Simple Nature - Light and Matter

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f / The energy levels of a particlein a box, contrasted with those ofthe hydrogen atom.Approximate treatmentRather than leaping straight into a full mathematical treatment,we’ll start by looking for some physical insight, which will lead toan approximate argument that correctly reproduces the form of theBohr equation.A typical standing-wave pattern for the electron consists of acentral oscillating area surrounded by a region in which the wavefunctiontails off. As discussed in subsection 13.3.6, the oscillatingtype of pattern is typically encountered in the classically allowedregion, while the tailing off occurs in the classically forbidden regionwhere the electron has insufficient kinetic energy to penetrateaccording to classical physics. We use the symbol r for the radiusof the spherical boundary between the classically allowed and classicallyforbidden regions. Classically, r would be the distance fromthe proton at which the electron would have to stop, turn around,and head back in.If r had the same value for every standing-wave pattern, thenwe’d essentially be solving the particle-in-a-box problem in threedimensions, with the box being a spherical cavity. Consider theenergy levels of the particle in a box compared to those of the hydrogenatom, f. They’re qualitatively different. The energy levels ofthe particle in a box get farther and farther apart as we go higherin energy, and this feature doesn’t even depend on the details ofwhether the box is two-dimensional or three-dimensional, or its exactshape. The reason for the spreading is that the box is taken tobe completely impenetrable, so its size, r, is fixed. A wave patternwith n humps has a wavelength proportional to r/n, and therefore amomentum proportional to n, and an energy proportional to n 2 . Inthe hydrogen atom, however, the force keeping the electron boundisn’t an infinite force encountered when it bounces off of a wall, it’sthe attractive electrical force from the nucleus. If we put more energyinto the electron, it’s like throwing a ball upward with a higherenergy — it will get farther out before coming back down. Thismeans that in the hydrogen atom, we expect r to increase as we goto states of higher energy. This tends to keep the wavelengths ofthe high energy states from getting too short, reducing their kineticenergy. The closer and closer crowding of the energy levels in hydrogenalso makes sense because we know that there is a certain energythat would be enough to make the electron escape completely, andtherefore the sequence of bound states cannot extend above thatenergy.When the electron is at the maximum classically allowed distancer from the proton, it has zero kinetic energy. Thus when the electronis at distance r, its energy is purely electrical:[1] E = − ke2r884 Chapter 13 Quantum Physics
Now comes the approximation. In reality, the electron’s wavelengthcannot be constant in the classically allowed region, but we pretendthat it is. Since n is the number of nodes in the wavefunction, wecan interpret it approximately as the number of wavelengths thatfit across the diameter 2r. We are not even attempting a derivationthat would produce all the correct numerical factors like 2 and πand so on, so we simply make the approximation[2] λ ∼ r n.Finally we assume that the typical kinetic energy of the electron ison the same order of magnitude as the absolute value of its totalenergy. (This is true to within a factor of two for a typical classicalsystem like a planet in a circular orbit around the sun.) We thenhave[3]absolute value of total energy= ke2r∼ K= p 2 /2m= (h/λ) 2 /2m∼ h 2 n 2 /2mr 2We now solve the equation ke 2 /r ∼ h 2 n 2 /2mr 2 for r and throwaway numerical factors we can’t hope to have gotten right, yielding[4] r ∼ h2 n 2mke 2 .Plugging n = 1 into this equation gives r = 2 nm, which is indeedon the right order of magnitude. Finally we combine equations [4]and [1] to findE ∼ − mk2 e 4h 2 n 2 ,which is correct except for the numerical factors we never aimed tofind.Exact treatment of the ground stateThe general proof of the Bohr equation for all values of n isbeyond the mathematical scope of this book, but it’s fairly straightforwardto verify it for a particular n, especially given a lucky guessas to what functional form to try for the wavefunction. The formthat works for the ground state isΨ = ue −r/a ,where r = √ x 2 + y 2 + z 2 is the electron’s distance from the proton,and u provides for normalization. In the following, the resultSection 13.4 The Atom 885
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Fullerton, Californiawww.lightandma
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Contents0 Introduction and Review0.
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5.5 More About Heat Engines . . . .
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669.11.3 Magnetic Fields by Ampère
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The Mars Climate Orbiter is prepare
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temperatures and with many combinat
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idence that quarks have smaller par
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give clearer explanations.Finally,
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It can also be handy to have a rela
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The prefix centi-, meaning 10 −2
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goes like this:V = 1 3 Ah[1]A = πr
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the notation 10 0 to stand for one,
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calculation, 5.04 cm, was really no
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0.2 Scaling and Order-of-Magnitude
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to many times your own height. The
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g / 1. This plank is as long as it
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the front panels of the three violi
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Correct solution #4: The area of a
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here?” The scientific Mr. Spock w
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1. Don’t even attempt more than o
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Problem 10.mean, however, is define
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(d) Find the person’s acceleratio
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Albert Einstein, and his moustache,
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54 Chapter 0 Introduction and Revie
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a / Portrait of Monsieur Lavoisiera
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1.1.1 Problem-solving techniquesHow
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∆m = 0, where m is the total mass
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different substances will have diff
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c / Left: In a frame of referenceth
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f / Discussion question B.(discusse
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Self-check D.since the derivative o
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ProblemsThe symbols √ , , etc. ar
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difficult? ⊲ Solution, p. 932Key
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Heat energy can be convertedto ligh
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a new form of invisible “mystery
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f / A realistic drawing of Joule’
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numbers. With a purely numerical ap
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of gravity doesn’t change much if
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field. If the plane can start from
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m / Discussion question C.n / A hyd
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88 Chapter 2 Conservation of Energy
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A car drives over a cliff.new frame
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How long does it take to move 1 met
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a / Approximations to thebrachistoc
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a / An ellipse is circle that hasbe
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to deduce the general equation for
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to the mass of the object that inte
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The minimum velocity required for t
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and since sin θ dθ occurs in the
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that is deeper than r. Under the as
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2.3.6 ⋆ Evidence for repulsive gr
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a / A vivid demonstration thatheat
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ture. This is a very good question,
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Three functions with thesame curvat
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This looks like a cosine function,
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Problem 16.13 Anya and Ivan lean ov
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Problem 27.then we’d have U/m =
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37 Two springs with spring constant
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ExercisesExercise 2A: Reasoning wit
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128 Chapter 2 Conservation of Energ
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3.1 Momentum In One Dimensiona / Sy
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eing to the right. The initial mome
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3.1.3 Momentum compared to kinetic
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3.1.4 Collisions in one dimensiong
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could never do that again in a mill
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i / The highjumper’s body passeso
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where x 1 is the mass of the first
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3.1.6 The center of mass frame of r
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The airbag increases ∆tso as to r
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d / Two magnets exert forceson each
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x (m) t (s)10 1.8420 2.8630 3.8040
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3.2.4 Forces between solidsConserva
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Maximum acceleration of a car examp
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Discussion QuestionA Criticize the
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C A pool ball is rebounding from th
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forces, as if the hand were a pair
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This may not sound like an impressi
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stroke are both executed in straigh
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Accelerating a cart example 35If yo
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w / A wedge.x / Archimedes’ screw
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As in the preceding example, we hav
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that have the same frequency is a s
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around until I got this result, sin
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quality factor, Q, is defined as Q
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ight direction to add energy to the
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i / Example 45: a viola withouta mu
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Collapse of the Nimitz Freeway exam
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end up canceling out, however:Q =
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186 Chapter 3 Conservation of Momen
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c / Bullets are dropped and shot at
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the asteroid’s energy and boostin
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coordinate axes. Even though ∆x,
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of the three components,∆p x = 0
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A useless vector operation example
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Solving for the unknowns gives∆x
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o / Example 64.p / Adding vectors g
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How to generalize one-dimensional e
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needed to support the object in fig
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The easiest method is the one demon
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Incorrect solution #4:(same notatio
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The magnitude of the acceleration i
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dropped out like a trap door, showi
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af / Breaking trail, by WalterE. Bo
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know how to integrate with respect
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Problem 16.Problem 19.resistance.)1
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of gravity on Mars.(a) Find the tim
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partner force. (a) A swimmer speeds
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Problem 45the 0.4-gram masses would
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53 If you walk 35 km at an angle 25
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(b) Interpret this equation in the
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Problem 71.232 Chapter 3 Conservati
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77 A car accelerates from rest. At
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Problem 81.81 Complete example 71 o
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ExercisesExercise 3A: Force and Mot
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Exercise 3C: Worksheet on Resonance
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Exercise 3D: Vectors and MotionEach
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244 Chapter 3 Conservation of Momen
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An overhead view of apiece of putty
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inward toward the hinge will have n
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special case, we can choose to visu
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j / Two asteroids collide.k / Every
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Note that although the factors of 2
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Relationship between force and torq
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is straight down, which is perpendi
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⊲ All three objects in the figure
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aa / Example 10.to either side of e
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momentum tells us that L = mrv sin
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In the absence of any torque, a rig
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Radial acceleration at the surface
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The parallel axis theorem example 1
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differential. The result isarea ===
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of inertia as if the object was smo
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h / Moments of inertia of somegeome
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4.3 Angular Momentum In Three Dimen
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14 deg, and it points along an axis
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manner like this, then the definiti
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k / Example 27.torque to the left w
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We can also generalize the plane-ro
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Problem 1.Problem 6.Problem 8.Probl
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14 (a) The bar of mass m is attache
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Problem 22.22 The sun turns on its
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35 The nucleus 168 Er (erbium-168)
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ExercisesExercise 4A: TorqueEquipme
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pesky set of constraints on heat en
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mit more air into the cavity behind
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Pressure of lava underneath a volca
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h / A simplified version of anideal
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for us to construct a simple connec
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For the first time we have an inter
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a / 1. The temperature differencebe
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Carnot engines operating between a
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from A to B, he lets it by, but whe
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B When we run the Carnot engine in
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f / A phase space for a singleatom
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time, the energy sharing is very un
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will the the one that maximizes the
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If the gas is monoatomic, then we k
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5.4.4 The arrow of time, or “this
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5.4.6 Summary of the laws of thermo
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may be in the form of microscopic d
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significant amount of heat to flow
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while the smaller area under the bo
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7 (a) Determine the ratio between t
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338 Chapter 5 Thermodynamics
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end of this book, we’ll even see
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c / As the wave pattern passes the
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the wave move ahead faster and get
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k / Hitting a key on a pianocauses
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Our final result for the speed of t
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care, because the delay is the same
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An easy way to visualize this is in
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can be constructed as a superpositi
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⊲ Looking up the speed of light i
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scientists, to speak of the Big Ban
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6.2 Bounded WavesSpeech is what sep
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d / An uninverted reflection. There
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our intuitive expectation of strong
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valid solutions. In the following s
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j / In the mirror image, theareas o
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m / A pulse bounces backand forth.i
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q / Graphs of loudness versusfreque
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s / Standing waves on a rope. (PSSC
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“cavity” and “neck” parts o
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Problem 5.Problem 8.5 The figure sh
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Problem 16.16 A Fabry-Perot interfe
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c / Newton’s laws do not distingu
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f / The correspondence principlereq
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d / A Galilean version of therelati
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g / Three types of transformations
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system, velocities are always unitl
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p / Apparatus used for the testof r
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sumption, the assumption that it ma
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at rest relative to the water. But
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7.2.4 No action at a distanceThe Ne
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so that Alice can complete her moti
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time into different regions accordi
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position relative to the sun at exa
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7.2.7 ⋆ Four-vectors and the inne
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would depend on the relative veloci
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7.3 DynamicsSo far we have said not
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In this frame, as expected, the sma
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But this whole argument was based o
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meters per second, so converting to
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Expressing γ as ( 1 − v 2 /c 2)
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7.3.4 ⋆ ProofsThis optional secti
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these lines. For example, a car sit
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An Einstein’s ring. Thedistant ob
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two-dimensional universe as if it w
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h / 1. A ray of light is emittedupw
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object that that isn’t influenced
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it impossible for any observer to b
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a variant on the Penrose singularit
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in it. Instead, it is currently spe
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2 Astronauts in three different spa
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(d) Simplify your answer to part c
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Problem 25b. Redrawn fromVan Baak,
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ExercisesExercise 7A: The Michelson
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Exercise 7B: Sports in Slowlightlan
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448 Chapter 7 Relativity
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Exercise 7D: Misconceptions about R
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452 Chapter 7 Relativity
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sun, moon, stars, and planets were
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Two types of chargeWe can easily co
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the other acquires an equal amount
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to get the beam up to speed in the
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telling us that we know about matte
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the muddle. The row-and-column sche
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the force of air friction canceled
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ectly evaluate the implications of
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that they were indeed electrically
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C Thomson found that the m/q of an
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the object with a net positive char
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thickness of material the radioacti
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It was already known that although
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particle. It turned out that it was
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For example they could easily strip
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cording to Mosely, the atomic numbe
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that fly off to see what was inside
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solution was found by measuring the
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o / A nuclear power plant at Catten
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neutrons. In a nuclear fission bomb
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We can now list all four of the kno
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1. Our sun’s source of energy is
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a / The known nuclei, represented o
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killed, because the DNA becomes una
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e / Wild Przewalski’s horsesprosp
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Problem 1. Top: A realisticpicture
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12 The subatomic particles called m
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ExercisesExercise 8A: Nuclear decay
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saxophone, every technological tool
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Example 1Ions moving across a cell
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ut the original meaning was to trav
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Force depends only on position. Sin
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Here are a few questions and answer
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flow through it.For many substances
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superconductivity in metals that wo
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j / Example 9. In 1 and 2,charges t
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look like the usual resistor. The f
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9.1.5 Current-conducting properties
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attery acid becomes depleted of hyd
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9.2 Parallel and Series CircuitsIn
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to each resistance, resulting inI t
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where “...” means that the sum
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the two resistors in figure h/3.We
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Choice of high voltage for power li
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A complicated circuit example 20⊲
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Problem 2.ProblemsThe symbols √ ,
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Problem 16.only count that as one u
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knob turned all the way clockwise,
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A printed circuit board, likethe ki
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ExercisesExercise 9A: Voltage and C
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Exercise 9C: Reasoning About Circui
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556 Chapter 9 Circuits
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a / A bar magnet’s atoms are(part
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defined. The ship’s captain can m
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Reduction in gravity on Io due to J
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j / Example 3.self-check AFind an e
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a dipole moment — they are define
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10.2.1 One dimensionVoltage is elec
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The interpretation is that if you b
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Figure c shows some examples of way
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For large values of d, this express
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where Q is the total charge of the
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g / Example 12: variation of the fi
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corners to the disk and transform i
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10.4 Energy In Fieldsa / Two opposi
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self-check DWe can think of the qua
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of the inward field contributed by
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space, 2 while charge doesn’t, we
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a / The symbol for a capacitor.b /
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ounding each capacitor will be half
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in time:x ↔ qv ↔ Iself-check GH
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due to its own momentum. It perform
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or|V | =∣ LdIdt ∣ ,which in man
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the inductor resists such a sudden
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Example 26.Death by solenoid; spark
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ordering.( 1 √2+√ i ) 2 = √ 1
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x / The complex number e iφlies on
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Figure aa shows a useful way to vis
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Inductors tend to be big, heavy, ex
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The reason for using the trig ident
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a purely inductive or capacitive lo
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Resonance with damping example 33
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orE outward, on side 1 A + E outwar
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|E| = kq totalr 2 ,where r is the r
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y considering its point charges ind
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Discussion Questionsg / Discussion
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with charge, change the Coulomb con
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The symmetry between the two sides
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where dv is the volume of the cube.
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The three terms in the divergence a
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Problem 19.Problem 20.proton, for e
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the lightning strike.(b) Based on y
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units.(b) Verify that RC has units
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lating freely (without any driving
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ExercisesExercise 10A: Field Vector
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5. Now hook up the two solenoids in
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A large current is createdby shorti
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time I used it implicitly was in fi
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f / A standard dipole madefrom a sq
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and substituting q = λh and v = m/
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o / Magnetic forces cause abeam of
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electric field, a magnetic one, can
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u / In this scene from SwanLake, th
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so the total field in the z directi
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For the y component, we havee / A s
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We can pin down the result even mor
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i / The field of a dipole.where r i
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circulates around the y axis, so at
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to be, not where it is now. Coulomb
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We have found one specific example
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to a flagpole, we can cancel out a
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11.4 Ampère’s Law In Differentia
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y evaluating the field at the midpo
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i.e.,F = axˆx + byˆx + cˆx + dx
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k / A summary of the derivative, gr
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c / Detail from Ascending andDescen
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the magnet. Are these atomic curren
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field in her region of space has be
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A changing magnetic flux makes a cu
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nearly zero. By Faraday’s law, th
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c / An Ampèrian surface superimpos
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and Maxwell’s equations becomeΦ
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k / Red and blue light travelat the
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second, the zero-point is located a
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is the magnitude of the momentum ve
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Discussion question A.Discussion qu
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only penetrates to a very small dep
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elationship D = ɛE would actually
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esulting in cancellation.The opposi
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a low permeability, while the other
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n / A fluxgate compass.is externall
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6 Two parallel wires of length L ca
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an instant at the upper right, but
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Problem 25.20 Four long wires are a
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Problem 35.Problem 37.32 Verify Amp
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Problem 42.beam of light usually co
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(c) Discuss the relationship betwee
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(f) Use conservation of energy to r
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5. Now position yourself with your
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12.1.1 The nature of lightThe cause
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An image of Jupiter andits moon Io
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d / Two self-portraits of theauthor
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ultimate truth about light, but the
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• There is a tendency to conceptu
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self-check AEach of these diagrams
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m / Discussion question B.Discussio
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12.2 Images by ReflectionInfants ar
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Discussion QuestionA The figure sho
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down magnification is AB/DE. A repe
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i / A Newtonian telescopebeing used
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B Locate the images formed by two p
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12.3 Images, QuantitativelyIt sound
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⊲ The object and image angles are
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12.3.2 Other cases with curved mirr
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signs also have to be memorized for
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h / A diverging mirror in the shape
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j / Spherical mirrors are thecheape
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general type of eye that we share w
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shown on this graph and then attemp
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the mechanical model would predict
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that your calculator will flash an
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D Classify the examples shown in fi
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12.4.6 ⋆ Microscopic description
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12.5 Wave OpticsElectron microscope
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of a crystal? Sound waves are used
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j / Thomas Youngk / Double-slit dif
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object that diffracts it, so the tr
Page 790 and 791:
Although the equation λ/d = sin θ
Page 792 and 793:
things we’ve learned about diffra
Page 794 and 795:
apidly changing distances; on reuni
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6 The figure on the next page shows
Page 798 and 799:
14 Here’s a game my kids like to
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27 Suppose a converging lens is con
Page 802 and 803:
same depth, but not quite. [Check:
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Problem 42.42 Panel 1 of the figure
Page 806 and 807:
46 The figure below shows two diffr
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53 The beam of a laser passes throu
Page 810 and 811:
Problem 59.the ancient problem of i
Page 812 and 813:
3. Now imagine the following new si
Page 814 and 815:
Exercise 12B: Object and Image Dist
Page 816 and 817:
Exercise 12D: Double-Source Interfe
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Exercise 12E: Single-slit diffracti
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Exercise 12F: Diffraction of LightE
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energy, instead of being spread out
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correlated. If they have been playi
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small.The statement that the rule f
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But the y axis can no longer be a u
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for a long time once it gets there,
Page 832 and 833:
j / Calibration of the 14 C dating
Page 834 and 835: given by(probability of a ≤ x ≤
Page 836 and 837: k / In recent decades, a huge hole
Page 838 and 839: A wave is partially absorbed.c / A
Page 840 and 841: f / The hamster in her hamsterball
Page 842 and 843: How would you extract h from the gr
Page 844 and 845: F Does E = hf imply that a photon c
Page 846 and 847: of approaching this issue.)j / Bull
Page 848 and 849: We assume v is small enough so that
Page 850 and 851: the probability distribution will b
Page 852 and 853: trons that we have already used for
Page 854 and 855: equations of general validity are t
Page 856 and 857: wave carries high probability and w
Page 858 and 859: An infinite sine wave can only tell
Page 860 and 861: momentum implies a definite kinetic
Page 862 and 863: of snapshots would amount to a desc
Page 864 and 865: close are the electrons to the limi
Page 866 and 867: dimensional particle in a box, and
Page 868 and 869: Applying this to conservation of en
Page 870 and 871: the probability of making it throug
Page 872 and 873: Three dimensionsFor simplicity, we
Page 874 and 875: 1. Oscillations can go backand fort
Page 876 and 877: C The figure shows a skateboarder t
Page 878 and 879: a / Eight wavelengths fit aroundthi
Page 880 and 881: As shown by these examples, the unc
Page 882 and 883: e / The three states of the hydroge
Page 886 and 887: ∂r/∂x = x/r comes in handy. Com
Page 888 and 889: 50% of its time in each atom. It’
Page 890 and 891: tus to the z axis) and more memorab
Page 892 and 893: ut why does that have anything to d
Page 894 and 895: the data are only inaccurate due to
Page 896 and 897: 14 The photoelectric effect can occ
Page 898 and 899: Problem 25.(b) Sketch a graph showi
Page 900 and 901: conservation of energy and momentum
Page 902 and 903: Problem 43.43 On pp. 884-885 of sub
Page 904 and 905: ExercisesExercise 13A: Quantum Vers
Page 906 and 907: ⊲ First we convert the equation i
Page 908 and 909: Programming With PythonThe purpose
Page 910 and 911: Appendix 2: MiscellanyUnphysical
Page 912 and 913: 18 bestt = t19 c1 = bestc120 c2 = b
Page 914 and 915: p i + ∑ iThe spin theoremTheorem:
Page 916 and 917: Appendix 3: Photo CreditsExcept as
Page 918 and 919: Appendix 4: Hints and SolutionsHint
Page 920 and 921: Hints for Chapter 6Page 377, proble
Page 922 and 923: Answers to Self-Checks for Chapter
Page 924 and 925: Page 301: (1) Not valid. The equati
Page 926 and 927: Page 584:N −1 m −2 C 2 V 2 m
Page 928 and 929: the dashed ray.Page 748: You should
Page 930 and 931: direction it was initially going (i
Page 932 and 933: Page 52, problem 38: The cone of mi
Page 934 and 935:
Page 224, problem 37: (a) Spring co
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object, we have static friction, wh
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Page 549, problem 29:(a) Conservati
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Page 798, problem 19: (a) The objec
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.0.8 Notation and unitsquantity uni
Page 944 and 945:
surfaces. For comparison, a typical
Page 946 and 947:
elations:a = ∆v∆tx = 1 2 at2 +
Page 948 and 949:
we are moving along with the projec
Page 950 and 951:
the number of oscillations required
Page 952 and 953:
In general, the cross product of ve
Page 954 and 955:
Sound waves consist of increases an
Page 956 and 957:
signs for charge is that with this
Page 958 and 959:
know that there is a delay in time
Page 960 and 961:
is related byΦ = 4πkq into the ch
Page 962 and 963:
to use sophisticated models such as
Page 964 and 965:
Chapter 13, Quantum Physics, page 8
Page 966 and 967:
oth position and momentum, the Heis
Page 968 and 969:
Schrödinger’s, 864cathode rays,
Page 970 and 971:
unstable, 87equipartition theorem,
Page 972 and 973:
Joule, James, 73paddlewheel experim
Page 974 and 975:
Einstein’s early theory, 838energ
Page 976 and 977:
Nikola, 178tesla (unit), 652thermal
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Simple Nature - Light and Matter

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