Define Quantum mechanics and Einstein quantum mechanics

applications of quantum mechanics in real life applied quantum theory to atoms define quantum physics and a brief history of quantum mechanics
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Mathematical Tripos Part 1B Dr. Eugene A. Lim Michaelmas Term 2012 eal40damtp.cam.ac.uk QUANTUM MECHANICS Lecturer: Dr. Eugene A. Lim Michaelmas 2012 Oce : DAMTP B2.09 http://www.damtp.cam.ac.uk/user/eal40/teach/QM2012/QMroot.html  Introduction : Quantum Mechanics with Qubits The Postulates of Quantum Mechanics for Qubits. Dynamics of Qubits.  The Failure of Classical Mechanics and Wave-Particle Duality The Classical Wave Equation. Double Slit Experiment. Photoelectric E ect. Wave-Particle Duality.  The Mathematics of Quantum Mechanics The Postulates of Quantum Mechanics. Operators. Eigenfunctions and Eigenvalues. Observables and Hermitian Operators.  Schr odinger's Equation Dynamics of Non-relativistic Particles. Principle of Superposition. Probability Current. Free Particles. Degeneracies. Stationary States.  Solving Schr odinger's Equation in One Dimension Quantization of bound states. Scattering and Tunneling. The Gaussian Wavepacket. Parity Oper- ator.  The Simple Harmonic Oscillator Quantization in Position Basis. Quantization in Energy Basis.  Commutators, Measurement and The Uncertainty Principle Expectation Values. Commutators and Measurements. Simultaneous Eigenfunctions. The Uncer- tainty Principle.  The Hydrogen Atom The Bohr Model. Schr odinger's Equation in Spherical Coordinates. S-waves of Hydrogen Atom. Angular Momentum. The Full Spectrum of Hydrogen Atom.  Epilogue : Love and Quantum Mechanics Entanglement. Teleportation. 1Recommended Books  S. Gasiorowicz, Quantum Physics, Wiley 2003.  R. Shankar,Principles of Quantum Mechanics, Kluwer 1994.  R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, Volume 3, Addison-Wesley 1970.  P. V. Landsho , A. J. F. Metherell and W. G. Rees, Essential Quantum Physics, Cambridge University Press 1997.  P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford University Press 1967, reprinted 2003.  J. J. Sakurai, J. Napolitano, Modern Quantum Mechanics, Addison-Wesley 2011 Acknowledgements Firstly, thanks to Prof. David Tong and Prof. Anne Davis for advice and words of wisdom when I taught this class last year. Also thanks to the students of Part IB QM 2011 Michaelmas who had to sit through my lectures and gave a lot of feedback. Finally and most importantly, many thanks to Prof. Nick Dorey who provided much advice and whose lecture notes of the same class these notes are based on. No cats were harmed during the production of these notes. 21 Introduction : Quantum Mechanics with Qubits This is Serious Thread. Serious Cat When you studied Classical Mechanics in the Dynamics and Relativity lectures last year, you were told that a particle is an object of insigni cant size. Then you spent eight weeks studying the dynamics of this particle and took an exam. One of the things you learned is to describe the state of the particle in terms of its positionx and momentump (and given its massm, one can deduce its velocityx _ =p=m), both which take de nite real values at any given moment in time. Let's think about a non-relativistic particle of mass m. In many of the problems and calculations, you often assumed that once you know information of these two variables (x(t );p(t )) of this particle at 0 0 some initial time t , using Newton's Laws of Motion, 0 dp =F (1) dt you can calculate and predict to any arbitrary accuracy the position and momentum (x(t);p(t)) of this particle at some later time t t . In addition, it is implicit that one can at any time measure with 0 arbitary accuracy the values of variables as we please. In words we say that we \know the state of the particle" at any time t. The key phrases we have used in the above description is \Classical Mechanics" and \arbitrary accuracy". It turns out that, in Quantum Mechanics, one of the ideas that we have to abandon is the notion that we can \predict to any arbitrary accuracy" the position and momentum of any particle. In fact, it is worse than this: another notion we have to abandon is the idea of that we can measure with arbitrary accuracy both variables at the same time t. These two notions are not only those we will abandon of course, but giving these up already begs a bunch of questions: How do we describe the state of a particle, and how do we describe its dynamics? Hence, in the study of how quantum particles move, or more generally how dynamical systems behave: rocks, electrons, Higgs Bosons, cats, you name it, we have to start with an entire new notion of how dynamical states of systems are described mathematically. Indeed, once we give up the notion of absolute knowledge about the state, we can start to introduce even more abstract states which has no classical analog such as the \spin" of an electron, and even more abstractly, how information is encoded in quantum mechanical systems. In this rst section of the lectures, we will use the simplest possible dynamical system a system with only two possible states as an introduction into the weird world of Quantum Mechanics. The goal of this introduction is to give you a broad overview of the structure of Quantum Mechanics, and to introduce several new concepts. Don't worry if you don't follow some of the details or you nd that there are a lot of unexplained holes, we will go over the same ground and more in the coming lectures. 1.1 Classical Bit vs Quantum Qubit As children of the computer revolution, you must be familiar with the idea of a bit of information. The bit is a system that can only has two possible states: 1/0 or up/down or on/o or dead cat/live cat etc. Let's use up/down for now. Such binary systems are also called (obviously) two-state systems. We can endow this bit with some set of physical rules which when acted upon the system, may change it from one state to another. For example, in Newton's Law of motion, the dynamics of (x;p) are described by Eq. (1). In words it means \When we act on the particle with a force described by F (x) for an in nitisimal time dt, the value of p changes by Fdt". What kind of rules can we write down for a bit? 3INPUT OUTPUT down up up down Table 1: A NOT gate The set of rules for a bit can be something simple like a NOT gate. This rule simply ips an up to a down, and a down to an up. A NOT gate rule is shown in Table 1. Another rule we can write down is the \do nothing" gate, which just returns up if acted on up, and down if acted on down. Mathematically, we can de ne the following column matrices to represent the up/down states 1 0  = ;  = ; (2) up down 0 1 so a NOT gate can be described by the 2 2 matrix 0 1 P = ; (3) 1 0 while a \do nothing" gate is obviously the identity 1 0 I = : (4) 0 1 \Acting" then means usual matrix multiplication of the column vector from the left by the gate matrix result = gate matrix state: (5) You can check that acting from the left with P and I on an up/down state gets you the right results, e.g. acting on up state with NOT gate yields a down state  =P : (6) down up A bit is a classical quantity, so we can measure with arbitrary accuracy whether it is up or down. For example, a classical cat is either dead or alive (just check its pulse). We can also predict with arbitrary accuracy what would happen when we act on the bit with the rules: if we start with a up, acting on it with a NOT gate we predict that it will become a down (and then we can measure it to con rm that our prediction is true). What about a quantum two-state system? Such a quantum state is called a qubit, for \quantum bit" obviously. What are the properties of a qubit and what kind of real physical system is modeled by one? You might have heard about the sad story of Schr odinger's Cat. The cat is put inside a closed box. There is a vial of poison gas in the box. A hammer will strike the vial if a certain amount of radioactivity is detected in the box, thus killing the cat. An observer outside the box has no way of nding out if this sad a air has occured without opening the box. Hence the cat is in the curious state of being both alive and dead at the same time according to the observer: the information about the deadness or aliveness of the cat is carried by a qubit. You probably have realized that I have shoved a ton of things under a carpet of words here, and words are not well de ned there are many equally good ways to implement those words but Nature chose the path of Quantum Mechanics. Let's now be a bit more precise, and introduce the Postulates of Quantum Mechanics for two-state systems. We will elaborate on each of these postulates for more general cases in future lectures. 4Figure 1: Schr odinger's Cat and its sad/happy fate. Stolen from Wikipedia. Postulate 1 (State): A qubit, , is described by the state vector =  +  ; where ; 2C: (7) up down and are called probability amplitudes for nding the up and down state, for reasons we will soon see. The important point here is that the coecients and are complex numbers this means that the 1 information encoded in the state has been enlarged when compared to the binary classical bit . Postulate 1 tells us that the state can be neither up nor down; it is some linear superposition beteween two possible states hence the cat can be both dead and alive. T  2 2 By convention, we normalize the state vector ( ) = 1, hencej j +j j = 1, where the superscript T denotes transpose and denotes complex conjugration. The combination of these two operations is y called Hermitian Conjugation, which we denote with a i.e. for any complex matrix A T  y (A ) A (8) This operation occurs so often in Quantum Mechanics that we will de ne the inner product (or \dot product") of two state vectors the following way. Given two state vectors  and , the inner product is then de ned as y   : (9) Postulate 2 (Born's Rule): The probability of measuring an up/down state is the absolute square of the inner product of the desired outcome with the state, i.e. 2 2 Probability of measuring up state =j  j =j j ; (10) up 2 2 Probability of measuring down state =j  j =j j : (11) down 2 2 Note that since the qubit has to be in some state, the probability must add up to unityj j +j j = 1 this is the reason why the state vectors are normalized to one. More generally, state vectors must be normalizable to be valid quantum mechanical states. A note on jargon: note that probability amplitudes are complex, while probabilities are real. 1 Technically, the space in which a two-state quantum mechanically system live in is a S sphere called the Bloch 2 Sphere where the up/down state reside at the North and South poles of this sphere. 5Postulate 3 (Measurement): Once a measurement has been made and up/down has been obtained, the state vector collapses into the measured state measure  : (12) up=down While Postulate 1 tells us that a qubit can be neither up nor down, Postulate 2 tells us the probability of measuring either of the two states. Postulate 3 then tells us that once the measurement has been made, follow up measurements will yield identical results (as long as we have not act on the state other than make a measurement). In particular, Postulate 3 implies that the very act of measurement a ects the system. This is often called the Collapse of the State Vector. So the story of the cat is now the following: the state of aliveness/deadness of the cat is carried by a qubit due to the quantum mechanical nature of radioactivity, and the probability of nding the cat to 2 2 be dead or alive when we open the box is given byj j orj j . Once the box is opened, the cat's state will collapse into one of the two states depending on which is measured. 1.2 Operators on the State and Observables Knowing how to describe a state is not enough of course; we want to know ultimately how states evolve with time. In the case of Classical Mechanics, the force F (x) acts on a particle of mass m for an in nitesimal timedt to change the statepp +Fdt. In Quantum Mechanics, we will soon discover that the equivalent law of motion for a quantum mechanical particle is given to us by Schr odinger's Equation which we will spend much of our time in these lectures studying. For now, however, let's abstract the notion of \acting on". Recall that a NOT gate Eq. (4) ips up/down to down/up. What happens when we act on the qubit with the NOT gate? Viz. 1 0 P = P +P (13) 0 1 0 1 = + ; (14) 1 0 i.e. we ip the probability amplitudes of measuring the up and down. Now P and I are very simple operations, and in classical information theory with bits, these are the only two possible actions you can act on a single bit. However, in the case of a qubit, the additional complex structure of the state vector allows for a much more varied kinds of actions any 2 2 non-singular unitary matrix O with complex 0 coecients that takes to another normalizable state is ne, i.e. 0 =O : (15) We will now introduce the high-brow word operator, i.e. O is an operator, and it operates on to give 0 us another state it is a map from the space of all possible states to itself. As an aside, in these lectures, we will only deal with linear operators, which you will learn a lot about in your Linear Algebra class. Also, jargonwise we use \act" and \operate" interchangeably (although we don't use \actor".) There is a special class of operators which corresponds to observables, i.e. things that we can measure, such as position and momentum of a particle. In the case of the qubit, the observable is the information of whether the state is up or down. How do we mathematically describe such an operator? This will lead us to Postulate 4. First, we introduce some more math you may remember some of this from your IA class. (De nition) Eigenvectors and Eigenvalues: If the operatorO acting on state returns the state multiplied by some 2C, O = ; 2C (16) 6 then  is called an eigenvector of O and  its associate eigenvalue. (De nition) Hermitian Matrices: Furthermore, suppose O obey y T  O =O = (O ) (17) then 2R and O is a Hermitian Matrix. In other words: Hermitian Matrices have real eigenvalues. Proof : From the de nition of eigenvalues O =  (18) O = : (19) It is easy to show that O is also Hermitian, so y  (O) =O =  (20) hence  must be real. We are now ready to state Postulate 4. Postulate 4 (Observables): An operator associated with an observableO is Hermitian. The result of a measurement of such an observable on a state yields one of the eigenvalues, and the state collapses (via Postulate 3) into its associated eigenvector. Returning to the qubit, and we want to associate the result of such a measurement with 1 being up, 2 and -1 being down . One such operator (called a Boolean operator) is given by 1 0 N = : (21) 0 1 y This operator is clearly Hermitian N = N and you can easily check that operating N on an up/down state returns the up/down state back with its associated eigenvalue1 N = 1 ; N =1 ; (22) up up down down i.e. the eigenvectors of N are the up/down states with their associated eigenvalues1. In the simple case of the qubit, it is clear that any state can be described by some linear combination of the eigenvectors of N, i.e.  and  , hence the space of eigenvectors is complete. You might up down also have noticed that the two eigenvectors are orthonormal to each other   = 0: (23) up down In fact, it can be shown that the eigenvectors of a Hermitian operator are complete and orthonormal to each other, but for the moment let us plow on. Here we emphasise that the physical act of measurement is not described by the mathematical act of operating on the state with a Hermitian operator, even though it is tempting to think that way Uncertainty Principle of the qubit : One of the things that we have not gone into a lot of details in this introduction, is the notion of \measuring to arbitrary accuracy". We have asserted that one of the main idea of Quantum Mechanics of a single particle is that one cannot measure both its position x and momentum p to arbitrary accuracy at the same time. But this does not mean that we cannot measure the position x to arbitrary accuracy we simply pay the price that we lose accuracy on the p measurement. This is the called the Uncertainty Principle, and we will formalize it in the coming lectures. However, you might ask, in the context of the qubit: can't we measure the up-ness or 2 We can also do 0 and 1, but that would be confusing. 7down-ness of a qubit to arbitrary accuracy? For example, once we open the box, the cat is dead what is the price we pay for this accuracy? The answer to this paradox is to realize that the qubit actually has more than one Boolean operator. In fact it turns out that there exist a one (compact) parameter family of operators with eigenvalues1, viz cos sin N = : (24)  sin cos The uncertainty principle of the qubit pertains to the fact that one cannot measure, to arbitrary accuracy, the observables of all Boolean operators at the same time. 1.3 Dynamics of a qubit Finally, to close our whirlwind introduction to Quantum Mechanics, we turn to the dynamics of the qubit we want to study its evolution. We want to still work with the eigenvectors of N, so the time dependence is encoded in their coecients ( (t); (t)), i.e. 1 0 (t) (t) = (t) + (t) = : (25) 0 1 (t) We have argued that operators act on states to give us another state. Colloquially, operators \do stu to states". To give qubits dynamics, we can construct an operator U(t + t;t), let's call it the \wait for time t" operator, i.e. (t + t) =U(t + t;t) (t): (26) Physically, if t = 0, then U(t;t) =I must be the identity. We can expand around this, to obtain i U(t + t;t) =I H(t)t +::: (27) 27 34 where 2 =h is the Planck's Constant and has the value 6:626 10 erg s (or 1:055 10 Joule s), while H is some Hermitian matrix with (possibly) time-dependent coecients. Whyi= and what is? Here we will cheat and simply say the reasons which will be a bit unmotivated  has dimensions, so we extract it out of H to get H to have the right dimensions of Energy.  i is extracted so that H is Hermitian as it corresponds to an observable as we will see. Rearranging Eq. (27) into something we are familiar with in calculus, and taking the limit of t 0 (t + t) (t) i lim = H(t) (t) (28) t0 t or d i =H(t) (t): (29) dt This is known as the Schr odinger's Equation for a qubit. The Hermitian operator H(t) is called the Hamiltonian, because it is a generator of motion for (t). Note that it is rst order in time derivative, unlike Newton's Law of motion, so a speci cation of (t ) at some initial time t is all you need to evolve 0 the system. The Hamiltonian H(t) is time-dependent in general, but for most of the lectures we will consider the special and very important case of a time-independent H. Consider a simple Hamiltonian H of the following form E 0 1 H = : (30) 0 E 2 8 Since H = E  and H = E  ,  are eigenvectors of H with eigenvalues E and up 1 up down 2 down 1 up=down E . You might have learned from other courses (don't worry if you have not) that the Hamiltonian is 2 also associated with the energies of the system here E and E are the energy eigenvalues of H. Since 1 2 actingH on does not change but simply give us their energies, sometimes we also call up=down up=down the eigenvectors of H Energy Eigenvectors. This Hamiltonian is very simple, and we can easily nd the solution to d E 0 1 i = ; (31) dt 0 E 2 or     iE t iE t 1 2 (t) = (0) exp ; (t) = (0) exp : (32) 2 2 Since the probabilities of  and in this case,j (t)j andj (t)j do not evolve with time, we also up down call these states Stationary States. In particular, if we have started with the up/down state, we will stay in the up/down state forever. ThisH is not very interesting as it is at the moment. We can make it more interesting by adding the NOT operator to it, and for simplicity, let's assume E =E =E for the moment 1 2 E  H =HP = ; (33)  E where  is some real constantjjE. Physically, since the action of P is to ip an up state to a down state, one can think of this new operatorH as giving the dynamics to the qubit where it has a non-trivial chance of ipping states. (The conditionjjE ensures that the chance is small.) Let's start with an up state at time t = 0, i.e. 0 1 0 (t ) = (t ) + (t ) (34) 0 0 0 0 1 with (t ) = 1 and (t ) = 0. The question is: what is the probability of measuring the down state at 0 0 some time t 0? Using Schr odinger's Equation Eq. (29) and Eq. (25), we nd the following pair of rst order di erential equations d i =E (t) (t); (35) dt d i =E (t) (t); (36) dt 3 which we can solve, using the boundary conditions Eq. (34) to nd the solutions   t iEt= (t) =e cos ; (37)   t iEt= (t) =ie sin : (38) Using Postulate 3, the probability of nding the down state at time t 0 is then the amplitude square of the inner product of  with (t), down   t 2 2 P =j  (t)j = sin : (39) down down The result is plotted in Fig. 2. In words, the presence of theP operator inH means that the probability 3 Hint : de ne new variables A(t) = (t) (t), B(t) = (t) + (t). 9P 2 1.0 0.8 0.6 0.4 Figure 2: The probability of measuring 0.2 the down state as a function of time in t 5Π Π Π 3Π units of=. Π 4 4 2 4 of measuring a down state oscillates with time. We emphasise that it is the probability that is oscillating say at time t = 3=, we have a 50-50 of measuring the state being up or down. Notice though, at t = n for n = 0; 1; 2;::: , there is zero probability of measuring the state being down. Hence sometimes you might hear people say \Quantum Mechanics mean that we cannot predict anything with certainty and hence it is not deterministic". That is certainly wrong we can calculate the state vector to arbitrary accuracy, it is the notion of simultaneous measurements of multiple incompatible observables to arbitrary accuracy that is lost. But we are rushing ahead, so we leave this discussion for the future when we have developed the necessary mathematical tools to describe them. 1.4 Summary Congratulations, you have just learned most of the structure of Quantum Mechanics If this sounds too simple to be true is Quantum Mechanics all about manipulating matrices and doing linear algebra the secret answer is that yes it is true. Quantum Mechanics at its core is underlaid by linear algebra. Often, many people nd Quantum Mechanics hard because physically interpreting the results is extremely counter-intuitive, and not because the calculational details are complicated. In the coming lectures, we will abandon the qubit, and study the Quantum Mechanics of a single non- relativistic particle which is a more complex system than a qubit. When we do that however, do keep this rst lecture in your mind often you will nd that a concept that is hard to understand in a particle system can be easily grasped when we strip everything down to a qubit. The study of the Quantum Mechanics of a non-relativistic particle, while seemingly a simple system, have remarkable explanatory power indeed at the end of the lectures this very system reproduces the entire energy spectrum of the Hydrogen Atom. Once we have done that, we will return to the qubit, and talk about love. 102 The Failure of Classical Mechanics and Wave-Particle Duality This lecture is more qualitative than the rest of the class. Very roughly speaking, in Classical Mechanics, one can describe motion in terms of either particles or waves. Classically, they are distinctly di erent things. In our every day life, we intuitively think of some things as particles (like bullets, cars, cats etc) while some other things as waves (sound waves, water waves, radio etc), because they seemingly behave very di erently. In this lecture, we will show that quantum objects can neither be described as waves or particles, but has features of both. You have studied particle motion under a system of Newtonian forces in your Dynamics and Relativity class in great detail they obey Newton's Laws of Motion. Particles carry energy and momentum in in nitisimally localized small chunks, hence one often call them \point particles". Waves, on the other hand, describe motion of entities which are spreaded out and not localized. Some common day examples are:  Sound is carried by the compression and decompression of air, and the displacement of some parcel of air molecules from its original undisturbed position obeys the Wave Equation.  Throw a pebble into a pond, and the disturbance caused by the pebble will caused a circular wave front to propagate outwards from the point of contact. The height of the water of the pond and its motion is described by waves.  Classical Electric elds and Magnetic elds are described by waves. You will study this in great detail in the Part IB Electromagnetism class. In the rst two examples above, the waves are disturbances of some medium, while in the third example, electric and magnetic elds are waves themselves and do not need to propagate in a medium. 2.1 Wave Mechanics : The Wave Equation Consider some function f(x) that describe some pulse, with a maximum centered around the origin f = f(0) as shown in Fig. 3. This pulse for the moment is static since f(x) is just a function of max space and not time. Now, how do we give it motion? Suppose we want to describe the pulse traveling with constant velocity v to the right, in such a way that it preserves its original shape. At some time t later, the pulse is now centered around the location x =vt. Since we know that f(0) =f , it is clear max that a traveling pulse can be describe by the same function f(x), but with the argument xxvt, i.e. f(xvt). Similarly, a pulse with some other functional form g(x) traveling to the left will be described by some function g(x +vt). Let's call them right-moving and left-moving respectively. Notice that the right moving pulse f(xvt) satis es the di erential equation   v + f(xvt) =L f(xvt) = 0; (40) 1 x t and the left moving pulse f(x +vt) satis es   v g(x +vt) =L g(x +vt) = 0: (41) 2 x t Since the result of the action of any di erential operator on zero is zero, we can operateL on Eq. (40) 2 andL on Eq. (41) to get 1    v + v f(x;t) = 0; (42) x t x t and    v v + g(x;t) = 0; (43) x t x t 11f(x) v Figure 3: A pulse described by some functionf(x) moving with speedv to the x right. Figure 4: Circular waves caused by throwing a pebble into a pond. or 2 2 2 =v ; (44) 2 2 t x where (x;t) = f(xvt) +g(x +vt). Notice that both f and g satis es the same equation of motion Eq. (44), which is known imaginatively as the Wave Equation. Some features of the Wave Equation:  Linearity: If and are both solutions of the Wave Equation, then so isa +b wherea;b2C. 1 2 1 2 This is known as the Superposition Principle.  2nd order in time derivative: As in Newton's Law of motion Eq. (1), the equation is second order _ in time derivative. This means that we need to specify two initial conditions (x; 0) and (x; 0), as in particle dynamics. There is a very special solution to the Wave Equation which will be important in our study called plane waves. These are solutions which are periodic in space and time    2 2 (x;t) =A exp i x t ; (45)  T where is the wavelength andT is the period, andA2C is known as the amplitude of the plane wave. The 2's are annoying, so we often use the related quantities wave number k 2= and angular frequency = 2=T instead, i.e. (x;t) =A exp i (kxt): (46) It is easy to show that Eq. (46) is a solution to the Wave Equation Eq. (44) provided that v = : (47) k v is sometimes known as the phase velocity of the wave, and quanti es the propagation velocity of the wave. Like particles, waves can carry energy and momentum. Since waves oscillate, we can de ne a quantity called intensity, 2 I =j (x;t)j ; (48) 12which is proportional to the time averaged (over the period) energy and momentum ow. Note that we have to put in the right constants to make the dimensions right, but let us ignore that for the moment. In 3-dimensions, the Wave Equation is given by 2 2 2 =vr ; (49) 2 t while plane waves generalize simply from their 1 dimensional counterpart Eq. (46) to ikxit (x;t) =Ae ; =jkjv (50) where k is known as the wave vector. 2.2 Two Slit Experiment with Waves and Particles We assert early on that particles and waves are distinct entities, with di erent behaviors. A simple and famous experiment to illustrate the di erence is the Two Slit experiment. In this section, we will talk about the results of this experiments with waves and particles, and then we will describe the results of an experiment with when quantum e ects are important to show that quantum systems possess qualities of both. 2.2.1 Two Slit with Waves : Sound waves One of the consequences of the Superposition Principle in wave behavior is interference, which you might have studied in high school. Since linearity implies that the sum of two solutions and to 1 2 the Wave Equation is also a solution, if we add two identical plane waves in terms of A, andT but are o -phase by , then the interference is destructive and the result is zero, i.e.       2 2 2 2 (x;t) =A exp i x t +A exp i x t + = 0: (51)  T  T Likewise, constructive interference can occur when the two plane waves are in phase. Let us now discuss the Two Slit experiment with waves. Sound waves are described by displacement of the position of air molecules from their original \undisturbed" position due to a pressure di erence. Let (x) the displacement of some parcel of air from its original position at x. They obey the 2-D Wave Equation 2 2 2 1 + = : (52) 2 2 2 2 x y v t As the molecules \empty" out or \rush" back into space, this changes the local density which in turns changes the pressure. And this pressure inequality generates further motion of the molecules. Our ears pick up the motion of air, and the greater the amplitude of the motion of the air molecules, the greater the pressure and hence the louder we hear the sound. Human ears are sensitive to sound frequencies from 20 Hz to 20 kHz, so a good measure of the \loudness" would be to use Intensity Eq. (48) since our senses do not normally pick up stimuli changes of such high frequency. Imagine a wall with two holesS andS separated by distancea, see Figure 5. On the left of the wall, 1 2 is a mono-frequency sound source (say a piano playing the note middle C, 1=T = 261 Hz) suciently far away such that by the time the sound waves arrive at the wall, we assume that the crests are parallel to the wall. First, we block the hole S . The incoming sound wave will come through S , and then 2 1 propagate radially outward. Since it is radial, it is clear that the energy and hence the intensity of the sound waves is dissipated. An observer then walk along parallel to the wall at a constant distance d. As expected, the closer she is to the open hole, the more intense the sound she hears, and she plots the intensity as a function of 13S1 a S2 Figure 5: A double slit experiment with sound waves. The observer measures the left intensity d plot when one of the slits is closed, and the right one slit closed both slits opened intensity plot when both slits are opened. position and call it I . Similarly, she then closes S and opens S , and the resulting intensity plot is I . 1 1 2 2 Finally, she opens both S and S , and make an intensity plot I . The plot she will obtain is the one 1 2 1+2 on the right of Fig. 5, which is obviously I 6=I +I . 1+2 1 2 This is because of the linearity of the wave solutions. Let describe the wave fromS and describe 1 1 2 2 waves from S . With S closed, the intensity is given by I =j j , and similarly for S . However, if 2 2 1 1 1 both holes are opened, then both waves and will propagate towards the observer, and by linearity 1 2 2 2 2 the = + . She then measures I =j + j 6=j j +j j . The oscillatory pattern is easily 1 2 1+2 1 2 1 2 explained by the fact that, depending on the distance of the observer from the S andS , each wave will 1 2 have arrived with a di erence phase and hence can be constructive or destructive. You can easily show that the spacing between adjacent maxima of I is given by d=a. 1+2 2.2.2 Two Slit with Particles : Machine Gun What about particles? We replace the peaceful piano playing middle C with a more violent machine gun. The machine res equal mass bullets at equal velocities (and hence each bullet has equal kinetic energy). This machine gun is also designed such that it res bullet at all directions at an equal rate. See Fig 6. Since each bullet carries equal kinetic energy, the energy ux or intensity is then de ned by the number of particles arriving per second at any given location x. As above, the intensity plots with one slit closed is given on the left of Fig. 6. When both slits are opened, it is not surprising that the total intensity I =I +I , as expected since particles clearly do not exhibit wavelike behavior. 1+2 1 2 What about bullets hitting each other in mid-air, for example say a bullet hitting the edge of S 1 and gets de ected onto the path of another bullet coming through S , wouldn't this cause some form of 2 \interference"? To eliminate this possibility, we can tune down the rate of the ring of the machine gun such that only one bullet is in the air at any time, and make the wall so thin that there is no chance of de ection o the inside of the hole. Then we will recover the result we asserted. 2.3 Is light particle-like or wave-like? Now we want to repeat this experiment with light. Historically, this is known as the Young Two Slit Experiment and was credited in proving the wave nature of light in the 1800s. The set up is again similar to Fig. 5, except that the incoming sound waves are now replaced by some monochromatic light source with frequency . Instead of an observer with ears, we set up a row of light 14S1 a S2 Figure 6: A double slit experiment with bullets. The observer measures the left intensity plot d when one of the slits is closed, and the right one slit closed intensity plot when both slits are opened. both slits opened − e Figure 1: Incident light expels electron from metal. Figure 7: The Photoelectric E ect. The energy threshold required to expel an electron from a piece of metal is E 0. One can do this by shining a monochromatic light onto it. We can vary both the 0 intensityI and the (angular) frequency = 2c= of the light. We nd that the liberation of an electron 1Introduction requires  =E = but is independent of I. On the other hand, the rate of electron emitted is/I. 0 0 QM introduces a single new constant of fundamental nature: Planck’s constant detectors at a distance d from and parallel to the wall. −34 =1.055 × 10 Joule s We then do the same experiment, do we see the wave pattern of Fig. 5 or the particle pattern of Fig. 6? You probably know the answer : it is the former. So light must be wave-like, as you will study in We will also use Planck’s original constant h=2π your Electromagnetism class, right? But wait Let us slowly dial down the intensity of the incoming light. At rst, the intensity registered 2 −2 2 −1 Dimensions: =ML T ×T =ML T by the light detectors fall as expected. As we keep dialing down, a strange thing begins to happen at some very low incoming intensity, the detectors are not being activated continuously. Instead, they are being activated individually one detector goes o at location x , and then another goes o at some 1 Photoelectric effect other location and so forth. It seems that light is coming in in localized chunks Perhaps an explanation is that at low intensity, light become particle-like? We can redo the experiment • To liberate electron from metal requires energy E≥ E 0. Threshold energy E is 0 0 with a very low intensity such that the detectors are activated one at a time, and then plot out the number different for different metals. of times each detector is activated and the result is still the same as in Fig. 5 So is light wave-like as it demonstrates interference behavior, or particle-like as it activates detectors locally? • Shine monochromatic light at a metal plate (see Fig 1), The answer is of course, light has features of both. This is known as the wave-particle duality of light. Historically, while the Young Two Slit experiment is credited with \con rming" the wave-like – Intensity I nature of light, it is the Photoelectric e ect (see Fig. 7) that shows that light also exhibits particle-like behavior we call light particles photons. – Angular frequency ω.Here ω=2πc/λ where λ is the wavelength of the light. To observe this e ect, we take a piece of metal, and shine a monochromatic on it. We can change the frequency and the intensity of this light, and we observe that at certain frequenices, electrons will Find, be emitted from this metal. Surprisingly, whether electrons are emitted or not depends only on the frequency, while the rate of emitted electrons depends on the incident intensity of light Experimentally, 1. Liberation of electron requires ω≥ω where, 0 15 ω = E 0 0 Independent of intensity, I 2. Number of electrons emitted∝ I. 2 Copyright © 2008 University of Cambridge. Not to be quoted or reproduced without permission.the frequency of the light required to liberate an electron is E 0 = (53) 0 where E depends on the metal used. We have introduced Planck's constant h = 2 in the previous 0 lecture. Older books on quantum mechanics often like to use h, but we will use in these lectures as most modern physicists now do. This was puzzling to many people, until Einstein in his lunch break (1905) came up with the ex- planation with the crucial insight that if one thinks of light as localized bundles instead of a wave, the Photoelectric e ect is completely natural. He stated that:  A photon of frequency carries the energy E =; (54) and the momentum p =k: (55)  The energy and momentum of each photon is related to each other by E =pc; (56) where c is the speed of light. From your Dynamics class, relativistic kinematics imposes 2 2 2 2 4 E =p c +m c ; (57) so the photon is massless.  Intensity of light corresponds to the rate of photons emitted.  After liberation, conservation of energy implies that the electron has kinetic energy E =E =( ); (58) K 0 0 which agrees well with experiments. 2.4 Everything is Quantum : De Broglie Waves Is this strange behavior limited to light/photons? After all, light is massless so maybe it is special in some way. How about electrons, bullets, cats do they exhibit wave-particle duality? Historically, it was de Broglie who proposed that all matter exhibits this behavior. So for a particle moving at some momentum p possess a de Broglie wavelength given by 2  = ; (59) jpj or sometimes simply p =k. When he proposed this (it was his PhD thesis), there was no experiment that can test for this 4 conjecture . Indeed, for a bullet of mass 1g and moving at 1 cm/sec would have the de Broglie wavelength of 2 26  =  10 cm (60) p so any interference pattern would be impossible to see But technology has moved on, and indeed there are now many experiments that veri ed this the entire Universe is quantum mechanical For those 4 And hence demonstrating that the standard for theory PhDs in physics has not changed over the years. 16who have taken Dr. Baumann's Concepts in Theoretical Physics class last term, he showed you a video of interference pattern of a two slit experiment done with electrons which conclusively demonstrated the wave-particle nature of matter in the Universe (the link to the Youtube video is on the course webpage). Now we are in a conundrum neither Newton's Law of motion of particles, and the Wave Equation of waves can describe things that exhibit wave-particle duality which inconveniently turns out to be everything. What should we do? 173 The Mathematics of Quantum Mechanics What has been seen cannot be unseen. Previously we have introduced a two-state system to illustrate broadly the mathematical structure of Quantum Mechanics. Of course, the world is made out of more than just two-state qubits. In particular, qubits are discrete systems while most things we know and love are continuous systems for example, how do we describe the position x and momentum p of a quantum mechanical object? In this section, we will generalize what we have learned in the rst lecture to how a non-relativistic particle move in a continuous coordinate space. As we move from discrete to continuous systems, we will take some liberties in the rigor of the mathematics unfortunately we have limited time but hopefully it will not be too much of a jolt to your senses. 3.1 Quantum Mechanics of a Particle We started this class with the Classical equation of motion for a single particle, Newton's Law of motion Eq. (1) but then went o to discuss qubits. Let us now come back to the particle and ask \how do we describe a quantum mechanical particle?" We begin by restating the Postulates of Quantum Mechanics, but this time in the context of describing a particle. Postulate 1 (State and Wavefunction): A particle is described by a state . Furthermore, the probability amplitude of nding the particle at position x is given by a complex function (x) 3 : R C; (61) called the wavefunction. The space of is the space of all possible states in the system called the Hilbert Space. 3 The wavefunction is normalizable or square integrable over all spaceR Z Z  2 (x) (x)dV = j (x)j =N 1: (62) 3 3 R R Compare this to the state vector of the qubit in section 1.1. In the qubit, there are only two possible states up or down. But a particle can be anywhere in space, i.e. an in nite number of possible points. So instead of just two complex coecients and describing the state, we have a continuous complex 3 n function which mapsR (orR for an n-dimensional space) intoC. We have intentionally introduced the notion of the state as an individual entity, without con ating it with the wavefunction (x). In the high-brow way of thinking about quantum mechanics, (x) is really the (complex) coecient of the state in the continuous and complete basis of x. In words, we say that (x) is the state expressed in the x representation. We can also represent in the p momentum representation. We will discuss representations when we introduce Hermitian Operators for the moment you can think of (x) as some complex function. Having said all that, we will often interchange the words \wavefunction" and \state" in these lectures. Postulate 2 (Born's Rule): The probability of nding the particle in some in nitisimal volume dV is given by 2 j (x)j dV: (63) It is nice to normalize total probability to unity so we normalize the wavefunction 1 (x) =p (x) (64) N 18such that Z 2 j (x)j dV = 1: (65) 3 R This means that 2 (x)j (x)j 2R (66) is a probability density function in the usual sense. A little note on jargon: sometimes we sloppily 2 callj (x)j the probability, which is nomenclaturely heinous but I hope you won't mind too much. We will also drop tildes to denote normalized wavefunctions from now on it will be clear from the context which wavefunctions are normalized or not.  Since (x) is a scalar function, the probability density function is simply (x) (x). Comparing this to the inner product of the qubit state we introduced way back in Eq. (9), we obviously do not need the transposition operation. To keep our notation consistent, we can also introduce the inner product of two wavefunctions  and Z y (x)(x) (x)(x) dV; (67) 3 R y where denotes Hermitian Conjugation as before. Of course, if is a scalar then this is simply usual R 2 conjugation. The scalar product of with itself  = j (x)j dV is called the norm, so normalized 3 R has unit norm (doh). In general, if (x) is the desired outcome, then the probability of measuring such an outcome given the wavefunction (x) is given by Born's Rule Z 2 2 y Probability of measuring state (x) in (x) =  (x) (x) dV =j j : (68) 3 R R We say (x) (x) dV is the probability amplitude of  to be found in it measures the overlap 3 R of the two wavefunctions. You might note that the form Eq. (68) bears a lot more resemblance to the Born's Rule we encountered when we studied the qubit. From the viewpoint of Eq. (68), one can go super pedantic and write the probability amplitude (x) as Z 0 3 0 (x ) =  (x x ) (x) dV; (69) 3 R so Eq. (68) becomes Z 2 2 3 0 3 0 0 Probability of measuring state  (x x ) in (x) =  (x x ) (x) dV =j (x )j (70) 3 R recovering Eq. (63). This is cumbersome, but it allows us to be completely general about representation. You will study this in the Part II class Principles of Quantum Mechanics, but with snazzier notation than carrying all those integrals around. Equivalence class of states. If the wavefunctions are related to each other by a multiplicative non-zero complex number, (x;t) = (x;t) ; 2C 0; (71) then both wavefunctions describe the same state. To check for this, we need to ensure that both wave- functions will yield the same probability density function, viz: Suppose (x;t) is normalizable, then (x;t) also normalizable, Z Z 2 2 2 2 j (x;t)j dV = j j j (x;t)j dV =j jN 1: (72) 3 3 R R 19Normalizing both wavefunctions (x;t) (x;t) = p = (x;t); (73) 2 j j j jN we see that the di erence only depends on through the complex phase =j j and therefore yields the same probability density function, 2 2 (x;t) = j (x;t)j =j (x;t)j (74) for all values of . Postulate 3 (Measurement): Once a measurement has been made, the wavefunction collapses into its normalized measured state  measure ; (75) the very act of measurement has a ected the wavefunction which would \jump" into its measured state. This is often called Collapse of the Wavefunction. Why does a measurement \collapse" the wavefunction? One way to think about this is to consider the act of measurement we would need an apparatus which must somehow interact with the system. For example, if we want to measure the position of a moving particle, we would need to see where it is, so we shine a light on it. The light bounces o the particle, and enter our eyes so we say \Aha, the particle is here." But the light has a ected the particle in some way. In Feynman's words, the disturbance is necessary for the consistency of the viewpoint of making observations. But this does not mean we understand how or why the interaction occurs in such a way to \collapse" the wavefunction, since the apparatus themselves are made out of quantum mechanical things too. This is known as the Measurement Problem. Let's think about this what it means to measure the position of the particle. If we have an amazingly accurate detector which can pinpoint the position of the particle, then once we found its position, say it is at position x , then the wavefunction must collapses to something that will have support only at x , 0 0 3 i.e. a Dirac Delta Function (x)  (x x ) you have previously encountered in your IA class. 0 Since the Dirac Delta is not normalizable so in principle it is not a valid wavefunction so we have to be a little bit more careful about calling it such. The root cause of this is that the probability of nding a 3 particle exactly any point x is zero since there is an in nite of points in a continuous space R , the 0 probability of nding the particle at any particular point cannot be nite. Fortunately in practice, the best a detector can do is to pinpoint the position of the particle within some small but nite region V . In this case, the wavefunction (x) then collapses into some other normalized wavefunction (x) which has support only around this region V , i.e. Z 2 j(x)j dV 1: (76) V 3.2 Operators on the Wavefunction Recall that in the qubit system, we can ip the states with the NOT gate. How do we \do stu to the wavefunction"? Instead of complex matrices acting on vectors, we introduce operators. An operator O eats a complex valued function f and returns another such complex function, so it is a map from the space of to itself, i.e. 3 g =Of ; where f;g :R C: (77) Fortunately, in Quantum Mechanics, we only deal with linear operators, so the corresponding map is also linear O f + f = Of + Of (78) 1 1 2 2 1 1 2 2 20