Understanding Waves and Quantum Mechanics: A Review of Classical and Quantum Waves, Part 1, Study notes of Quantum Mechanics

Download Understanding Waves and Quantum Mechanics: A Review of Classical and Quantum Waves, Part 1 and more Study notes Quantum Mechanics in PDF only on Docsity! Ch. 1 notes, part1 1 of 26 8/27/2008 © University of Colorado, Michael Dubson (mods by S. Pollock ) Quantum Mechanics Quantum Mechanics Introductory Remarks: Q.M. is a new (and absolutely necessary) way of predicting the behavior of microscopic objects. It is based on several radical, and generally also counter-intuitive, ideas: 1) Many aspects of the world are essentially probabilistic, not deterministic. 2) Some aspects of the world are essentially discontinuous Bohr: "Those who are not shocked when they first come across quantum theory cannot possibly have understood it." Humans have divided physics into a few artificial categories, called theories, such as • classical mechanics (non-relativistic and relativistic) • electricity & magnetism (classical version) • quantum mechanics (non-relativistic) • general relativity (theory of gravity) • thermodynamics and statistical mechanics • quantum electrodynamics and quantum chromodynamics (relativistic versions of quantum mechanics) Each of these theories can be taught without much reference to the others. (Whether any theory can be learned that way is another question.) This is a bad way to teach and view physics, of course, since we live in a single universe that must obey one set of rules. Really smart students look for the connections between apparently different topics. We can only really learn a concept by seeing it in context, that is, by answering the question: how does this new concept fit in with other, previously learned, concepts? Each of these theories, non-relativistic classical mechanics for instance, must rest on a set of statements called axioms or postulates or laws. Laws or Postulates are statements that are presented without proof; they cannot be proven; we believe them to be true because they have been experimentally verified. (E.g. Newton's 2nd Law, € Fnet = ma , is a postulate; it cannot be proven from more fundamental relations. We believe it is true because it has been abundantly verified by experiment. ) Actually, Newton's 2nd Law has a limited regime of validity. If you consider objects going very fast (approaching the speed of light) or very small (microscopic, atomic), then this "law" begins to make predictions that conflict with experiment. However, within its regime of validity, classical mechanics is quite correct; it works so well that we can use it to predict the time of a solar eclipse to the nearest second, hundreds of year in advance. It works so well, that we can send a probe to Pluto and have it arrive right on target, right on schedule, 8 years after launch. Classical mechanics is not wrong; it is just incomplete. If you stay within its well-prescribed limits, it is correct. Ch. 1 notes, part1 2 of 26 8/27/2008 © University of Colorado, Michael Dubson (mods by S. Pollock ) Each of our theories, except relativistic Quantum Mechanics, has a limited regime of validity. As far as we can tell (to date), QM (relativistic version) is perfectly correct. It works for all situations, no matter how small or how fast. Well... this is not quite true: no one knows how to properly describe gravity using QM, but everyone believes that the basic framework of QM is so robust and correct, that eventually gravity will be successfully folded into QM without requiring a fundamental overhaul of our present understanding of QM. String theory is our current best attempt to combine General Relativity and QM (some people argue "String Theory" is perhaps not yet really a theory, since it cannot yet make (many) predictions that can be checked experimentally, but we can debate this!) Roughly speaking, our knowledge can be divided into regimes like so: In this course, we will mainly be restricting ourselves to the upper left quadrant of this figure. However, we will show how non-relativistic QM is completely compatible with non- relativistic classical mechanics. (We will show how QM agrees with classical mechanics, in the limit of macroscopic objects.) In order to get some perspective, let's step back, and ask What is classical mechanics (C.M.)? It is, most simply put, the study of how things move! Given a force, what is the motion? So, C.M. studies ballistics, pendula, simple harmonic motion, macroscopic charged particles in E and B fields, etc. Then, one might use the concept of energy (and conservation laws) to make life easier. This leads to new tools beyond just Newton's laws: e.g. the Lagrangian, L, and the Hamiltonian,H, describe systems in terms of different (but still conventional) variables. With these, C.M. becomes more economical, and solving problems is often simpler. (At the possible cost of being more formal) Of course, what one is doing is fundamentally the same as Newton's F=ma! Mechanics (non-relativistic) speed c 0 big small 1/size Relativistic Mechanics QM (non-relativistic) Relativistic QM Ch. 1 notes, part1 5 of 26 8/27/2008 © University of Colorado, Michael Dubson (mods by S. Pollock ) Postulate 1: The state of a physical system is completely described by a complex mathematical object, called the wavefunction Ψ (psi, pronounced "sigh"). At any time, the wavefunction € Ψ(x) is single-valued, continuous, and normalized. The wave function € Ψ(x) is not "the particle", or "the position of the particle", it is a mathematical function which carries information about the particle. (Hang on!) In this course, we will mostly be restricting ourselves to systems that contain a single particle (like one electron). In such a case, the wavefunction can be written as a function of the position coordinate of the particle, and the time: € Ψ = Ψ( r ,t). Often, we will simplify our lives by considering the (rather artificial) case of a particle restricted to motion in 1D, in which case we can write € Ψ = Ψ(x, t). We may also consider a particular moment in time, and focus on just € Ψ(x). In general, € Ψ(x) is a complex function of x; it has a real and an imaginary parts. So when graphed, it looks something like. In fact, it can look like anything, so long as it is continuous and normalized. Definition: A wavefunction is normalized if . There are many different ways to write the wavefunction describing a single simple (spinless) particle in 1D at some time: , and others, to be explained later. (Here x is position, and p is momentum). If the particle has spin, then we have to include a spin coordinate m, in addition to the position coordinate in the wavefunction . If the system has 2 particles, then the wavefunction is a function of two positions: . Postulate 2 has to do with operators and observables and the possible results of a measurement. We will just skip that one for now! Re[Ψ] x Im[Ψ] x Ch. 1 notes, part1 6 of 26 8/27/2008 © University of Colorado, Michael Dubson (mods by S. Pollock ) Postulate 3 has to do with the results of a measurement of some property of the system and it introduces indeterminacy in a fundamental way. It provides the physical interpretation of the wavefunction. Postulate 3: If the system at time t has wavefunction , then a measurement of the position x of a particle will not produce the same result every time. does not tell where the particle is, rather it give the probability that a position measurement will yield a particular value according to € Ψ(x,t) 2dx = Probability (particle will be found between x and x+dx, at time t) An immediate consequence of Postulate 3 is Since the particle, if it exists, has to be found somewhere, then Prob(particle will be found between –∞ and +∞ ) = 1. Hence the necessity that the wavefunction be normalized, This QM description is very, very different from the situation in classical mechanics. In classical mechanics, the state of a one-particle system at any given instant of time is determined by the position and the momentum (or velocity): €  r ,  p . So, a maximum of 6 real numbers completely describes the state of a classical single-particle system. (Only 2 numbers, x and p, are needed in 1-D) In contrast, in QM, you need a function . To specify a function, you need an infinite number of numbers. (And it's a complex function, so you need 2 × ∞ numbers!) In classical mechanics, the particle always has a precise, definite position, whether or not you bother to measure its position. In quantum mechanics, the particle does not have a definite position, until you measure it. The Conventional Umpire: "I calls 'em as I see 'em." The Classical Umpire: "I calls 'em as they are." The Quantum Umpire: "They ain't nothing till I calls 'em." In quantum mechanics, we are not allowed to ask questions like "What is the particle doing?" or "Where is the particle?" Instead, we can only ask about the possible results of measurements: "If I make a measurement, what is the probability that I will get such-and- such a result?" QM is all about measurement, which is the only way we ever truly know anything about the physical universe. Ch. 1 notes, part1 7 of 26 8/27/2008 © University of Colorado, Michael Dubson (mods by S. Pollock ) Quantum Mechanics is fundamentally a probabilistic theory. This indeterminacy was deeply disturbing to some of the founders of quantum mechanics. Einstein and Schrödinger were never happy with postulate 3. Einstein was particularly unhappy and never accepted QM as complete theory. He agreed that QM always gave correct predictions, but he didn't believe that the wavefunction contained all the information describing a physical state. He felt that there must be other information ("hidden variables"), in addition to the wavefunction, which if known, would allow an exact, deterministic computation of the result of any measurement. In the 60's and 70's, well after Einstein's death, it was established that "local hidden variables" theories conflict with experiment. Postulates 1 and 3 are consistent with experment! The wavefunction really does contain everything there is to know about a physical system, and it only allows probabilistic predictions of the results of measurements. The act of measuring the position changes the wavefunction according to postulate 4: Postulate 4: If a measurement of position (or any observable property such as momentum or energy) is made on a system, and a particular result x (or p or E) is found, then the wavefunction changes instantly, discontinuously, to be a wavefunction describing a particle with that definite value of x (or p or E). (Formally, we say "the wavefunction collapses to the eigenfunction corresponding to the eigenvalue x." ) (If you're not familiar with this math terminology, don't worry - we'll discuss these words more soon) If you make a measurement of position, and find the value xo, then immediately after the measurement is made, the wavefunction will be sharply peaked about that value, like so: (The graph on the right should have a much taller peak because the area under the curve is the same as before the measurement. The wavefunction should remain normalized. ) Postulate 1 states that the wavefunction is continuous. By this we mean that Ψ(x,t) is continuous in space. It is not necessary continuous in time. The wavefunction can change discontinuously in time as a result of a measurement. Because of postulate 4, results of rapidly repeated measurements are perfectly reproducible. In general, if you make only one measurement on a system, you cannot predict the result with certainty. But if you make two identical measurements, in rapid succession, the second measurement will always confirm the first. |Ψ|2 x x |Ψ|2 xo Before measurement After measurement Ch. 1 notes, part1 10 of 26 8/27/2008 © University of Colorado, Michael Dubson (mods by S. Pollock ) Let us disassemble and reassemble: The deviation from the mean of any particular result x is . The deviation from the mean is just as likely to be positive as negative, so if we average the deviation from the mean, we get zero: . To get the average size of , we will square it first, before taking the average, and then later, square-root it: It is not hard to show that another way to write this is . There are times when this way of finding the variance is more convenient, but the two definitions are mathematically equivalent: Proof: _________ Now we make the transition from thinking about discrete values of x (say x = 1, 2, 3, ...) to a continuous distribution (e.g. x any real number). We define a probability density ρ(x): ρ(x) dx = Prob( randomly chosen x lies in the range x → x+dx ) In switching from discrete x to continuous x, we make the following transitions: € Pi → ρ(xi)dx Pi i ∑ =1 → ρ(x)dx =1 -∞ ∞ ∫ x = xiPi i ∑ → x = x ⋅ ρ(x)dx =1 -∞ ∞ ∫ Please look again at these equations, (on left and right): think about how they "match up" and mean basically the same thing! (We'll use both sides, throughout this course.) From Postulate 3, we make the identification and we have So in QM, the expectation value of the position (x) of a particle (with given wave function Ψ) is given by this (simple) formula for <x>. It's the "average of position measurements" if you had a bunch of identically prepared systems with the same wave function Ψ. More generally , for any function f = f(x), we have . Ch. 1 notes, part1 11 of 26 8/27/2008 © University of Colorado, Michael Dubson (mods by S. Pollock ) Griffiths gives an example (1.1) of a continuous probability distribution. Let's redo that example, just slightly modified, to help make sense of it. (Take a look at it first, though) A rock, released from rest at time t=0, falls a distance h in time T. € x = 1 2 gt 2, h = 1 2 gT 2 . A move is taken as the rock falls (from t=0 → T), at 60 frames/sec, resulting in thousands of photos of the rock at regularly-spaced time intervals. The individual frames are cut out from the film and then shuffled. Each frame corresponds to a particular x and t, and a particular dx and dt. (dx might show up visually as a smear, since the rock moved during the short time that picture frame was taken) All frames have the SAME dt, but different frames have different dx's: dx/dt = gt => dx = g t dt. We can define the probability distribution in space, ρ(x), and the probability distribution in time, τ(t): ρ(x)dx = Prob (frame chosen at random is the one at x → x+dx) τ(t)dt = Prob (frame chosen at random is the one at t → t+dt) Here's a little picture that might help: (To be precise, I should really be writing Δx instead of dx, and Δt instead of dt. In the end, I'll take the limit Δt →0) Notice all the dt's are the same size, but the dx's start out short and get longer and longer. Now: τ(t)dt = dt/T , that is, τ(t) = constant = 1/T. Convince yourself! That's because any random frame is equally likely to be at any given time (early, middle, late). So the probability τ(t) needs to be constant. But why is it 1/T? That's to ensure that the total probability of the frame being somewhere between 0 and T is exactly one: € τ(t)dt 0 T ∫ = 1T dt0 T ∫ = 1T T =1 Each frame is equally likely, and the probability of grabbing one particular frame is proportional to 1/T. (It is also proportional to dt, if the frames are all longer, there are fewer overall, and the probability scales accordingly. Convince yourself!) dx h T x t dx dt dt Ch. 1 notes, part1 12 of 26 8/27/2008 © University of Colorado, Michael Dubson (mods by S. Pollock ) If you pick a particular t and dt (i.e. some particular frame) then corresponding to that (t,dt) is a particular (x,dx). The probability that that particular frame will be picked is what it is (all frames are equally likely, after all): Prob(t → t+dt) = Prob(x t→ x+dx) which means € τ(t)dt = ρ(x)dx € ⇒ ρ(x) = τ(t)dt /dx = τ (t) / dx /dt( ) = (1/T) / gt( ) But we know € T = 2h /g, and t = 2x/g (see our kinematics equations above) So € ρ(x) = (1/T) / gt( ) = g/2h g 2x /g = 1 2 h x That's what Griffiths got (thinking about it slightly differently). Check out his derivation too! _____________ The key formula in this problem is € τ(t)dt = ρ(x)dx ⇒ ρ(x) = τ(t) (dx /dt) It is vital to remember that, when using this formula, x and t are not independent. The x is the x which corresponds to the particular t (and dx is the interval in x which corresponds to the dt of that "frame") _____ By the way, you might be uncomfortable treating dx/dt as though it was just a fraction Δx/Δt. But, we often "pull apart" dx/dt and writ things like € dx dt = f (x) ⇒ dx = f(x)dt , or € dt dx = 1 (dx /dt) This makes sense if you remember that € dx dt = Δt→0 lim Δx Δt To physicists, dx/dt really is a tiny Δx (dx) divided by a tiny Δt (dt). Ch. 1 notes, part1 15 of 26 8/27/2008 © University of Colorado, Michael Dubson (mods by S. Pollock ) Classical Waves Review: QM is all about solving a wave equation, for ψ(x,t). But before learning that, let's quickly review classical waves. (If you've never learned about waves in an earlier physics class - take a little extra time to be sure you understand the basic ideas here!) A wave = a self-propagating disturbance in a medium. A wave at some moment in time is described by y = f(x) = displacement of the medium from its equilibrium position Claim: For any function y=f(x), the function y(x,t) = f(x-vt) is a (1-dimensional) traveling wave moving rightward, with speed v. If you flip the sign, you change the direction. (We will prove the claim in a couple of pages, but first let’s just make sense of it) Example 1: A gaussian pulse y = f(x) = € Ae−x 2 /(2σ 2 ) (If you are not familiar with the Gaussian function in the above equation, stare at it and think about what it looks like. It has max height A, which occurs at x=0, and it has "width" σ. Sketch it for yourself, be sure you can visualize it. It looks rather like the form shown above) A traveling gaussian pulse is thus given by y(x,t) = f(x-vt) = € Ae−(x−vt ) 2 /(2σ 2 ). Note that the peak of this pulse is located where the argument of f is 0, which means (check!) the peak is where x-vt=0, in other words, the peak is always located at position x=vt. That's why it's a traveling wave! Such a wave is sometimes called a “traveling wave packet”, since it’s localized at any moment in time, and travels to the right at steady speed. Example 2: A sinusoidal wave y = f(x) = € Asin(2π x λ ) (This one is probably very familiar, but still think about it carefully. ) “A” is the amplitude or maximum height. The argument changes by 2π, exactly one "cycle", whenever x increases by λ. (That's the length of the sin wave, or "wavelength", of course!) Now think about the traveling wave y(x,t) = f(x-vt) - try to visualize this as a movie - the wave looks like a sin wave, and slides smoothly to the right at speed v. Can you picture it?) Ch. 1 notes, part1 16 of 26 8/27/2008 © University of Colorado, Michael Dubson (mods by S. Pollock ) Review of sinuosoidal waves: For sine waves, we define k = € 2π λ = "wave number". (k has units "radians/meter") k is to wavelength as angular frequency (ω) is to period, T. Recall (or much better yet re-derive!) € ω = 2π /T = 2πf = angular frequency = rads/sec Remember also, frequency f = # cycles/ time = 1 cycle/(time for 1 cycle) = 1/T. In the previous sketched example, (the traveling sin wave) y(x) = A sin(kx) => y(x,y) = A sin(k(x-vt)) Let's think about the speed of this wave, v. Look at the picture: when it moves over by one wavelength, the sin peak at any given point has oscillated up and down through one cycle, which takes time T (one period, right?) That means speed v = (horizontal distance) / time = λ / T = λ f So the argument of the sin is € k(x − vt) = 2π λ (x − λ T t) = 2π ( x λ − t T ) = (kx - ωt) (Don't skim over any of that algebra! Convince yourself, this is stuff we'll use over and over) Summarizing: for our traveling sin wave, we can write it several equivalent ways: € y(x, t) = Asin k(x − vt)( ) = Asin 2π ( x λ − t T )       = Asin kx - ωt( ) The argument of the sign changes by 2π when x changes by λ, or t changes by T. The wave travels with speed v = λ / T = ω/k. (We’ll use these relations all the time!) Please check units, to make sure it’s all consistent. Technically, this speed v =ω/k is called the phase velocity, because it’s the speed at which a point of constant phase (like say the “zero crossing” or “first peak” or whatever) is moving. Soon we will discover, for some waves, another kind of velocity, the group velocity. Never mind for now!) I said that f(x-vt) represents a traveling wave – it should be reasonable from the above pictures and discussion, but let’s see a formal proof – (next page) ________________________________________________________ Ch. 1 notes, part1 17 of 26 8/27/2008 © University of Colorado, Michael Dubson (mods by S. Pollock ) Claim: y(x,t) = f(x ± vt) represents a rigidly shaped ("dispersionless") traveling wave. The upper "+" sign gives you a LEFT-moving wave. The - sign is what we've been talking about above, a RIGHT-moving wave. Proof of Claim: Consider such a traveling wave, moving to the right, and then think of a new, moving coordinate system (x',y'), moving along with the wave at the wave's speed v. Here, (x,y) is the original coordinate system, And (x’,y’) is a new, moving coordinate system, traveling to the right at the same speed as the wave. Let’s look at how the coordinates are related: Look at some particular point (the big black dot). It has coordinates (x,y) in the original frame. It has coordinates (x',y') in the new frame. But it's the same physical point. Stare, and convince yourself that x=x'+vt, and y=y' That's the cordinate transformation we’re after, or turning it around, x'=x-vt, and y'=y Now, in the moving (x',y') frame, the moving wave is stationary, right? (Because we're moving right along with it.) It's very simple in that frame: (In this frame, the (x,y) axes are running away from us off to the left at speed v, but never mind…) The point is that in this frame the wave is simple, y'=f(x'), at all times. It just sits there! If y'=f(x'), we can use our transformation to rewrite this (y'=y, x'=x-vt), giving us y=f(x-vt). This is what I was trying to prove: this formula describes the waveform traveling to the RIGHT with speed v, and fixed "shape" given by f. Ch. 1 notes, part1 20 of 26 8/27/2008 © University of Colorado, Michael Dubson (mods by S. Pollock ) Wave Nature of Light, and Interference Because of the superposition principle, waves add just as you would expect. That is, if you send two waves "down a string", they just add (or cancel) as simply as y(total) = y1+y2. This leads to constructive and destructive interference, one of the defining characteristic properties of waves. Following, you will find some notes from a freshman course reviewing interference, in case you’ve forgotten. (For instance, how is that you get an interference pattern from two slits?) The math and physics here will apply directly in quantum mechanics, because particles also exhibit wavelike behaviour. In general, wave-like effects with light are difficult to detect because of the small wavelength of visible light (400 nm (violet) → 700 nm (red)) The problem is even tougher with particles, it requires very special circumstances to demonstrate the wave nature of matter. So, in many situations, light behaves like a ray, exhibiting no obvious wave-like behavior. Newton (late 1600's) did not believe that light was a wave since he always observed ray- behavior. Wave-like behavior was not clearly observed until around 1800, by Young. Wave-like behavior of particles was not clearly observed until around 1925, in Davisson and Germer's experiment (using a crystal of nickel as a "grating") big hole D >> λ: ray-behavior Light passing through hole in wall: tiny hole D ≈ λ: wave-behavior wavefronts λ Ch. 1 notes, part1 21 of 26 8/27/2008 © University of Colorado, Michael Dubson (mods by S. Pollock ) Review of Constructive/Destructive interference of Waves: Consider 2 waves, with the same speed v, the same wavelength λ, (and therefore same frequency f = c / λ ), traveling in the same (or nearly the same) direction, overlapping in the same region of space: If the waves are in phase, they add ⇒ constructive interference If the waves are out of phase, they subtract ⇒ destructive interference If wave in nearly the same direction: Huygen's Principle: Each point on a wavefront (of given f, λ ) can be considered to be the source of a spherical wave. To see interference of light waves, you need a monochromatic (single λ) light source, which is coherent (nice, clean plane wave). This is not easy to make. Most light sources are incoherent (jumble of waves with random phase relations) and polychromatic (many different wavelengths). add subtract = + = + plane wave λ spherical wave (same λ, f ) speed c c c wall with infinitesimal hole Ch. 1 notes, part1 22 of 26 8/27/2008 © University of Colorado, Michael Dubson (mods by S. Pollock ) Young's Double slit experiment (1801) : What do you expect to see on the screen? If you believe light is a ray, then you expect to see 2 bright patches on the screen, one patch of light from each slit. But here is what you actually see: A series of bright and dark fringes: wave interference How do we explain this? Consider the 2 slits as 2 coherent point sources of monochromatic light. Two sources are coherent is they have the same wavelength λ (and therefore the same frequency f ) and they emit peaks and troughs in sync, in phase. Each slit (source) emits light in all forward directions, but let us consider only the parts of the waves heading toward a particular point on the screen. monochromatic plane wave λ d 2 slits screen c L d position x on screen Intensity I d x I Ch. 1 notes, part1 25 of 26 8/27/2008 © University of Colorado, Michael Dubson (mods by S. Pollock ) The intensity pattern on the screen looks like this: The angular width of the "central maximum" is € θ = 2λ /D . Notice that in the limit, D → λ (slit width becomes as small as the wavelength of light), the central max becomes so broad, that we get spherical wave behavior. Diffraction Grating A diffraction grating is an array of many narrow slits with a uniform inter-slit spacing d. A grating with "500 lines per cm" has a slit separation of A typical diffraction grating has thousands of slits. With exactly the same argument we used in the double-slit case, we see that maximum brightness occurs when € p.d.= d sinθ = mλ , m=0,1,2,…. The maxima occur at the same angles as with a double slit of the same d, but the peaks are much sharper and much brighter. sin θ ≅ θ I (intensity) λ / D 2λ / D −λ / D 0 2λ / D λ D > λ λ D ≈ λ d θ d sinθ grating Ch. 1 notes, part1 26 of 26 8/27/2008 © University of Colorado, Michael Dubson (mods by S. Pollock ) As N (the number of slits) increases, the width of each peak decreases (and gets brighter). Why? With just 2 slits, when we are near the maximum at the angle θ = λ / d, then the waves from the two slits are nearly in phase and we have nearly complete constructive interference and nearly maximum brightness. But with N-slits, when we are near the angle θ = λ / d, any two adjacent slits are nearly in phase, but the next slit over is a little more out of phase, and the next one over is even more out of phase. With many slits, if you are just a bit off the special angle for maximum brightness, the phase differences among the slits quickly add up and gives destructive interference. Another nice feature of the grating is that, with many slits for the light to get through, the pattern on the screen is brighter than in the double-slit case sin θ ≅ θ I (intensity) 0 λ / d 2λ / d −λ / d 3λ / d sin θ I 0 λ / d 2λ / d −λ / d 3λ / d d d ...... 2 slits: N-slit grating: 1st order 2nd order 3rd order