Photons and the loss of determinism

determinism and it all begins with photons Einstein was reluctantly came up with the idea that light was made of quanta Quant of light called photons now when you think of photons uh we think of a particle so it everybody knew that light was a wave maxw equation have been so successful nevertheless photoelectric effect Plank's work all were leading to the idea that in some ways uh photons were also particles for photons or light so when you think of a particle however you know there's a kind of an important difference between a particle in the sense

of Newton which is an object with zero size that carries energy that has a precise position and velocity at any time and the quantum mechanical idea of particle which is just some indivisible amount of energy or momentum that it propagates um so light was made of photons called pockets of energy and a photon is a particle a quantum mechanical particle not in the sense that maybe has position and velocity determined or it's a point particle but more like a packet that is indivisible you can't decompose it in further packets so um Einstein realized that for

a photon the energy was given by H new where new is the frequency of the light that this photon is helping build up so if you have a beam of light you should think as billions of photons and according to the frequency of that light that is related to the wavelength by the equation frequency times wavelength is velocity of light you typically know for light the wavelength and you know the frequency and then you know the energy of each of the photons the photons have very very little energy they have very very little energy but

your eyes are very good Detectors of photons if you're in a totally dark room uh you your eye probably can detect as little as five photons if they hit your routina so it's a pretty good detector of photons anyway uh the thing that I want to explain here is what happens if the a beam of light hits a polarizer so what is a polarizer is a sheet of plastic or some material it has a preferential Direction let me align that preferential Direction with the X AIS and that's a polarizer and if I send light that

is linearly polarized along the x- axis it all goes through if I said light linearly polarized along the Y AIS nothing goes through it all gets absorbed that's what a polarizer does for a living in fact if you send light in this direction the light that comes out is identical to the light that came in the frequency doesn't change the wavelength doesn't change it's the same light the same energy so far so good now let's imagine that we send in light linearly polarized at some angle Alpha so we send an El electric field e Alpha

which is e0 cosine alpha x hat plus e0 sin alpha y hat well U you've studed electromagnetism and you know that this thing basically will come around and say okay you can go through because you're a line in the right direction but you are orthogonal to my preferential Direction orthogonal I absorb so this disappears so after the polarizer polarizer e is just e0 cosine alpha x hat that's all that is left after the polarizer well here is uh something interesting you know that the energy on a electromagnetic field is proportional to the magnitude of the

electric field Square that's what it is so the magnitude of this electric field if you can notice it's the square root of the sum of the squares will give you e not as the magnitude of this full electric field but this electric field has magnitude e not cosine Alpha so the fraction of power fraction of energy through is coine Alpha squared is propor the energy of is proportional to the square so the square of this is e^ s cosine squ Alpha and for this one the magnitude of it is e not so you divide by

E not and cosine Alpha is the right thing this is the fraction of the energy if Alpha is equal to zero you get cosine of 0 1 you get all the energy one if Alpha is equal to Pi / 2 the light is polarized along the y direction nothing goes through indeed cosine of Pi / 2 is zero and nothing goes through so the fraction of energy that goes through is cosine squ Alpha but now think what this means for photons what it means for photons is something extraordinarily strange and so strange that it's almost

unbelievable that we get so easily in trouble here is this light beam over here and it's made up of photons all identical photons maybe billions of photons but all identical and now think of sending this light beam over there a billion identical photons you send them one by one into this thing and see what happens you know what has to happen because classical behavior is about right this fraction of the photons must go through and one minus that must not go through you see it cannot be that comes a photon and half of it goes

through because there's no such thing as half of it if there would be half of it it would have half the energy and therefore different color and we know that after a polarizer the color doesn't change so here is the situation you're sending a billion photons and say one3 has to get through but now the photons are identical how can that happen in classical physics if you send ident identical photons whatever happens to a photon will happen to all but the photon either gets absorbed or goes through and if it gets absorbed then all should

get absorbed and if it goes through all should go through because they're all identical and now you have finally a situation in which an identical set of experiments with identically prepared objects sometimes give you different results it's it's a debacle it's a total disaster what uh seems to have happened here you suddenly have identical photons and sometimes they go through and sometimes they don't go through and therefore you've lost predictability it's so simple to show that if photons exist you lose predictability and that's what drove Einstein crazy he knew when he introduced photons that he

was getting in trouble he was going to get in trouble with classical physics so possible ways out people speculate about people said well yes the photons are identical but the polarizer has substructure if it hits in this interatomic part it goes through and in that interatomic body it doesn't go through people did experiment many times it's not true the polarizer is like that and then came a more outrageous proposition by Einstein and others that there are hidden variables you thought the photons are identical but you think they are identical but a photon has a hidden

variable a property you cannot you don't know about if you knew that property about the photon you would be able to tell if it goes through or it doesn't go through but you don't know it so that's why you're stuck with probabilities it's because the quantum the is not complete there are hidden variables and once you put the hidden variables you'll discover the photon has more something inside it and they are not the same even though they look the same and that's a hidden variable Theory and it sounds so philosophical that you would think well

if you don't know about them but they are there these properties how could you ever know they are there and the great uh progress of John Bell with the Bell inequalities is that he demonstrated that that would not fix the problem quantum mechanics cannot be made deterministic with hidden variables was an unbelievable result the result of John Bell so um that's something we will advance towards in this course but not quite get there 805 discusses this subject in detail so at the end of the day we've lost determinism uh we can only predict probabilities so

photons either gets through or not and can only predict Le only predict probabilities now um we write in classical physics a beam like that but how do we write the wave function of a photon well this is quite interesting we think of a states of a particle as wave fun functions and I will call them sometimes States I will call them sometimes wave functions and I sometimes will call them vectors why vectors because the main thing you do with vectors is adding them or multiplying them by numbers to scale them and that's exactly what you

can do with a linear equation so that's why people think of states or wave functions as vectors so um and IR invented a notation in which to describe a photon polarized in the X Direction you would simply write something like this Photon column X and this object it you think of it as some vector or wave function and it represents a photon in the X Direction and we're not saying yet what kind of vector this is but it's some sort of vector is not just a symbol represents a vector and that's a possible State this

is a photon polarized along X and you can also have if you wish a photon polarized along Y and linearity means that if those photons can exist the superposition can exist so there can exist a state called cos Alpha Photon X Plus sin Alpha Photon Y in which I've superposed one state with another created a sum and this I call the photon State polarized in the alpha Direction so this is how in quantum mechanics you think of this photon um we will elaborate that compare with this equation it's kind of interesting uh what you lose

here is this ease there's no ease there because it's a one Photon when you have a big electric field I don't know how many photons there are I would have to calculate the energy of this beam and find the frequency that I didn't specify and see how many photos but each Photon in this beam quantum mechan mechanically can be represented as this superposition and we'll talk more about this superposition now because our next subject is superpositions and how unusual they are well the hidden variable explanation uh failed because Bell was very clever and he noted

that you could design an experiment in which the hidden variables would imply that some measurements would satisfy an inequality if the existed hidden variables and the world was after all classical the results of experiments would satisfy a bell inequality and then a few years later the technology was good enough that people could test the Bell inequality with an experiment and they figure out it didn't hold so the hidden variables lead to Bell inequalities that are experimentally shown not to hold and we will touch a little bit on it when we get to entangle but after

the polarizer the polarizer the photon is in the state Photon X it's always polarized along the X Direction so it's kind of similar that this doesn't go through this goes through but uh at the end of the day as we will explain very soon the cosine Alpha is not relevant here when it goes through the whole Photon goes through so there's no need for a coine alpha so that's what goes out of the polarizer