But isen’t it weird that while the road to becoming a theoretical physicist includes numerous rigorous math courses like Algebra, Measure theory, Geometry, and Analysis, then most physicists still opt for the shortcut and get away with it?
Here we will take an interest in the field of mathematical analysis, that investigates mathematical functions, their derivatives and integrals by dealing with the infinitely small. The point of this post is to emphasize that rigor can take on more or less pedagogical forms by exploring a little known but perhaps more intuitive reformulation of mathematical analysis, known as the hyperreals…
Historically, the idea of taking something quite small, but finite, and then turn it infinitely small has been employed by many mathematicians throughout the ages, including works by Archimedes, Fermat, Euler, Bolzano, and Cauchy. The greatest result of this approach is calculus, where first Newton and especially Leibniz introduced positive quantities smaller than any real number and called them “fluxions” or “infinitesimals”.
But, infinitesimals ended up making serious formalistic mathematicians, aka David Hilbert, uneasy. Infinity and infinitesimals do not represent real numbers. So, how can we do calculations with numbers that do not exist? It was Karl Weierstrass, who followed up on the work of Bolzano, and answered this riddle by getting completely away with the infinitesimals. Instead he introduced the concept of limits and the method of epsilondelta reasoning – a method so strong that it is currently being celebrated in every calculus textbook and in many Analysis101 courses.
However, I’m sure that most people (phycisits included) have felt that this rigid approach to calculus could not be the full story. How was a successful tool like calculus built on such a shaky ground? And how did it manage to stand there majestically for centuries until somebody eventually took the time to secure the foundations? And how can it be that all the physics shortcuts have not left physicists in a logicless ruin?
Abraham Robinson was the first to consider taking the concept of infinitesimals seriously. He expanded the field of real numbers with unlimited numbers (the infinite) as well as infinitesimals. He needed a fancy name for his contrsuction, and today it is known as the field of the hyperreal. We are going to take a close look at it, following the most common approach based on something called ultrafilters.
For the construction of the hyperreals we need ultrafilters. They will help us ensure that the hyperreals is a wellordered field of numbers.
A free ultrafilter $U$ on the natural numbers, $\mathbb{N}$, is a set of subsets of $\mathbb{N}$ with the properties that,
As a simple example consider the empty set $\emptyset = \{\}$. The empty set is finite and does not belong in an ultrafilter (per statement 1). The complement of the empty set (which is the full set $\mathbb{N} \setminus \emptyset = \mathbb{N}$) then instead belongs to $U$ (per statement 4).
On the other hand: Consider the infinite subset containing all the even numbers, $\mathbb{N}_e$, and the subset containing all the odd numbers, $\mathbb{N}_o$. The intersection of those two subsets $\mathbb{N}_e \cap \mathbb{N}_o = \emptyset$, meaning that they cannot both belong to our ultrafilter (per statement 2). Because the two sets are the complements of each other, exactly one of them must belong to $U$ (per statement 4), but we are free to choose which one!
So an ultrafilter is not unique, but it can be shown that two ultrafilters on the same set are equivalent, and one can be constructed from the other simply by exchanging some elements with their complements.
After this small intermission, we are ready to construct the hyperreals. This particular construction uses sequences. A sequence is a function on the positive integers, and we write them in the following way,
The hyperreals are constructed from infinite sequences on the reals, $\underline{a} \in \mathbb{R}^{\mathbb{N}}$. The real numbers themselves can easily be represented as infinite sequences with constant elements,
So the number 2 is written,
But we may dream up many other “nonreal” sequences, like these:
We want the hyperreal numbers to be ordered, so any two hyperreal numbers can be compared and found to be either larger or smaller than each other. The main problem with our sequences is that they cannot be compared to each other in a consistent way. Is greater or smaller than ? And what about compared to ? If I change a single element of a sequence, is this new sequence then greater, smaller or equivalent to the original sequence?
In order to extract something usable, we must “thin out” in the forest of possible sequences.
We do this by letting two sequences represent the same hyperreal number if “most” of their elements are identical. For a consistent definition of “most” we return to the ultrafilter construction. We say that two sequences are equivalent, $\underline{r} \equiv \underline{s}$, if and only if the places, at which they share elements, form an infinite set that belongs to our ultrafilter, $U$:
Note that we sneakily introduced the parenthesis $\langle \cdots \rangle$, to signify that we are transforming a relation between two sequence representations of hyperreal numbers into a set of indices for which the sequence elements obey that same relation.
Using the properties of the ultrafilter, it is easy to show that this construction is transitive and defines an equivalence relation. The resulting equivalence classes define the hyperreals:
Hyperreal numbers may be manipulated by first choosing sample sequences from the relevant equivalence classes, then performing elementwise operations on those sample sequences, and finally representing the resulting hyperreal as the equivalence class of the resulting sequence. If that was too convoluted, here comes the simple definitions of addition and multiplication,
We are now ready to show that this careful construction allow us to order the hyperreals. Assume that one sequence, $r$, for “a large part” is smaller than another sequence, $s$. Again we take “for a large part” to mean that the sequence indices at which $r$ is smaller than $s$ belong to our ultrafilter, so $\langle r < s \rangle = \{ j : r_j < s_j \} \in U$. It follows almost automatically that $r \not\equiv s$, because $\langle r = s \rangle \subset \mathbb{N} \setminus \langle r < s \rangle \not\in U$ (by use of ultrafilter properties 1 and 4).
When applied a little more rigorously this shows that the hyperreals, $(*\mathbb{R}, \oplus, \odot)$, indeed form an ordered field.
As we already anticipated, the real numbers are embedded in the hyperreals represented by classes equivalent to the constant sequences. Let us introduce a map from the reals to the hyperreals, $* : \mathbb{R} \rightarrow *\mathbb{R}$, such that
We refer to these hyperreal numbers as standard. They directly equip the hyperreals with neutral elements for addition, $*0$, and multiplication, $*1$.
You may remember this alternating sequence, we looked at earlier:
It directly exhibits the cleverness of the ultrafilter construction. This sequence overlaps with the real number $*0$ on all the even sites, and with the real $*1$ on all the odd sites. From our discussion of ultrafilters, we know that the set of even numbers, $\mathbb{N}_e$, and the set of odd numbers, $\mathbb{N}_o$ cannot belong to the same ultrafilter. This means that for some choice of ultrafilter, the alternating sequence will belong to the $*0$ equivalence class, and in others it may belong to the $*1$ class.
The hyperreals also contain a lot of numbers in the vicinity of the standard numbers. Consider for example the hyperreal $s = [\underline{s}]$ defined by the sequence,
This sequence only intersects sample sequences from the standard number classes a finite number of times. This makes the corresponding hyperreal number, $s = [\underline{s}]$, decidedly nonstandard. It is easy to show that $\langle *x < s \rangle = \mathbb{N}$. More interestingly $s$ squeezes in between $*x$ and any other standard number $y > x$, because $\langle s < y \rangle \in U$.
The difference between those two hyperreal numbers, $s  *x$, is now smaller than any positive real number, and we may refer to it as infinitesimal. There exists a whole plethora of wellordered infinitesimals, and they can be compared to each other, so one is smaller and one is bigger, even though they are all infinitesimal. When two hyperreal numbers, $s$ and $r$, are (only) separated by an infinitesimal, we write that $s \simeq r$.
We can then introduce two useful functions. The halo of a hyperreal contains the infinitesimal cloud around the closest standard number,
Similarly, we say that two hyperreals, $s$ and $r$, are limited separated whenever their distance is not an infinitesimal, and we write $s \sim r$. The classes of this equivalence relation are called the galaxies, and we write.
We may take a look at the hyperreal infinitesimals through the infinitesimal microscope—a pedagogical representation of the hyperreal number line originally introduced by Jerome Keisler. Focusing on a particular hyperreal, $r$, the microscope magnifies its halo:
The unlimited numbers form another interesting family of hyperreal numbers. Consider for example the hyperreal number represented by,
It is clearly nonstandard (its sample sequences only overlap real number sequences at a finite number of places), and it is also greater than any real number: All properties which we normally associate with the infinite. However, like the infinitesimals, there are many different unlimited numbers within the hyperreals. Consider for example:
Here $m$ is also unlimited, and in addition $m > n$, which may confuse or comfort you, depending on your mathematical standpoint.
What have we just done? It seems we have taken the Cauchy sequences and turned those sequences into numbers themselves… or entities? Is that all there is to it?
Well, yes, in part.
But if you glance back, you may notice that it is not at all obvious. The ultrafilter construction is a necessary complication, and this complication partly explains why the hyperreals was not built earlier. Whether or not you like this construction is of course a subjective matter. Currently, the main advantage of nonstandard analysis, is that it allows you to think differently about calculus.
Instead of thinking about a process where finite elements shrink to zero, the hyperreals allow for a direct construction. In essence it makes it easier to extend properties of finite systems into continuous cases.
Let me note that there exists many other approaches to nonstandard analysis: axiomatic, like Internal Set Theory or Alternative Set Theory, as well as constructionist, like the surreal numbers or the superreal numbers.
We now leave the safe shore of pretended rigor and jump into computation. We will do so with two simple examples, but first we need:
The transfer principle
Any appropriately formulated statement is true of $∗\mathbb{R}$, if and only if it is true of $\mathbb{R}$.
This principle is a necessity in nonstandard analysis. For the ultrapower construction of the hyperreals—which we have considered here—the principle follows from Łoś’s theorem.
In order to transfer our results from the hyperreals and back to the reals, we define the shadow, $\mathrm{sh}(r)$, which maps a hyperreal number onto its closets real number. The function effectively removes any stray infinitesimals or infinites. We may do all our work within the hyperreals, and only return to the reals at the very end of our computation.
We start simple. Assuming—somewhat haphazardly—that simple functions may easily be transferred to the hyperreals, we define the derivative of a function as the shadow,
where $\epsilon$ is a hyperreal infinitesimal. We now drop the star extensions for brevity.
Take the cubic function, $f(x) = x^3$. The nonstandard derivation follows then almost automatically:
Canceling terms and performing the overall division gives the wellknown result,
In order to contrast the standard epsilondelta approach with our novel nonstandard method we turn to the proof of the intermediate value theorem (also known as Bolzanos theorem). If you want to, you can take a look at the standard proof before continuing.
The theorem considers a continuous function $f$ on the interval $[a, b]$. It states that for any $d$ in between $f(a)$ and $f(b)$, there exists $c \in [a,b]$ such that $f(c) = d$.
Without loss of generality we assume that $f(a) < f(b)$. Initially, we divide the interval $[a,b]$ into an integer, $N$, number of pieces of equal length, $\delta_N = (ba)/N$. Consider now the first division point, $s_N$, where $f(s_N) > d$. Then logically $f(s_N  \Delta_N) \leq d$.
Consider the division of the interval into a number of segments given by the unlimited hyperinteger, $M \in *\mathbb{N}$. Due to the transfer principle there still exists a smallest division point, $s_M$, where $f(s_M) > d$. However, $\Delta_M$ is infinitesimal, and this shows that $s_M \simeq s_M  \Delta_M$. By continuity $f(s_M) \simeq f(s_M  \Delta_M) \simeq d$. So the real number we are looking for is actually $c = \mathrm{sh}(s_M)$. Q.E.D.
This proof also ends this short section on the application of the hyperreals. As a sidenote we introduced continuity on the hyperreals as $r \simeq s \Rightarrow f(r) \simeq f(s)$. If you are interested in a rigorous derivation, you can find it in the source material.
I am indebted to the wellwritten article “Infinitesimals: History & Application” by Joel A. Tropp. If you want to delve further into nonstandard analysis, you should give it a good read.
Also the textbook “Elementary Calculus: An Infinitesimal Approach” by H. Jerome Keisler provided a sound foundation for my understanding of the hyperreals.
Wikipedia also has several articles about nonstandard analysis, although I’d rather recommend the two sources above for further study.
I hope you have enjoyed this small excursion into the hyperreals. Now, at least you have an answer ready when pesky mathematicians question your logic. I have also read that people are working on applying nonstandard reasoning to distributions and to concepts in mathematical physics like the Feynman Pathintegral. I may (or may not) cover that in a later post…
]]>The setup – which have been realized in many labs around the world – involves a nanometersized gold bridge, which is very carefully broken in order to create a nanometersized gap. Pour molecules onto this golden gap and pray that one of them will be caught in the gap, forming a very delicate molecular bridge. The transport properties of the molecule is then characterized by applying a voltage drop across this molecular junction and measuring the current response.
While electrons are normally thought of as particles, they will – due to quantum mechanics – also behave as waves. And what we seek to explain in this post, is how the wave nature of the electrons affects the transport properties of the molecular junction.
The most wellknown wave property is interference, and our aim is to understand how interference can affect the electronic properties of molecular bridges. The main example of interference is the popularized twoslitexperiment:
In the double slit experiment a wave originating on the left, passes through two slits and the wave intensity at some point in space is then detected by a detector. Here we will investigate what happens as we simultaneously change the position of the detector and the wave source on the vertical axis.
In the middle between the two slits, the two paths are of equal length. The wave passing along those two paths share the same phase meaning that the waves add, and this phenomena is known as constructive interference.
If we move both the source and the detector we encounter a different scenario. When the difference in path lengths exactly matches half a wavelength, the wave passing along the two paths is out of phase, and the waves substract – an effect known as destructive interference.
In fact, the wave intensity measured by the detector can be plotted as a function of the vertical displacement, y. The resulting transmission shows a characteristic interference pattern with valleys and hills corresponding to destructive and constructive interference effects. You may also note that when the source and the detector aligns with one of the slits, we may have perfect direct transmission:
We are interested in transport through molecules. As we shall see, the molecular bridge can be thought of more or less like the twoslitexperiment that we just discussed. However, in molecular electronics we cannot control the actual vertical displacement of the electronic wave passing through the molecular bridge. Instead we have – through the bias voltage and perhaps a small electrostatic gate – another very precise handle which we can control: The energy. In the following we will only look at situations where the energy of each electron passing through the molecule is conserved. That means that we cannot gain or loose energy by interacting with the electrodes, the molecule, the substrate or anything else in the setup.
Now, an electron enter the molecule enter with a certain energy, and it leaves with the exact same energy. This makes the energy equivalent to the vertical displacement, we discussed in the double slit experiment. This also means that it does not help us very much to look at only the molecular structure in realspace. In fact it is highly nontrivial to understand the wavecoherent transport through a molecule by just looking at the physical positions of the atoms in the molecule.
For simplicity we will constrict ourselves to a certain class of molecules: conjugated organic molecules. Conjugated organic molecules have a backbone of carbon atoms, which are strongly bound together by sigma bonds. In conjugated molecules those sigmabonds all lie in the plane, and perpendicular to this plane a single $p_z$ orbital sticks out from each carbon atomic site.
This $p_z$ orbital carries a single electron. The $p_z$ electrons are allowed to hop and delocalize over the entire system of $p_z$ orbitals. We call this system of orbitals and electrons for the pisystem. It is those pisystems that we will consider in the following.
We may diagonalize the Hamiltonian governing the electronic pisystem, and find the eigenenergies and the eigenstates. The eigenstates can for simplicity be thought of as molecular orbitals, where each orbital can accommodate up to two electrons (depending on their spin).
Let us make these concepts more tangible by considering the pisystem of 1,3butadiene. In the following figure we show how this molecule is put together, we show the schematic and the molecular orbitals as a function of energy. Note that the molecular orbitals have different colors, which actually correspond the sign of the wavefunction  actually the phase.
You may also note that we have filled up the molecular orbitals with the four available electrons (here drawn as up and down arrows). This fact will become important as we enter the next section.
We are now able to to look at our molecule in energy space, and we are now ready to perform a simple transport experiment. Am electron source sends in electrons with a certain energy. The electrons are then then transmitted through the molecule and obtain a phase given by the molecular orbital, which it passes through. Like in the double slit experiment, the electrons are allowed to pass through slits which does not match the energy, as long as they end up with the same energy after the process.
Lets splash some electronwaves on 1,3butadiene and see what we get:
As you can see, the electronic wavepackets have been colored according to the phase they have at each stage of the transport process. This is obviously true for the transport processes which happen through the unoccupied orbitals (with positive energy). Here the electron is added to the molecule, propagates, and is finally removed.
However, the electrons cannot move through the orbitals already occupied by electrons (negative energy). Instead one must take out an electron, allow the hole to propagate and then add the incoming electron. Because electrons are fermions this reversal of the process gives an additional sign relative to transport through unoccupied orbitals.
Looking at the phases, we expect to find constructive interference at zero energy. Here comes the transmission as a function of energy:
Not many surprises there. This would be almost indistinguishable to the case where we interpreted the electrons as being classical particles. The hallmark of wave interference is really the presence of destructive interference. So let us look at a slightly different molecular junction.
Again the colors correspond to the wavefunction phase, and the hole transport processes have gotten an extra sign because of fermionic exchange. In this case we see that the contributions are out of phase, and so we expect the transmission to vanish at zero energy. Lo and behold…
Now, fermionic exchange refers to how the exchange of two fermions (electrons are fermions) produces an additional sign. In this scattering setup two different processes take place. If the electronic wave moves through the unoccupied (unfilled) orbitals, the electron first enters the molecular orbital on the left and then leaves it on the right. For the filled orbitals the
involving the unfilled orbitals we first add and then remove an electron on the other side. For the filled orbitals we must switch the process around and remove an electron before we can add one from the electron source.
So, we have shown how quantum interference happens in molecular systems by creating a clear analogy with the wellknown double slit experiment. You may wish to try your understanding a bit, so let me ask some questions to help you along:
You may also wonder how this picture changes when you add Coulomb repulsion between the electrons. The funny thing is that not much changes, because in this case one can paraphrase the explanation in terms of the FeynmanDyson manybody orbitals. You can read more about those generalized molecular orbitals in this publication of mine.
]]>While I could have focused on the results of my thesis, I instead choose to talk about the more general topic: “Quantum mechanics  how hard can it be?”… or it was in fact done in danish: “Kvantemekanik  hvor svært kan det være?”.
While quantum mechanics is often considered quite difficult, the talk offered an intuitive introduction to the enthralling subject. The presentation emphasized the fact that many quantum mechanical phenomena are easily explained as wave phenomena.
A total of 80 teachers showed up for the presentation, filling the entire room. It was a pleasure giving the talk to such a dedicated crowd.
The presentation slides (including a number of vector illustrations created secifically for it) is available for download at a separate page.
]]>If you are interested, my thesis can be downloaded here:
This document explains how to access a sshenabled server by using another server as a bridge. This may be relevant if you must access server S2, say, in order to to carry out some calculations, but S2 is only connected to a university network S1. From your home computer you would have to first ssh into S1 and then ssh into S2. This quickly becomes messy, when you wish to do anything remotely complicated like transferring files between different machines.
In the end of this tutorial you should be able to simply write ssh S2
. And then end up immediately in S2. Similarly you can use any sshbased technology like the terminalbased scp S2:file .
, or draganddrop based solutions like MacFusion without ever thinking about the complicated two stage sshconnection.
The ssh program must of course be installed on your home computer and both servers, and publickey authentication must be enabled.
You also need server names, usernames and passwords for both servers S1 and S2.
Generate public keys on your home computer, by issuing the command:
sshkeygen t rsa
Leave everything blank. If you wish, you can generate different keys for logging into S1 and S2, and in that case you should save the keys in different files likes id1_rsa
and id2_rsa.
Move the generated keys to the server:
scp ~/.ssh/id_rsa.pub username1@S1:id_rsa.pub
Create the .ssh
directory and append the key to the authorized_keys
file.
1 2 3 4 5 

Repeat this procedure for S2, making sure to move the relevant files all the way through S1 first. Remember to delete the id_rsa.pub
file after you have added it to the authorized_keys
file.
rm id_rsa.pub
You should then be able to log in to S1 from your home computer without using your password. Test that this work seamlessly before moving on.
Create ~/.ssh/config
file to manage your connections. In the config file you should put the following:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 

The variables S1nick and S2nick are some short nick names for the two servers. This is convenient because servernames tend to be long and tedious to type. If you are using a university network you could e.g. set S1nick to “uni” and S2nick to “calc”.
Now you can access S1 with the command
ssh S1nick
And server S2 with,
ssh S2nick
Voila!
]]>If you indeed happened upon this blog in its infancy, when this is the only post… Then I am sorry to inform you that there are no more posts, and this post is also quite boring. Move along, nothing to see here.
To really get started with this first post I would like to entertain you with some bits and pieces of homespun philosophy:
Having started on this rather silly note… Allow me to contrast with a very serious picture of me:
I am a physicist by education, and the blood of the natural sciences runs deep in my veins. However, as a scout I have flirted quite a bit with project management, leadership and administration.
I have a strange fascination with writing computer code. I have been told that my grandfather – who was an engineer using early mainframe computers – could suddenly stop whatever he was doing, and then try to solve a computer problem for hours before returning to the task at hand… like eating or clothing. I do not think I have inherited that character flaw, but the fascination is definitely there.
I also have a small family. My girlfirend and I have a 2year old daughter, which in the most wonderful way takes up most of our spare time.
I tell you all this to warn you that any subsequent posts are probably going to center around the four themes: Physics, Scouting, Programming and Parenting. In what proportions these themes will feature, I am not quite sure as of yet.
To finish this post let me tell you the reason for actually writing this blog. It is in fact somewhat lame (as is the reasons for most real world projects). So let me divulge you in my lameness:
I used to have a static html homepage which mainly hosted my wishlist. I then stumbled upon Octopress which works by compiling all posts to static content, and in my case serves it to you via github. I became fascinated with the setup and one sleepless night later my old site was migrated to Octopress. The blog was baked into Octopress, but I do not know why I did not remove it then.
Over time I got annoyed with the empty homepage… so slowly, in the back of my brain, I began to consider the possibility of actually writing a blog… and this night then just happened to be the time where this possibility was violently born into the world as a fact…
I know that this is quite a silly first post, and I do apologize. Instead I hope that you may value the incoherent and silly character of this piece. I believe when pushing a text towards the incoherent and rambling, there is a turning point where meaning will change and suddenly the test makes sense on a completely new level and in a completely new way.
If not then hopefully the post is silly enough to force me to quickly write more posts to cover this up…
See you later.
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