Wave Phenomena Physics 15c

Transcription

1 Wave Phenomena Physics 15c Lecture 4 Introduction to Quantum Mechanics (H&L Chapter 14)

2 Administravia! This is our last lecture! No meeting during the Reading Period! Problem sets being graded! Will be returned soon Bear with us! Practice problems for the Final will be posted next week! Final Exam: 5/17 (Fri) 9:15 1:15, Sever 10! Covers Lectures #1 through #3, but not this one! 3/4 from post-midterm part! Make a cheat sheet! Bring a calculator

3 What We Did Last Time! Discussed coherence! Laser = (almost) coherent light source! Holography! Coherence length of incoherent light! Wave packet! Real waves are neither fully coherent nor incoherent! One can always use Fourier to find out! Understanding when waves behave coherently (and when they don t) is a key to mastering wave phenomena

4 Goals For Today! Introduction to Quantum Mechanics! Particle nature of light " Photon! Why do we think light is made of photons?! How can light be waves and particles at the same time?! How does this theory extend to normal particles?! Mainly historical background for the development of Quantum Mechanics! Wait for Physics 143a for the real stuff

5 Photoelectric Effect! When light hits a metal surface, electrons come out! It s called photoelectric effect! Taking an electron out of metal requires energy W! Light must be supplying this energy! You need certain intensity to get any electron out! Energy E e of the electron that came out should be larger for higher light intensity Energy received by an electron Ee + W I! Problem: experiments don t agree with our theory light Light intensity e

6 Experiment vs. Theory! Experiments have shown! Energy E e of electrons is independent of light intensity I! Expected to increase with I! At low I, fewer electrons come out, but some do come out! Expected no electrons below a certain intensity! Totally unexpected fact: E e depends on the frequency ν of the light as! Planck s constant Ee + W = hν h = J sec! Einstein s explanation: smallest unit of light (= photon) carries an energy E = hν

7 Photon as a Reality! R.A. Millikan s careful experiment confirmed the correctness of Einstein s theory beyond doubt! Existence of photon was clearly established! Einstein (19) and Millikan (193) won Nobel! But now we are back to the original particle-vs-wave argument! Young s experiment and countless other evidences support that light is made of waves! Millikan considered the photon theory quite unthinkable because of it s inconsistency with the interference and diffraction phenomena

8 Particle and/or Wave! You have probably heard/read that light behaves sometimes like particles and sometimes like waves! Is that a clear statement, or what?! We must know better than sometimes or like! Photoelectric effect gives us a starting point! Absorption of light by matter (electrons) occur with the minimum unit of energy E = hν! Generalize this into a hypothesis Light is emitted and absorbed as particles (photons), each particle carrying energy E = hν

9 Energy, Momentum, Intensity Energy Momentum Velocity =! Velocity = c for light hν h! If each photon has E = hν, then it should have p = = c λ! Using! h π, we can say E =! ω p =! k! We know for any waves! Intensity of light is determined by the number of photons emitted/received per unit time! Example: a laser pointer with P = 5 mw and λ = 650 nm 3 9 P Pλ = = = photons sec 34 8 hν hc ! That s quite a few photons

10 Young s Experiment! I said in Lecture #19 that Young s experiment had demonstrated light s wave-ness! Let s review it assuming that light is made of photons y d θ D! Number of photons arriving at screen is I π dy = N0 cos photons sec m hν λd Intensity I I cos π dy = 0 λd

11 Young s Experiment! Darken the light until photons arrive one at a time y! Photons hit the screen at random locations, one by one! Distribution still follows the interference! If you wait long enough, you get the same pattern! Each single photon is affected by the interference! Interference determines the probability P(y) of each photon arriving at location y π dy P( y) cos But how? λd

12 Single Slit Experiment! As a particle, each photon can go through only one slit! With one slit, however, interference does not occur y! Each photon is affected by the presence of the both slits! Conclusion: Each photon goes through both slits! OK with waves, but not with particles! Fractional photons have never been observed! We are going in circles P( y) P ( y) + P ( y) " upper " lower

13 What Is a Particle, Anyway?! Exactly what do we mean when we say something is a particle?! It is undivisible! It has a location (x, y, z) at any time t! The concept is an idealization/generalization from our experiences in observing ordinary objects! Watch a ball fly. Imagine it were infinitely small! There are implicit assumptions in it! We must reexamine its validity! Keywords: observing and watch

14 Observing a Ball! A ball flies as xt () = v0 xt 1 y() t = v0 yt gt! We watch the ball continuously " x(t), y(t) are continuous functions of time t! Watching = detecting the light scattered by the ball! Since light is made of photons, we know the location of the ball only when a photon hits it! Observation is not continuous! We may relax the definition of a particle a little A particle is found at a single location (x, y, z) whenever it is observed

15 Photons! From photoelectric effect, we assumed that light is emitted and absorbed as particles! We can detect emission/absorption of a photon by its energy! Photon source loses E = hν! Photon absorber gains E = hν! These processes occur at a particular point in space! Between the two events, we cannot detect photons! To detect a photon, we must absorb it! What a particle should do when nobody is watching?! More accurately: when no observation is possible even in principle?

16 Back to a Ball! Newtonian (continuous) view of motion is intuitive! Ball does what it does whether or not anybody is watching! But we know it s an approximation! We use light for observation! Light carries momentum " Pushes the ball! Trajectory is affected (very slightly!) by the light! We can ignore this effect in the limit of weak light! Photon theory breaks this approximation! Light cannot be made weaker than single photon xt () = v0 xt 1 y() t = v0 yt gt

17 Small Ball " Electron! Let s make the ball very small " An electron! Small target " Fewer photons hit it! Mass m is small " Photon s momentum is relatively large! Each hit by a photon changes the electron s momentum " Electron staggers around! You don t know its exact location until the next photon hits it! Continuous picture is an approximation of this random-walk! Valid only when p = mv of the object is much larger than the photon momentum h/λ

18 New Particle View! Once we accept that light is made of photons, we find that we don t know any better than A particle is found at a single location whenever it is hit by anther particle! What happens between two interactions is unknown! Our experience and intuition offer no help! Only way to know " Experiment! For light, we have enough experimental data that support wave-like propagation! New photon theory must satisfy them We ve pretty much painted ourselves into a corner

19 Photon Rules! Light is made of photons that obey the following rules:! A photon is emitted or absorbed in unit of E = hν! Each process takes place at a specific point in space! Exact location at which a photon is found is random! i.e. it cannot be predicted even in principle! Probability of finding a photon at a given location can be calculated using the wave equation! This is the light intensity! We reinterpret the EM wave equation as a probabilistic description of how photons travel from one place to another

20 Young s Experiment! Apply the rules to Young s experiment y! An electron vibrates and emits a photon! It s localized, i.e. position known accurately " Diffraction makes the direction of photon uncertain " Spherical waves! EM waves travel through the slits and interfere! A photon hits the screen with P( y) cos π dy λd We know this part

21 Occam s Razor William of Ockham (185? 1349)! Classical physics contains waves (light, sound, etc.) and particles that build up the objects! We found light was in fact made of particles whose motion is described by waves! Terribly awkward complication compared with the clean and intuitive Newtonian physics Plurality should not be posited without necessity Occam s razor! Theory should be as simple as possible to explain things! Can we restore some simplicity?! What if we assume that the motion of any particle is described by waves?

22 Particles as Waves! Try to describe the motion of a particle using waves! Suppose E =! ω and p =! k holds! Kinetic energy of a particle with mass m is! For plane waves Ψ ( x, t) = e i( kx ωt)! Solution has dispersion relation! Group velocity of such waves is ω k p cg = = = = v k m m t E = mv = p EΨ=! ωψ= i! Ψ p Ψ =! k Ψ =! Ψ 1 1 m 1 i! t! Ψ m Wave equation for a free particle Ψ= k!ω =! m! A wave packet will move with the velocity v

23 Wave Function! Represent a particle with a wave packet! Let s call it the wave function Ψ(x, t)! What is the physical meaning of Ψ?! Analogous to E(x, t) for photons! For photons, probability P( x, t) S E! Suppose P( x, t) =Ψ( x, t)! i.e. the square of the amplitude gives the probability of finding the particle at a particular point! For the wave packet above, 1 e < x< l Ψ ( xt, ) = l 0 x > i( kx ωt) l l P( xt, ) 1 l = 0 λ l Integrates to 1 < x < l l l x >

24 Uncertainty Principle! How does this wave packet behave?! It moves with c g = v! It has a finite length l! Position is not exactly known! Its Fourier integral gives uncertainty of wavenumber! Momentum p =! k is not exactly known! Again, we find the Uncertainty Principle x p = l! k = h! Good sign We are on the right track λ l k = π l Lecture #3

25 Scale of Uncertainty! In QM, nothing s position is exactly known! Sounds a bit scary! Doesn t it break good old Newtonian mechanics?! Consider a 0.1 kg ball rolling on a flat surface! Let s say we know the initial location down to x = 1 nm h! Using uncertainty principle p = 34 x h v = = = m s 9 m x ! This uncertainty in velocity accumulates to another 1 nm in 14 T = seconds = 4.8 million years

26 De Broglie Wavelength! Since particle = wave packet, it has a wavelength! From p =! k π h λ = = k p! It s sizable only for very small momentum " Very light particles, e.g. electrons De Broglie wavelength! Even with electrons, De Broglie wavelengths are too short to observe any wave phenomena! Well, that s why we never suspected that particles could also be waves, right?! We need to look at electrons moving in a very small space! How about a hydrogen atom?

27 Hydrogen Atom! Electron is circling around a proton! Electrostatic force = centripetal force r q 4πε r 0 = mv r! Waves circle around the orbit! Must close the loop at full circle = πr h ε0 π r = nλ r = n n is any integer π mq! Energy of the electron is E 1 q 1 0 q +q mq h 4πε 0r p = mv = λ = = h 4πε r p mq mq 4 = mv = = 8πε0r n 8ε 0h

28 Hydrogen Atom! Radius r is quantized to! Smallest radius is h ε0 11 r0 = = m π mq 0! This solves the problem of atom instability! Since there is a minimum orbit radius, the electron cannot lose all energy and fall into the proton! It can still absorb/emit photons by moving between different allowed radii! We can calculate the wavelengths from r = n h ε π mq +q r 4 E 1 mq = n 8 ε h 0 q

29 Hydrogen Lines! Electron moves from orbit n to m and emit a photon! Energy of the photon is! Wavelength satisfies 4 1 mq = R 3 H 8ε0 hc n m n m λ! R H is known as Rydberg constant! Spectral lines of hydrogen matches this formula! Formula was found by Johann Balmer in 1885! Niels Bohr explained it with waves in 1913! First sign of particle being waves E n = 1 n 4 mq 8ε h 0 4 hc mq 1 1 E = = En Em = λ 8ε h n m R H = m 0 7 1

30 Schrödinger s Equation! Take the wave equation for a particle! Let s think in 1-dimension! Suppose there is a potential V(x)! Total energy of the particle is 1 E = p + V( x) m!! 1 i t Ψ = m! We can rewrite the wave equation as 1 ( E V( x) ) Ψ = m! Ψ Schrödinger s Equation! Schrödinger s equation describes motion of a particle under the influence of any potential! Much of Physics 143a concerns how to deal with this Ψ V( x) x

31 Particle in a Box! Easy example: a box potential 0 0< x< L V( x) = elsewhere! Inside the box EΨ = 1 m! Outside the box! Require continuity of Ψ(x)! d dx Ψ Ψ = 0 ikx Ψ= Ae me k =±! Ψ (0) =Ψ ( L) = 0! Same as the guitar string " Standing waves ( ) sin n π Ψ x = A x me nπ = E = L! L L n π! ml x

32 QM and Wave Phenomena! Quantum Mechanics deals with the wave nature of everything! QM and the classical wave mechanics are separated by just one logical leap: Position of an object is not necessarily defined when it is not observed! Reevaluation of what it means to be a particle! Probabilistic interpretation of wave intensity! After this leap, all quantum phenomena are derived using the same tools we have learned in this course! Of course, you will have to learn the mathematical formalism (which is rather cool)! But the essence is already here

33 Summary! Much of the exciting/mysterious behavior of QM comes from basic nature of waves! Phase and interference! Real (plane wave) and virtual (tunneling) wavenumbers! Standing waves (bound states)! Spherical waves (scattering cross sections)! And more! Solid understanding of wave phenomena will help you through the more advanced subjects you will study in the near future