ok, ok… it’s easy when you know how!) to prove that \zeta(-2n)=0 for all positive n. Those are called the trivial zeroes of the Riemann zeta function (amazing!)… So, what are the non-trivial ones? Riemann found a few zeroes which were not for negative even numbers. But all of them had something in common: their real part was 1/2. And here comes the Riemann hypothesis: maybe (maybe) all the non-trivial zeroes of the \zeta function will have real part 1/2.

OK, I hear you say. I got it. But I still don’t get the fun about the title of the post, and why so much fuss about it. Here it comes…

Euler himself (all praise be given to him!) found an amazing relation, which I encourage you to prove by yourselves:

\displaystyle{\zeta(z)=\sum_{n=1}^\infty {1\over n^z} = \prod_{p \hbox{ prime}} {1\over 1-p^{-z}}}

Ahí comienza el link verdadero. Una pista para la demostración: expandimos el producto:

\displaystyle{\prod_{p \hbox{ primes}} {1\over 1-p^{-z}} = {1\over 1-2^{-z}} {1\over 1-3^{-z}} {1\over 1-5^{-z}} \cdots}

Wonderful. But 1/(1-x) can be easily recognized as the sum of a geometric series, right?

OK, in a few days, I’ll post the second part, explaining why there’s music in the primes, and how quantum mechanics might save the day…

Chameleons

The old man stared at us and spoke thus: “Those were the days, in Mod Island… We lived surrounded by beautiful colorful chameleons… I remember that, the day when I reached, 17 of them were blue, 15 were red and 13 were yellow. But, whenever two chameleons of different colour met, they would start a funny dance after which both would change to the colour which neither of them had. For example, if a blue and a red chameleon met, after the dance they both would become yellow. These dances took place for a long time, until all the chameleons became the same colour, and we knew it was time to leave…”.

“You’re lying”, said Alice. The old man grinned… “How are you so sure?” And Alice replied back, “Because, with your numbers,  it’s impossible that all chameleons become the same colour”.

Is Alice right?

 

Three chameleons for the math-kings under the sky...

More about Euler’s crazy sums

Recently we talked about how Euler managed, through some magic tricks, to find that the sum of the inverse of the square numbers is \pi^2/6… That was a piece of virtuosismo, but even geniuses sometimes slip up. This one is funny. You may know the sum of a geometrical series:

1+x+x^2+x^3+\cdots = {1\over 1-x}

(There are many ways to understand that formula, I’ll give you a nice one soon). Now, using the same formula, sum the inverse powers:

1+x^{-1}+x^{-2}+x^{-3}+\cdots = {1\over 1-x^{-1}} = {x\over x-1}

So far, so good. Now, imagine that we want the sum of all powers, positive and negative:

\cdots+x^{-3}+x^{-2}+x^{-1}+x^0+x^1+x^2+x^3+\cdots = {1\over 1-x} + {x\over x-1} -1 = 0

Amazing!!! The sum of all powers is zero!! Nice, ein? Euler concluded that this proved the possibility of the creation of the world from nothing, ex nihilo. So, this is easy… what is the problem with this proof?

BTW, I read this story and many others in William Dunham’s book The master of us all

I'm better than you all even when I'm wrong!