Download Einstein`s Postulate of Relaavity

2/8/11 PH300 Modern Physics SP11
Last *me: •  Rela*vis*c momentum and energy •  Review EM Waves Today: •  Review SR •  Intro to quantum/photoelectric effect Thursday: Exam I (in class); Homework 05 assigned(?) “Quantum Mechanics is the greatest intellectual accomplishment of the human race.” -­‐ Carl Wieman, Nobel Laureate in Physics 2001 2/8 Day 8: Ques*ons? Review SR Intro to Quantum, Photoelectric Effect Next: Exam I (In-­‐class) 1 Photoelectric Effect Review informa*on: h]p://*nyurl.com/PH300SP11 Don’t Stress: Can drop lowest exam score (see syllabus). 3 Space>me •  Galileo transforma*on: Classical rela*vity •  Michelson-­‐Morley à 'c' is same in all iner*al frames •  Einstein's postulate: Incompa*ble with Galilean rela*vity! •  Consequences were '*me dila*on' and 'length contrac*on' à Lorentz transforma*on à Velocity transforma*on •  Space*me interval: Invariant under Lorentz transforma*on Rela>vis>c Mechanics Re-­‐defini*on of important physical quan**es to preserve conserva*on laws under Lorentz transforma*ons: -­‐ Momentum -­‐ Force -­‐ Kine*c Energy -­‐ Rest Energy -­‐ Total Energy Velocity Transforma>on (a consequence of the Lorentz transforma*on) If S’ is moving with speed v in the posi*ve x direc*on rela*ve to S, then the space*me coordinates for two events in S’ are related to the space*me coordinates in S via: Lorentz transforma>on (rela*vis*c) Δx ′ = γ (Δx − vΔt)
Δy′ = Δy
Δz′ = Δz
v
Δt ′ = γ (Δt − 2 Δx)
c
Velocity transforma>on (rela*vis*c) ux=Δx/Δt … * An iner*al frame is a reference frame that is not accelera*ng. 2 What exactly did we do during the last couple weeks? Bring: •  pencil and eraser •  calculator •  ONE 3x5 card with whatever you want to write on both sides •  Some constants and equa*ons provided (see website) The speed of light in vacuum is the same in all iner>al frames of reference. Photoelectric effect, photons Special Rela*vity Format of exam: •  EM waves & special rela*vity (Lecture 07) •  10 M/C @ 2pts + 3 Long Answer @ 10 pts = 50 pts Einstein’s Postulate of Rela>vity Next Week: u 'x =
ux − v
1 − ux v / c2
1 2/8/11 Proper Time Proper Length Proper >me: Time interval Δt = t2 – t1 between two events (x1,y1,z1,t1) and (x2,y2,z2,t2) measured in the frame, in which the two events occur at the same spa*al coordinates, i.e. *me interval that can be measured with one clock. Proper length: Length of object measured in the frame where it is at rest (use a ruler) v ... -­‐3 -­‐2 -­‐1 0 1 2 3 ... Same spa*al coordinates means: x1 = x2 y1 = y2 z1 = z2 Simultaneity, >me dila>on & length contrac>on Space>me Interval As a consequence of Einstein's postulate of rela*vity ('The speed of light is the same in all iner2al frames of reference') we came to interes*ng conclusions: Say we have two events: (x1,y1,z1,t1) and (x2,y2,z2,t2). Define the space>me interval (sort of the "distance") between two events as: With: Δx = x2 − x1
Δy = y2 − y1
Δz = z2 − z1
Δt = t 2 − t1
-­‐ Rela*vity of simultaneity -­‐ Time dila*on -­‐ Length contrac*on Space*me interval All these effects are summarized in a set of equa*ons: The space>me interval has the same value in all reference frames! i.e. Δs2 is “invariant” under Lorentz transforma>ons. The Lorentz transforma*on Defini>ons: We redefined several physical quan**es to maintain the conserva*on laws for momentum and energy in special rela*vity. Rela*vis*c momentum: Rela*vis*c force: Rela*vis*c Energy: p=m
Important Rela>on γu =
(This results from the previous defini*ons) 1
1 − (u c )
dr
dr
= γ um = γ um u
dτ
dt
dp d
F=
= (γ u m u )
dt dt
E = γ u mc 2 = K + mc 2
(K: kine*c energy) 2
E = γ u mc 2
Total energy of an object: Rela*vis*c momentum of an object: Energy – momentum rela*on: p = γ um u
( E )2 − ( pc )2 = ( mc 2 )
2
E
Momentum of a massless par*cle: p =
c
Velocity of a massless par*cle: p=
E
c
→
γ mu =
γ mc 2
c
→
u=c
2 2/8/11 Example from some previous exam Velocity transforma>on A high-­‐speed train is traveling at a velocity of v = 0.5c. The moment it passes over a bridge it launches a cannon ball with a velocity of 0.4c straight up (as seen by the train conductor). What is the velocity of the ball right ater it was launched as seen by an observer standing on the bridge? A high-­‐speed train is traveling at a velocity of v = 0.5c. The moment it passes over a bridge it launches a cannon ball straight up (as seen by the train conductor) with a velocity of 0.4c. What is the velocity of the ball right ater it was launched as seen by an observer standing on the bridge? A]ach reference frame S to the train: Observer is in frame S' traveling from right to let (v is nega*ve!!) Situa*on seen by the onlooker on the bridge: Cannon ball right ater firing the cannon. Now use the velocity transforma*on: v ux = 0 uy = 0.4c y S x Velocity transforma>on Velocity transf. u'x = 0.5c u'y = 0.346c S' v = -­‐0.5c x' Part 2: Quantum Mechanics }
ux = 0 uy = 0.4c y y' S x y' S' v = -­‐0.5c x' Quantum Mechanics is the greatest intellectual accomplishment of human race. Courtesy of IBM -­‐ Carl Wieman, Nobel Laureate in Physics 2001 Light is a wave  interference! Is light a wave or a par*cle? How do we know? The definite check that light IS a wave Observe interference! The beginnings of quantum mechanics 3 2/8/11 Electromagne*c waves carry energy Emax=peak amplitude Light shines on a black tank full of water. How much energy is absorbed? c X E(x,t) = Emaxsin(ax-­‐bt) Intensity = Power = energy/time ~ (Eavg)2
area area
~ (amplitude of
wave)2
~ Emax
2
Intensity only depends on the E-­‐field amplitude but not 19 on the color (frequency) of the light! Classical waves: Intensity ~ Emax2 |Eave|2 X |Eave|2 X “Why do higher frequency gamma rays … carry more energy that lower frequency radio waves, but frequency has nothing to do with intensity?” Energy carried by a beam of light vs. Energy in a single quantum par*cle of light The photoelectric effect (~1900) The photoelectric effect is a phenomenon in which electrons are emi]ed from a metal as a consequence of the electrons absorbing energy from light. vs. The photoelectric effect The results from the photoelectric effect where inconsistent with the classical view of EM theory. (Discovered 1887 by Hertz, Explained by Einstein (using some of Plank's ideas) 1905. Nobel prize: 1921) The effect was only observed with UV light, but not so with red or IR light. à Effect is frequency dependent!? Einstein proposed: "…the energy in a beam of light is not distributed con*nuously through space, but consists of a finite number of energy quanta, which are localized at points, which cannot be subdivided, and which are absorbed and emi]ed only as whole units.” But Maxwell told us that the light intensity doesn’t depend on frequency! (Intensity only depends on |E|2) He took the energy of these single units to be hf, as proposed earlier by Planck. The Farmer and the Seeds The Photoelectric Effect (a parable of scien*fic reasoning) Photoelectric effect demonstrates that light also behaves like a par*cle. Energy comes in par*cle-­‐like chunks, the basics of quantum physics. (energy of one chunk depends on frequency, a beam of light has MANY chunks & the energy of the beam is the sum) Next two classes: I. Understand the P.E. experiment and what results you would expect if light were a classical wave (like physicists at the *me expected the experiment should give). II. What experimental results it actually did give. III. The implica*ons/interpreta*on of the results. •  A seed is a square with some dots on it. •  The farmer always plants 4 seeds in a group. First Group: Second Group: •
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•  Farmer observes # of sprouts each group produces. •
24 4 2/8/11 The Farmer and the Seeds First Group: •
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4 sprouts •
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The Farmer and the Seeds Another seed grouping the farmer tried: Second Group: • •
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6 sprouts •
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•  What do you think? What schemes can we come up with that would predict the number of sprouts based on the dot pa2ern showing when the seeds are planted? •
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What is the prediction of each of the different
schemes the class has come up with for this seed
grouping?
25 26 Ques*ons from this story: Another seed grouping the farmer tried: Possible Schemes 1.  Totally Random (???) 2.  (Number that is even) = 2 Sprouts 3.  [(Sum of all Numbers) – 3] / 2 = 2 Sprouts 1.  How could we decide if any of these three schemes is the correct one? 2 sprouts! 2.  If the farmer had to wait to plant more seeds, are there reasons we might in the mean*me favor one scheme over another? 3.  How do we know if we’ve figured out all the possible schemes? Possible Schemes 1.  Totally Random (Numbers aren’t related to # of sprouts) 2.  (Largest Number) = (# of Sprouts) 3.  4.  5.  6.  … 4.  Where did these schemes we’ve been discussing come from? (Note: This ques*on is not about the elements of the schemes, but the decisions as to what elements to use and how to use them.) 28 COMPETING THEORIES A MODEL EVEN NUMBER THEORY (# OF SPROUTS) = 2 SPROUTS 2 sprouts! COMPLEX MATHEMATICAL THEORY (# OF SPROUTS) = [(1 + 1 + 2 + 3) -­‐ 3] / 2 = 2 SPROUTS 29 30 5 2/8/11 INTERPRETATION Summary EVEN NUMBER THEORY (# OF SPROUTS) = 2 SPROUTS 2 sprouts! COMPLEX MATHEMATICAL THEORY (# OF SPROUTS) = [(1 + 1 + 2 + 3) -­‐ 3] / 2 = 2 SPROUTS •  Scien*sts “make up” theories to explain the evidence they see. •  These theories are constrained by experiment. •  We can’t always open up the seed and look inside. Have to make inferences from indirect evidence. •  A theory with a plausible mechanism is more convincing than a rote algorithm. •  The more different cases our theory works on, the more we believe it. •  But it could always be wrong… What do those numbers really represent? 31 32 6