Lecture - 1 Introduction to Quantum Physics;Heisenberg''s uncertainty principle

[Music] is [Music] [Music] [Music] ok we take off now from last semester where we left off with quantum physics this semester and before we begin the course proper i'd like to get a little background from you on what you already know how we should run this course what the modalities are and so on as far as the evaluation is concerned it's the usual two quizzes and a final exam so that's out of the way philosophical and interpretational aspects of quantum mechanics are still open in the sense that all questions haven't been answered in spite of

over 100 years about 100 years of effort there are unresolved questions there are deep mysteries there are central mysteries and issues some of them may turn out in the future to be non-questions in the sense that it might turn out you're asking the wrong question and so on but at the operational level it's actually easier than classical physics the mathematics is simpler than classical mechanics for instance and it works it works over a huge wide range of phenomena as far as we know there are no violations of quantum mechanics what we don't know is its

interpretation to complete satisfaction we don't know if it's a complete theory we don't know if it will answer all questions and so on and so forth but certainly at the level of as a calculational device it's certainly extremely successful and as a way of understanding natural phenomena it seems to be the most basic of all subjects in the sense that everything at the ultimate level turns out to obey quantum mechanics okay so with that kind of very sketchy preamble let me go on to state a few properties in quantum mechanics make a few statements and

then depending on what the response is whether it's familiar or not and so on then i start writing down the formalism because otherwise i could start by simply saying the postulates of quantum mechanics are as follows and as we go along when things become inconvenient i add one more postulate and then we go on and so forth and then you just say this is the set and that's it these are the rules and so on i shouldn't want to do that i'd like to first find out since it is a conceptually very rich subject i'd

like to find out what it is that you know what you don't know and so on before we decide how to approach this subject okay so the first point about quantum mechanics is that all of you have heard that quantum mechanics applies in the microscopic domain it applies to electrons atoms things of atomic size this is only partly true it applies to everything it applies to human beings it applies to galaxies it applies to the universe presumably and it applies to electrons it applies to photons and so on it's just that the manifestations of quantum

mechanics become very dramatic when you look at extremely small objects become even more dramatic when you look at extremely small objects moving very fast relativistically so that's the reason why one feels that quantum mechanics is applicable only in the microscopic domain not true it's applied everywhere it has some manifestations which are very dramatic even in real life even in the microscopic domain for instance my favorite example is the fact that solids are rigid objects the fact that i do not go through the floor here the fact that matter does not penetrate itself to reject those

all solids don't interpenetrate this is due to quantum mechanics it's due to the pauli exclusion principle which follows from quantum mechanics okay so there are these very deep consequences the phenomena you are used to in daily life like magnetism or electrical conduction or the propagation of sound in in solids these are all quantum phenomena not just super fluids superconductors and so on which are given as usual examples of macroscopic quantum mechanics but even ordinary phenomena like electrical conduction conduction of sound in in solids magnetism the existence of permanent magnets the existence of diamagnets paramagnets all

these things depend on quantum mechanical principles finally they're not explainable classically and of course you know that in the early days of quantum mechanics there were several ways in which people found that classical physics failed one of them was black body radiation so the very fact that a collection of photons behaves quantum mechanically and that's an obvious uh instance where you have quantum mechanics for large systems in this sense okay now of course when you go to the atomic domain one of the consequences of quantum mechanics becomes very apparent and it's called the uncertainty principle

now i should like to find out from you what you know about the uncertainty principle what's the uncertainty principle i need answers uh the statement let me take it one at a time says the position and momentum cannot be simultaneously and accurately measured so we will make this very precise now position and momentum of what one particle two particles a system well let's assume that systems are made up of particles let's start with we have individual particles and let us look at a given particle an individual particle then the statement is the position and momentum

of this particle cannot be simultaneously and accurately measured now let's take this word by word simultaneously means at the same instant of time at the same mathematical instant of time and by accurately to what precision to arbitrary precision to infinite precision in principle cannot be measured now what does that mean that means a failure of experiment is it just that we don't have a good enough piece of equipment it's a property it's fundamental it's more than just a statement about measurement it's nothing to do with measuring apparatus although that's the way it's normally looked at

it has nothing to do with that the position and momentum cannot even be precisely defined for such objects at a given instant of time the property of momentum and the property of position these cannot exist in the sense of being measurable with infinite being definable with infinite precision at a given instant of time so it is not a failure of experiment its not a failure of lack of resolution or technology or anything its an intrinsic property of nature for these objects what sort of position and what sort of momentum is it linear momentum angular momentum

position i understand conjugate canonically conjugate we already looked classically at canonically conjugate variables it's the same thing canonically conjugate coordinate and momentum pairs they would obey an uncertainty principle what about the x component of the position of a particle and the y component of the momentum do they satisfy an uncertainty principle they don't they don't they're not canonically conjugate to each other x with p x y with p y and so on what about non-cartesian coordinates about non-cartesian coordinates what about for example the radial distance in spherical polar coordinates in some coordinate system this is

the distance from the origin of a particle and this is the radial component of the momentum momentum along the radius vector the position vector these two are canonically conjugate to each other and its these two which in quantum mechanics would satisfy an uncertainty principle if they're not canonically conjugate this is not true so already you begin to see that the framework of hamiltonian mechanics classical hamiltonian mechanics is being translated into quantum mechanics so hamiltonians play a very very special role and the whole assumption is that you already have a hamiltonian for a system and you

have the hamiltonian framework of mechanics classical mechanics and then you translate to quantum mechanics the question of how do you quantize how do you do quantum mechanics for systems which are not hamiltonian in the classical limit that's a much deeper and much harder question we won't address that in this course at all it's not at all clear how one would do this in particular this immediately means that if you have friction in a system if you have dissipation in a system classically you know that it cannot be an autonomous hamiltonian system then you run into

trouble and the corresponding quantum mechanics is not clear at all it does not mean friction does not exist it is just that the extension of the usual formalism of quantum mechanics to these dissipative cases is not straightforward and the problem of dissipation and quantum mechanics is still an open problem the question of what happens in open systems which exchange energy and momentum and so on with the surroundings this is still an open problem so in that sense the quantum mechanics we are going to do is really very elementary kind of quantum mechanics for systems which

are essentially going to be at absolute zero of temperature so we won't bring in temperature till fairly late in the course and which do not have any dissipation at all so it is for those kinds of quads and momenta that you have the uncertainty principle now what is the uncertainty principle actually say let us look at x and p x for a particle say and this is a canonically conjugate pair so classically i know that this is equal to one in the sense of poisson brackets this is one if i look at a single particle

and i say x is a generalized coordinate and p x is a corresponding conjugate momentum the poisson bracket of x with p x is one now there is a similar relation written down quantum mechanically but what does the uncertainty principle actually say it says the uncertainty principle in quantum mechanics says delta x times delta p x yeah greater than equal to h cross over two okay this is certainly true cross is what i use for h over 2 pi this is a planck's constant now this is an experimental fact this has been around now since

1925 or so and it's not been contested it's an experimental fact but i should like to first find out from you what do you mean by delta x what's this delta x what's the meaning of this delta x what do you mean by uncertainty pardon me is it the least count of some ruler or measuring device well in principle if i have a sufficiently good microscope if i have in principle a sufficiently accurate meter rule length measuring device with an arbitrarily good least count i can measure position to arbitrary accuracy so what do you mean

by delta x i make many measurements and take the standard deviation of them but i can improve the standard deviation by making many many many measurements inherent in what yeah this is where we come to the fact that it's not definable with arbitrary precision a particle of this of the kind we are talking about cannot have a short value of its position and its conjugate momentum in any state of this particle it cannot have it period so it has nothing to do with the precision of the ruler or your experimental measuring device this well beyond

all that this quantity is a standard deviation so it says you take your measuring device no matter how accurate it is continue to make these measurements and you would still find a spread you would still find a standard deviation no matter how many measurements you make and that standard deviation is delta x now obviously you need a calculational device for delta x i need to be able to calculate it and i will show you that in quantum mechanics there there's a rule which helps you to calculate this delta x similarly for p p x delta

p x so these are really standard deviations but the moment i say standard deviation i have to define for you a probability distribution because then i find the standard deviation with respect to that probability distribution and the whole point of quantum mechanics the initial starting point is to tell you what that probability distribution is and there is an equation which a quantity related to the probability distribution satisfies and that's called the it's called the schrodinger equation this quantity which you call the wave function which we will introduce shortly will help you calculate the probability distribution

and after you find the distribution you then compute the average and of course you know that formally this quantity here is in fact x squared minus x average square to the power half that is the definition of the standard deviation this is the variance and the square root of the variance is the standard deviation the only difference is whereas when you normally do classical experiments or take any statistical quantity and measure it repeatedly you have a certain histogram that you obtain in quantum mechanics these angular brackets refer to averages with respect to a quantum mechanical

probability distribution and that distribution will be specified through the schrodinger equation once you do that then this is certainly true now in practice of course if you do this for a physical particle you will find an answer which is much much bigger than h cross over two the reason is over and above the quantum mechanical uncertainty which is intrinsic in nature there would be other uncertainties such as the fact that this system is in contact with surroundings there are thermal agitations there'd be noise there'd be vibrations there would be other effects from outside perturbations from

outside etcetera there'd be a passing comet and that would change your apparatus a little bit etcetera so all these effects add up on top of quantum mechanical uncertainty but in its pristine form this inequality here is the quantum mechanical product of quantum uncertainties here calculated in a very precise way ok but notice it is an inequality which means that it is conceivable that you can arrange matters such that it's inequality that's the least value it could take it could never become less than a certain non-zero positive quantity and of course you know that this is

extremely small in the units we are normally used to and that's the reason you don't see this very often unless you go to microscopic objects because this fellow here is of order 10 to the minus 34 in the normal unit standard international units okay so since macroscopic objects like this piece of chalk or a person or a rock the energies involved are of the order of joules or tens of joules and that's 10 to the minus 34 joule second out there because of that you do not see quantum uncertainties unless the objects themselves are extremely

small moving extremely fast etcetera now this is just one form of this principle here and it's an inequality whereas quantum mechanics are fairly precise subject in the sense that it should give you equalities always so this inequality is a very weak form of quantum mechanics it definitely cannot be violated but in any given situation you should be able to actually calculate what's delta x delta p x and find out what the right hand side is so you don't need the inequality it's just that no matter what you do this cannot be violated this could not

be violated incidentally is it possible to have delta x arbitrarily small from this inequality yes the price you pay for it would be that delta p x would become proportionately large larger okay but you cannot violate this yeah yeah we're going to derive this we are going to derive this inequality from the schrodinger equation from the postulates of quantum mechanics one of which would be the schrodinger equation we would derive this this inequality here it's a fairly straightforward thing incidentally this is not the most general form of the inequality you should immediately ask what if

i have two arbitrary canonically conjugate dynamical variables is there a poisson bracket is there is there an uncertainty principle between them the answer is yes you could in fact ask if i have two physical quantities doesn't have to be position and momentum two physical quantities associated with any object can i write the product of uncertainties can i write something like this and the answer is yes you could it's going to be generalized yeah there was a question we can derive this by saying that by what i meant by that was no matter what experiment you

do the inequality is never violated so in that sense it's an experimental fact however we know why that is so because from the postulates of quantum mechanics this follows as a consequence and its verified ok so it essentially says we verified quantum mechanics or the predictions of quantum mechanics okay you can never verify a theory you can only verify its predictions obviously so to the in that sense this is well established incidentally this is not the most general way of looking at it it will turn out it will turn out and i am going to

anticipate myself and write down the general form here it will turn out that possum brackets for physical quantities for canonical variables are replaced in quantum mechanics canonical variables are replaced in quantum mechanics by operators by some things called operators we will discuss this in great detail and then once you have operators on the right hand side you need to know what you mean by expectation values or average values of these operators we will go through that in detail and then it will turn out that if you had any physical quantity a and any other physical

quantity b where the a's and b's are functions of the canonical variables the q's and p's which we take over from classical mechanics then quantum mechanics will tell you that this is greater than equal to on the right hand side a certain quantity which is associated with a and b it will turn out to be this is greater than equal to one half the modulus of the expectation value of the poisson bracket of quantum bracket poisson bracket of a with b which turns out to be the commutator and i am going to define what this

means so that is the general uncertainty principle and this is something we are going to show explicitly and when you apply it all the way back to a certain position coordinate a cartesian component of position and its canonically conjugate momentum this right hand side will reduce to h cross over to a number in any case it is clear that you have the modulus of some number some quantity here this is some complex number in general and you have the modulus of it and the modulus cannot be a negative number it could be 0 this could

be 0 sometimes now you could ask where does planck's constant come in it will generally appear here in the calculation of this object planck's constant would appear very often but occasionally there are cases where it would become zero for example if you had x with p sub y this is zero here and then in quantum mechanics the corresponding term would mean something like delta x delta p y is in fact 0 could be 0 greater than equal to 0 greater than equal to zero and you could have a situation where it's actually zero this is

possible yeah do we have enough technology such that delta x into delta px is exactly equal to h cross but it's a very good question do we have enough technology he says are we sufficiently do we have sufficiently precise experiments where you could actually take a given particle take its x component of the position and the x component of its momentum measure delta x delta p put it in some state and measure this and then discover that delta x delta p is actually equal to h cross over 2. the answer is yes and that's called

a minimum uncertainty state because that's the least value the uncertainty could have in fact we can go better not in terms of x and p but in terms of quantities which satisfy a relation similar to that if i schematically plot delta x here with delta p here on this side these are standard deviations so they're not negative numbers they're both non-negative numbers if you plot this the minimum uncertainty occurs when delta x delta p is equal to h cross over 2 and what sort of curve is that it's a rectangular hyperbola because you have x

y equal to constant that's a rectangular hyperbola so in fact you have a situation like this on this curve you have minimum uncertainty okay and then in suitable units x and p do not have if x is a length then p is lt in mlt inverse so they don't have same physical dimensions but if you reduce them to both sufficiently dimensional dimensionless variables by dividing through a momentum scale and dividing by a length scale in the problem then it will turn out that the symmetric point here where delta x is equal to delta p this

here in some sense is the best you can do you redu optimal thing you can do you reduce the error in x to the least possible the uncertainty in p to the least possible and that's the least you can do such states are realizable and we will talk about it in some detail in fact now technology is such that not in the context of positions of particles but in the context of quantum optics states of the radiation field where these two quantities are a pair of variables which are analogous to x and p they are

the counterparts you can actually move along this curve here so you could get to a stage where you're here which implies that delta p is below the uncertainty limit but delta x is greater has increased these are possible they are called squeezed states so it means that you squeezed the uncertainty in one of them you practically go into zero but the price you pay is the uncertainty in the other one increases enormously theoretically of course you could even go down to zero in this case but then the other variable would go to infinity out there

so this is possible and of course most of the time the states of the system that you deal with are sitting out here where delta x delta p is greater than h cross over two so this is where the product is exactly equal to this but you could go to this region up there okay in general that is what happens unless you arrange matters extremely carefully so is this clear the uncertainty principle in quantum mechanics is nothing to do has nothing to do with the precision we have available there's nothing to do with that even

with the best of precision the uncertainty principle still operates and it says that the uncertainty in delta in x and the uncertainty in any variable and its conjugate variable the product of uncertainties has a least possible value and that least possible value is given by this kind of thing the equality here there are cases when it could be zero also but if the pair of variables is canonically conjugate classically then this possum bracket is not zero it is equal to one and correspondingly you it turns out that you have a relation of this kind out

here similarly delta r delta p sub r is greater than equal to h cross over two you guaranteed that and so we measure yeah momentum to very high precision yeah that's that delta x goes to the order of meters yeah then but we know how to measure yeah yeah yeah yeah yeah no no no no no that's the whole point simultaneously that's the crucial thing you cannot measure one at one time another at another time and then say ah i have accuracy simultaneously in the same at the same instant of time that's the operative point

okay that's not possible and has nothing to do with the equipment it is simply there as a fact of nature and the problem is you have to put on your quantum thinking cap and not think classically the moment you have a relation like this it implies that you cannot specify you cannot specify x and p x to arbitrary accuracy which means that for a particle which obeys quantum mechanics in the phase plane if this is x and p sub x i cannot say that the position of the that the state of the particle is there

i cannot put a point down which flies in the face of all that we did in the previous course but i talked about phase trajectories and a phase trajectory is a phase space point moving along and i said the point specifies the state of the system all the x's no all the coordinates and all the moment are specified but you can't do that now because the moment i put a point like this it implies that i know both x and p x to arbitrary precision so states of a system cannot be any longer represented by

points in phase space we have to replace it with something else which obeys which respects the uncertainty principle therefore there's an intrinsic fuzziness you can't do better than something like this that's the best you can do because you cannot say there's a point in face space the moment you can't you can't say there's a point in phase space as time evolves you cannot talk about a phase trajectory either because the earlier classical picture was that this point moved around as a function of time and gave you a trajectory at every instant of time you identified

a phase space point with the state of the system or the particle you can't do that now you have a little fuzzy ball and this fuzzy ball is moving around distorting its shape and you do not know what it is going to do so this entire classical description of particles with specific trajectories goes out of the window you simply cannot talk about it which is why when you study the hydrogen atom for instance you cannot talk about an orbit for the electron because the moment you say something is orbiting you immediately have told me its

position and its momentum because you write down an orbit here is the centre of attraction and here is an orbit then at this instant of time you said this is the position with respect to the center of attraction this is the precise position and the tangent here will give you the direction of the momentum you multiply the velocity by the mass or whatever and you get the momentum it is not possible so the idea of orbits is gone completely it may be a very convenient picturization even in quantum mechanics under certain circumstances but the fact

of the matter is electrons do not have orbits yeah how do we measure delta x and delta p simultaneously we will come to that we will talk about how we can possibly think of measuring it experimentally this property arise because particles have dimensions no its not because particles have dimensions at all no thats a good question he says you cannot specify a given point maybe because it got a finite extent no even if it were a point particle even if it were a mathematical ideal point mass even then this is true not so it has

nothing to do with the finite extent of the particle or anything like that it is an intrinsic property that's the way it is so we can't any longer speak about orbits okay now in quantum chemistry one is used to talking about molecular orbital theory and s orbitals and p orbitals and pi orbitals and so on they are convenient picturizations that's not the way the electron actually behaves in an atom at all a very convenient picturizations no more than that for instance let me give you another instance you are told that when you do the bohr

model of the hydrogen atom you are told that the ground state of the hydrogen atom the orbital distance is the bohr radius one times the bohr radius and it is moving in a circular orbit now that can't be true because if that were true you have this particle here this a center of attraction o and then you have something moving at the bore radius say this is a 0 the bohr radius if this were really true the orbital angular momentum of this particle about the center of attraction is not zero it cannot be zero right

it is not going through the particle at all so you calculate r cross b its supposed to be uniform circular motion and of course the orbital angular momentum is non-zero but you are the same breath you are told that in the ground state of the hydrogen atom l is equal to zero right you have the hydrogen atom in which you describe the state of the electron by a principal quantum number l an orbital or angular momentum quantum number m a l and a magnetic quantum number m and then in the ground state the lowest energy

state this n is equal to 1 this l is equal to 0 and this m is equal to 0 and of course you are also told that if you look at the component of the orbital angle of momentum perpendicular to the plane of the orbit it's l times h cross but if l is zero that's zero but that can't be true if it's orbiting at a finite distance so that alone should tell you that this picture is wrong immediately these facts are true this is certainly true but they can't be supported by a picture of

this kind so that's not true on the other hand it is true that if i calculate the average distance the mean distance most probable distance and so on in this case is in fact a zero of the electron from this quantity but that is a statistical and a quantum mechanical average here so the real fact of the matter is the electron does not have a position and momentum simultaneously and the conjugate momentum simultaneously it is not a point particle in that classical sense of the word at all you have to get out of this classical

way of thinking its not tenable just does not work at all it is some kind of crazy extended object now it is described by something you need something to describe it by what is called the state of the system a state vector described by an object called the state vector which is a generalization of the wave function and we will discuss it in great detail and i will tell you how the state vector is defined and how to calculate its properties here but that is what the electron is and as far as i am concerned

that's what the electron is in reality i don't answer questions like is it a wave is it a particle is it a wave on some days particle on other days etcetera these are these are terms which are meaningless when you apply it in the context of quantum mechanical particles the failure is not on the part of the quantum mechanical particle the failure is not on the part of the experiment list the failure is on the part of the english language words like wave particle and so on have been coined by us to paraphrase a set

of properties and these properties are the properties of objects with which we have daily experience macroscopic objects and then you say a particle in the mental picture you have a particle is like a billiard ball a hard rigid object very compact sometimes idealized even to a point massive something which is localized carries energy carries momentum and so on on the other hand the word wave is mentally as far as we are concerned based on our experience with waves in nature which we see around us like water waves or sound waves these are diffuse objects they're

not localized they don't have rigid boundaries they're not hard they're kind of smooth things which are you know gently undulating everywhere delocalized in space in time etc they carry wavelength they carry frequency and properties of that kind and in a sense we have said we put all classical objects into these two bins one of them would consist of particles which are compact localized hard objects and the other carrying properties like energy momentum etcetera and the other is waves which carry properties like wavelength frequency etcetera they delocalized extended objects and we know that in daily life

these are mutually exclusive a wave is not a particle a particle is not a wave and we can look and see and measure properties and decide whether it falls in this basket or in that other basket what we shouldn't do is to extrapolate this categorization of objects into waves and particles to the microscopic domain because in that domain it is conceivable that there exist objects which have some of these properties and some of those properties it is also conceivable that we have objects which have these properties all those properties depending on how you probe them

how you measure these properties and thats what happens for electrons its what happens for microscopic particles ok so the failure is not on the part of the electron or on the part of our ability to probe nature it's on the it's a semantic failure it's a failure of the language we shouldn't use terms which we have mutually exclusive sets of properties in the microscopic world to describe the microscopic world okay if you understand that then there is no wave particle duality mystery its just that this description fails completely and you need a better description it

so happens not unfortunately but fortunately that that description is not in terms of ordinary language that description is intrinsically mathematical and that's the language you need to describe these objects there and it's completely unambiguous once you use that description it's just that you cannot put it back in the normal words to 100 efficiency because these words have been coined by us to understand the world of macroscopic phenomena okay is this clear this is the reason why we need the mathematics and therefore the question of something being physical if i can say it in plain words

and something being unphysical or purely mathematical if i say it in equations shouldn't be there i mean this confusion shouldn't be there okay it's just that the words are insufficient it took a long time to recognize this but once you recognize it then it's like learning another language and that's the appropriate language for it it's not that we haven't coined enough words for this it's not that either because you'll just be coining arbitrary words to translate mathematical terms into this it's not that at all it's not that the language doesn't have enough words nothing much

is gained if i say something which is neither a wave nor a particle is a vehicle that's been used in the literature people have said this nothing is gained by it i'd rather say there is a state vector just as good in fact it's much more precise that way much much more precise but it's not just at the level of semantics alone it goes much much deeper than that the fundamental thing about quantum mechanics the real deep point about quantum mechanics will come out in the fact that even ordinary words like and and or have

different meanings in classical and quantum physics and you have to get used to it if you think a little bit about it the word and in ordinary language is always with reference to probabilities and it's always with respect to multiplication of probabilities you want this to happen and that to happen and then the probabilities multiply and the word or if you think about it is always with respect to addition of probabilities you have two mutually exclusive possibilities you want this all that to happen the total probability is addition so i'd like to say that or

means plus and and means multiply that's the classical way of looking at it in quantum mechanics the same thing is true except it's not true for probabilities but for objects called probability amplitudes so it's even a change of classical probability quantum mechanics could be looked at in that point point of view also that it is a change in the rules of classical probability completely quantum probability if you like and that's the way it is this is exactly the way nature operates now of course the immediate hard question that arises is when does something stop being

quantum mechanical and start being classical i know that a single electron is obviously a quantum mechanical object 10 electrons still quantum mechanical a molecule is quantum mechanical on the other hand this piece of chalk looks classical you can describe it by classical physics so when does it stop is there some sharp boundary and the answer is no there is no sharp boundary this is not something where there is a precise boundary between classical and quantum physics it is just that class quantum corrections as you go towards the quantum domain from the classical domain the quantum

corrections if you look if you look at it that way to classical physics become more and more significant then it is a question of what accuracy you want there will come a stage when the classical description breaks down completely it's like there's there's no there's no smooth boundary at all it's a very diffuse boundary this is like some of these paintings you've seen where you start with red on the right hand side and blue on the left hand side and as you go there is no sharp boundary it's just that things become more and more

blue as you move to the left and eventually there is no red everything is blue it is in the same sense the classical quantum boundary is diffuse its not a sharp boundary so all the quantum phenomena if you start with something fully quantum mechanical will start becoming more and more classical as you move towards the classical region but i must say here right away that classical mechanics is an approximation to quantum mechanics not the other way about the more fundamental theory is quantum mechanics more fundamental formalism and classical mechanics is a limiting case an approximation

to it just as relativistic physics is believed by us to be the right formalism and non-relativistic physics of phenomena as an approximation when v over c becomes negligible and so on it's in exactly the same way because planck's constant is not zero but when the effects that you are looking at are in some sense much much bigger than planck's constant then you would say its in the classical region ok so yeah can we have quantum mechanics formalism without having classical physics this is again a very deep question can you have a quantum formalism without a

classical formalism at all the answer is it looks like no you can't and it's different from relativity because it's conceivable that you live in a universe where everything moves at speeds comparable to speed of light in vacuum and then you do not have a non relativistic regime at all its conceivable and be perfectly consistent but the very interpretation of quantum mechanics by us at least because we are microscopic objects looks like you need to have a classical limit otherwise the interpretation becomes meaningless but i don't know the deep answer to the question of is it

consistent to have a universe in which you have only quantum physics and no classical regime at all i don't know i dont know but for us its a mood point the reason is we are large objects we are in the classical regime in many respects and therefore for us to be able to probe that universe of quantum physics and understand it it looks like its necessary essential for us to have a classical limit the very meaning of words like measurement and recording and so on so forth observation these are intrinsically classical notions so it looks

like we need it but whether it's consistent to have a fully quantum universe is not known it's not clear to me immediately on the face of it then why don't we have quantum gravity and why is this theory so distinct we don't know we don't know we don't know there are deep reasons why we have to go very very far down into length scales and time scales and so on before you see the effects of quantum gravity and the reason for it is simply that it's a very simple reason since we are spending time talking

about deep problems today let's continue with that you see the fundamental constants of nature that we know about are not constants in mass length and time that is not the way the universe is built it so happens it turns out that what we have available to us in the universe is a quantum of action h a c and newton's gravitational constant g these are the fundamental constants of nature everything these are the most basic constants of nature actually there are just two of them we could really say h cross and c we could set equal

to one and say there are fundamental these are really basic base units choice of units and then there is only g or something but quite apart from that you have these three constants of fundamental constants of nature and you can construct from them quantities of dimensions mass length and time by taking suitable combinations of hc and g so i request you to do this find combinations of h to the power alpha g to the power beta c to the power gamma find alpha beta gamma such that you get a length or a mass or a

time and these are the planck mass planck length planck time etc now clearly those would be the natural length scales and time scales when gravity as well as quantum mechanics would play a role because gravity is represented by this and quantum mechanics by that and it turns out that the length scale that you talk about l planck this is of the order of 10 to the minus 35 meters it turns out that's the way these numbers are in our universe and the planck time is of the order of 10 to the minus 42 seconds give

or take an order of magnitude so this is the reason why we don't know what's happening below that because concepts like length and time and so on may not even be definable continuously up to 0 below these lengths scales and time scales so the problem of quantizing gravity which is what i presume you are referring to is a difficult one we do not know we do not know what goes on there and part of the difficulty is that even concepts like length and mass and time length and time in particular space time whether it is

continuous all the way down to zero or not we do not know because the laws of physics as we know today do not presumably will not operate below this we don't know what replaces it so this is the difficulty there are other technical difficulties in quantizing gravity and so on even conceptually but the fundamental difficulty is simply that our notion of space time breaks down our words are not sufficient yeah semantics is not enough yeah are we sure that mathematics is sufficient well to the extent that if you make a cert of postulates based on

some observations which could not be explained by classical physics this is how quantum mechanics was initially put in and then you ask what are its predictions what are the results of its calculations what is predicted and then you observe once again do those experiments and test it out and so on you are testing this theory and this formalism and the mathematics associated with it it looks like there are no there are no violations as far as we know okay but of course you can never prove a physical theory two hundred percent accuracy does not exist

there will be a domain where it might perhaps break down at some stage we do not know at the moment now as to whether the mathematics is sufficient as far as we know yes this is sufficient but then we do not know what happens when you go to these regimes when you go to quantum field theory when you include relativity general relativity plus quantum mechanics and then try to do make calculations even the formalism of quantum field theory breaks down it yields nonsense you have to replace it with something else some other formalism there are

other candidates there are many candidates one of which is called string theory another one called loop gravity quantum gravity and so on but these are at a very deep conceptual level we do not know it is very likely ok now let me jump fast forward to 2007 we are now in the 21st century it's very likely it's very 2008 okay you're ahead of me it's very likely that this problem this is at the cutting edge it's really of reductionism it's very likely by knowledgeable people have guessed ah that is extremely likely that our present understanding

of space time itself as well as our understanding of quantum mechanics itself will undergo drastic revisions before we solve this problem no reason why you can ask how do you know and the answer is sort of not well established the answer is that it looks like we have not succeeded with existing technology existing tools now it is purely mathematical in some sense in constructing a completely consistent and satisfactory theory in spite of 20 or 30 years of effort and the indications are the difficulties the technical difficulties that arise in this program suggest that our very

ideas of space time what we mean by dimension what we mean by space length time etcetera as well as what we mean by normal quantum mechanics in quantum mechanics these i think will undergo drastic revision so it's still a house under construction in that sense it looks like the existing framework is not enough it's not powerful enough to do this you could ask now what about experiment that we lag far behind the shortest distances we can probe are of the order of 10 to the minus 17 or 18. this is all we have been able

to probe so far in the highest energy accelerators and there are 16 orders of magnitude or 17 orders of magnitude between these two we don't know what can happen in there similarly the length scales here the shortest length scales we've been able to probe are strong interaction length skills time scales which are of the order of 10 to the minus 23 seconds there are 20 orders of magnitude in between we don't know if things will change so we keep an open mind completely and we do not know that is the whole point about physics it

is not mathematics so it is not a set of axioms from which you derive results the rules of the game we prepared to change and subsume in an even broader framework and then say what we did so far is a special case limiting case etcetera and try to increase the true range of applicability of whatever theory you have so in that sense there is no dogma completely open to it but people have been thinking very hard for many many years now physics is 400 years old in its modern form and in the last 100 years

of course it really accelerated um so it's still by by you know any historical time scales or something this is a very small period of time and it's possible that in another 50 years you people will live to see this the whole complexion may change it may be very different but one thing is for sure that just as newtonian mechanics is now known to be an appropriate a very good excellent approximation under certain circumstances so is it true that today's quantum mechanics and our calculational tools will be applicable there will be applicable calculational tools in

a certain regime or physical parameters that is for sure the understanding of why this is so may get deeper should get deeper should get broader and so on the understanding of why this is an approximation why it works in this range and not in some other range that will become sharper but i think some fundamental issues themselves will become clearer namely what is the meaning of space time what determines the dimensions of space-time what's this business about gravity what are the fundamental forces of nature and so on after all its a sobering thought that we

do not know 95 percent of the constituents of our own universe we can only explain five percent of it in terms of matter has been that we know all the theories we have and all the phenomena we understand fully ninety five percent of it is a mystery we know it's there but we don't know what it is okay so that seems to be a kind of it just set us it's put put things in perspective right after the last after so many millennia of astronomy and real explosion in the last 50 years or so the

conclusion is that we really don't know 95 percent of our own universe and we don't know if it's one of an infinite number of possible universes so it's just that you know like wheeler said ah our knowledge is like an island and the boundary of this island is of course the frontier of ignorance so island is getting bigger all the time you know more but then its perimeter is also increasing so the level of ignorance is also increasing at the same time we don't know we don't know maybe there it will work and so on

but we have to keep trying i should end by what hilbert said in 1900 when he gave these famous 23 problems in mathematics and solve problems hilbert's 23 and solve problems in the international congress of mathematicians um in 1900 he listed what was then the major unsolved problems of which today only the riemann hypothesis remains unproved everything else has been taken care of one way or another some of them are not problems some of them were one of them was the axiomatization of physics could all of physics be written in terms of set of axioms

and everything follows and the answer to that is no that's not true physics is not aximitizable as far as we know in that sense but he made this statement and then it looked like there was you know a lot of problems ahead and so on but he made a very optimistic statement he said we must know and we will know said this in german so i think when one should have this note of optimism it is essential for us to be curious is essential for us to want to know many many reasons why it's not

part of the reason is simply that all physical laws as far as we know apply in certain regimes of physical parameters when you go out of that its a different set of rules the framework of quantum mechanics looks like its very general but its interpretation and the way it's set up it's not satisfying so it already looks like there are big black boxes in it which were pushed under the rug but they exist nevertheless we've not understood quantum non-locality have not understood quote unquote understood except in some mathematical form framework to our satisfaction we really

haven't come to grips with the meaning of measurement what happens to a system when you measure it so there are large gaps conceptual gaps here so its clear its an incomplete theory in that sense at least that understanding is not as deep as we would like it to be you could ask maybe this is the best you can do how do you know we don't know that too we don't know but we have to keep pushing ok so today has been a sort of general talk we'll start off with some specific issues and so on

next time [Music] you