Remember the discussion about superpositions? There we said that if some object, like an atom, can have states of spin up and spin down, it could also be in a superposition of the two, that is in the state of “spin up and down together”, which we wrote as . Well nothing stops us from also considering superpositions such as , that involve imaginary numbers. This can be made even more complicated by considering complex numbers, mixtures of imaginary an real numbers.

And where do we come across this kind of thing? As it turns out, the correct wave function (that is, state) of the electron in the hydrogen atom is a complex object, having both real and imaginary parts. The probability of finding the electron around the hydrogen is given by the square (absolute value squared, more precisely) of the wave function. Since it is difficult to draw, we present the picture of the wave function squared.

]]>When we speak of objects, we generally classify them by their possible states. for example, a cat is either dead or alive, or . A ball is rotating clockwise or anticlockwise, or . And remember our discussion about the Stern Gerlach experiment? Atoms have spin up or spin down, or . Any object that you can think of is essentially described by its possible states, and how they change when things happen to the object.

When you go about your physics business, you may insert into your equations that a particle is spin up, then go ahead and compute, say, how it may interact with other particles. Well, interestingly, the mathematics of quantum mechanics does not care if you plug in, instead of just a spin up particle, something that looks like this: , a **superposition** of two states!

Is this important? as it turns out, yes. First of all, when an object is in a superposition an we go to test if it is spin up or down, for example, it always comes up as **either **up **or **down. However, we can also go and retune our measuring apparatus, and detect this particle state. We must do a rotation and measure a long some new states, the plus state and minus state, . Note that adding these two newly defined states produces . Now if the particle was placed in the plus state, we would always detect it in the plus state.

Final question, so if we detect a particle in, say, the plus state, can we be sure it was in the plus state? Answer: Nope. Just like if we detect a particle in the spin up state, we cannot be sure it was in the spin up state, it could have been in spin up and down, in which case there is a fifty percent probability of detecting spin up or down. However, after the measurement we can be sure that if the particle is left alone, it is in the state that we detected it to be in!

]]>Problem 50 of this exercise sheet reads something in the spirit of “A circular field has diameter 9khet[approx. 50m]. What is its area?” and the written answer is “Take away thou 1/9 of it, namely 1; the remainder is 8. Make thou the multiplication 8 times 8; becomes it 64; the amount of it, this is, in area 64 setat”.

This was an application of a formula, which is an approximation to the area of a circle . Much easier to understand when we use our familiar symbols!

We skip across the pond and through many years to the ancient
Greeks, c.350BC, and examine what Eudoxus had to say;
“*Magnitudes
are said to be of the same ratio, the first to the second and the
third to the fourth, when, if any equimultiples whatever be taken of
the first and the third, and any equimultiples whatever of the second
and fourth, the former equimultiples alike exceed, are alike equal
to, or are alike less than the latter equimultiples taken in
corresponding order”.*
Which we can simply translate as:

“The ratios and are equal when for any integers and :

if then also,

if then also,

if if then also.”

This
is a rather significant definition, Huxley writes of it “*It
is difficult to exaggerate the significance of the theory, for it
amounts to a rigorous definition of real number.”*

Eudoxus was followed by perhaps the gretest mathematician of ancient times, Archimedes, who went as far as calculating the area under part of a parabola, a feat usually taught at schools when integration is introduced.

So
when did our familiar mathematical notation begin to appear? We know
the origin of the misnamed Arabic numerals, they were first seen by
europeans in the book “Concerning the hindu art of reckoning” by
al-Khwarizmi. Islamic mathematics built on the ideas from central
Asia, and spread
them to the west. But it was in the 14^{th}
century that things began to look familiar, when symbols such as “+”
began to be used for addition by Nicole Oresme. In the 16^{th}
century, equalis, minus, square root and parenthesise appeared, and
things took off.

Leibniz is famous for his developments of differential and integral calculus, introducing the , elongated S, for his infinite summations of small quantities. Although Newton famously also developed calculus, he appears to have been less consistent with notation. Leibniz’s notation is still in use today, as for example in the equations . It became clear to Leibnitz, that a clear and precise notation was necessary, and, this has been a major focus of mathematics. Not least because mathematics is about communication as well as solving problems, a good notation helps one see the problem and solve it more easily!

The 16^{th} century saw the arrival of great mathematicians such as Eurler, who’s notation provides the foundation for most of mathematics for the next 400 years. Much of Eurlers work is perfectly readable today, if you are willing to discard a few conventions. But many things will be perfectly familiar, the use of , denoting imaginary units as and summations with .

Matrix and vector notation was also an important development, just think, Maxwell wrote his equations component by component! We will not detail all the developments from here onwards, but simply give an example of some very compact notation, that of tensors. Suppose you have a grid of numbers called , which is by . The entry in cell is . Suppose also two lists of numbers each, and , where like before is the mth entry of . One writes to mean where repeated indices have sums on them, meaning,

Imagine writing out every element of this every time!

Closing remarks:

From words to compact notation for complicated and complex ideas in mathematics, we see the usefulness of the most precise language humanity has developed. For further reading, see a list of mathematical symbols here [http://jeff560.tripod.com/mathsym.html]. Another online reference for history of mathematics is [http://www-groups.dcs.st-and.ac.uk/~history/Indexes/HistoryTopics.html].

]]>The experiment was surprisingly simple (and under funded, but more on that later). It consisted of heating up silver atoms, and firing them between two magnets towards a screen where they could be observed. As we understand now, a silver atom has a magnetic moment, and so for this discussion, it can be thought of as a tiny magnetic marble. Since the silver atoms came flying from a hot oven, they had all sorts of speeds and their magnetic poles (north and south) pointed in random directions. One can imagine, that when such a tiny magnet flies between the larger magnets of the experimental apparatus (in red and blue in the figure above), its path would be deflected. For example, if the silver atom’s north pole points up in the **z** direction (towards the north pole of the top magnet), It will be repelled downwards. If its pole points downwards, **-z** direction, it should be deflected downwards. And if its pole points somewhere in between, it should be deflected somewhere in between. And hence, since all the atoms point in random directions, one could have expected to see silver atoms on the screen forming a kind of smear, since clearly, all atoms would be deflected in different directions. However, this was not observed. To the disappointment of Stern, the atoms formed two distinct smears, one above and one below the center line. No atoms were observed to fly out undisturbed by the magnets! Moreover, it was as though the atoms had only two possible magnetic orientations; they could point upwards in the **z** direction, or point downwards in the **-z** direction. Instead of observing the magnetic moment taking a value form a smooth distribution, *it was quantized, *and the atoms split evenly between the two quantized values.

This gets weirder still. Let us draw the experiment schematically, as below, so we could complicate it without burdening my drawing skills.

The oven and experimental magnets are represented by boxes, and the letter in the magnet box (**z** in this case) tells which way the North pole of the magnets points. The two outgoing lines form the **z** magnet represent the upwards deflected atoms and downwards deflected atoms. We can now consider putting a second set of magnets, still pointing in the same direction, so that they capture the beam that moved upwards only. Unsurprisingly, it does not split again.

But we are not restricted to orienting the magnets in the **z** direction. Consider discarding one of the two beams after each split (represented by the red stop sign below). Now what if another set of magnets in the **x** direction was put following the **z** magnet, capturing all upwards moving atoms? The atoms split again; one beam is deflected in the **x** direction (out of the page), the other in the opposite **-x** direction. And again about half of the atoms take each path.

And let us split the beam once again, this time in the **z** direction as before. What do you think happens? Does it split, or not?

Let us recap. Silver atoms, behaving like tiny magnets, are fired from an oven towards sets of magnets designed to split them according to their magnetic moment:

- First the beam is split along the
**z**axis, remarkably, half the atoms are deflected upwards, the other half deflected downwards. The down moving atoms are blocked. - The up-moving beam has another set of magnets put in its path, splitting along the
**x**axis. Two beams emerge, half deflected into the page, the other half out of the page. Those moving out of the page are blocked. - The remaining beam (atoms that deflected upwards, then into the page) has another set of magnets put in its path, again pointing in the
**z**direction.

And the beam splits again, half deflected upwards, the other half deflected downwards. As though it was a freshly heated beam from the oven. Somehow, measuring the magnetic moment of the atoms in the **x **direction has destroyed information about the magnetic moment originally measured in the **z** direction.

Being under funded, Gerlach could only afford cheap cigars, the type that contain a lot of sulfur. When examining glass slides on which the silver atoms should have been deposited, the experimenters originally could not see the very thinly deposited silver atoms. But as Gerlach breathed his sulfur filled breath on the glass slides, the pattern of the silver atoms began to emerge in black! The silver reacted with the sulfur to form black silver sulfide. To read more about this, take a look at “Stern and Gerlach; how a bad cigar helped reorient atomic physics” by B. Friedrich and D. Herschbach. For a more detailed and careful treatment of the experiment, check out “Modern quatum mechanics” by J. J. Sakurai.

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