Albert Einstein 1905
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- Marcia Thompson
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1 Albert Einstein 1905
2 How Small is Small? How Fast is Fast?
3 Small => e.g. atomic size Fast => v~c c= the velocity of light
4 Our First Topic
5 v<c
6 Einstein s Postulates of Relativity: (Albert Einstein, 1905) Postulated in an attempt to explain the laws of Classical Electromagnetism (Maxwell s Equations, the constancy of the speed of light)
7 There are many-many experiments that prove the consequences of the two postulates of relativity and no experiments that disprove, so far,
8 Einstein s Postulates of Relativity: Michelson- Morley Experiment, light soucre, and the medium (Discussion sessions, see Appendix A) Definition of an Event Some Consequences of Einstein s Postulates: 1. Relative Simultaneity 2. Time Dilation 3. Length Contraction
9 1 st Postulate: The form of each physical law is the same in all inertial frames of reference INERTIAL FRAME =? The reference frame in which an object, experiencing zero net force a free object moves at constant velocity v = constant
10 2 nd Postulate: Light moves with the same speed (c) relative to all observers Anna measures: Speed of light = c? Bob measures: Speed of light = c and not v+c
11
12
13 Einstein s Postulates of Relativity: Light Souce, Medium and Michelson- Morley Experiment (Discussion sessions, see Appendix A) Definition of an Event Consequences of Einstein s Postulates: 1. Relative Simultaneity 2. Time Dilation 3. Length Contraction
14 Einstein s Postulates of Relativity: Light Souce, Medium and Michelson- Morley Experiment (Discussion sessions, see Appendix A) Definition of an Event Consequences of Einstein s Postulates: 1. Relative Simultaneity 2. Time Dilation 3. Length Contraction
15 Consequence 1: Relative Simultaneity, or The absence of absolute simultaneity Simultaneous Flash
16 Simultaneous Arrival => AN EVENT
17 Simultaneous Arrival The same EVENT For Bob as well!
18 2 nd Postulate THEN: Simultaneous Arrival AN EVENT For Bob as well! Simultaneous Emission is impossible for Bob
19 2 nd Postulate BUT THEN: Simultaneous Arrival AN EVENT For Bob as well! Simultaneous Emission is impossible for Bob
20 Both emission and arrival are simultaneous Arrival is simultaneous but EMISSION is not simultaneous
21 QUESTIONS?
22 Consequence 2: Time Dilation, or The absence of absolute time
23
24 The time for the light to return: t = 2H/c The time for the light to return: T > 2H/c Longer Path + 2 nd Postulate
25 PRECISELY => T = 2 L/c Longer Path (L) + 2 nd Postulate L 2 = H 2 + (v T/2) 2 t = 2H/c The time for the light to return : T > 2H/c L T = t (1 (v/c) 2 ) -1/2
26 Light Clock
27 Light Clock MUON
28 Consequence 3: Lorentz Length Contraction, or The absence of absolute distance L = L 0 (1 (v/c) 2 ) 1/2
29 Consequence 3: Lorentz Length Contraction, or The absence of absolute distance
30
31
32 WHAT IS WRONG IN THIS CARTOON?
33 There are many-many experiments that prove the consequences of the two postulates of relativity
34 Muon Lifetime Muons are created abundantly in elementary particle showers in the atmosphere, initiated by energetic cosmic rays (photons, particles and nuclei). Muons originate from decays of particles called pions (p) that are the primary products in these showers
35 Muon Lifetime Muon (m) is an elementary particle similar to electron, but heavier (will learn more in Chapter 11)
36 Muon Lifetime Atmosphere Cosmic Ray Neutrinos Particle shower Earth
37
38 CONCLUSION: According to Classical Physics, muons should not be able to reach the ground! BUT THEY DO
39 Consequence 3: Lorentz Length Contraction, or The absence of absolute distance L = L 0 (1 (v/c) 2 ) 1/2
40 CLOUD CHAMBER
41
42
43
44 QUESTIONS?
45
46 The distance between the creation and detection points
47 Light Clock MUON
48 PRECISELY => T = 2 L/c Longer Path (L) + 2 nd Postulate L 2 = H 2 + (v T/2) 2 t = 2H/c The time for the light to return: T > 2H/c L T = t (1 (v/c) 2 ) -1/2
49 SUMMARY Einstein s Postulates of Relativity: Michelson- Morley Experiment NO AETHER! Consequences of Einstein s Postulates: 1. Relative Simultaneity 2. Time Dilation 3. Length Contraction
50 QUESTIONS
51 New Topics: 1. Twin Paradox 2. Lorentz Transformations 3. Examples
52 Viewed by Carl: - Bob moves towards right at v = c - Anna moves towards left, at v = c - Carl sends laser pulses towards Bob - Bob reflects those pulses towards Anna => What is the velocity of each laser pulse received by Anna, when measured by Anna?
53
54
55 IMPORTANT: space and time coordinates mix together! (Not the case in Classical Physics)
56 Another way of expressing the Lorentz Transformation Equations 4-vectors x x + y y + z z xx + yy + zz The LENGTH is not INVARIANT, i.e. it is not conserved under Lorentz transformations
57 4-dimensional LENGTH Is there anything that IS INVARIANT under Lorentz Transformations? x x + y y + z z xx + yy + zz x x + y y + z z (ct )(ct ) = xx + yy + zz (ct)(ct)
58 Another way of expressing the Lorentz Transformation Equations 4-vectors
59
60
61 Another Example: Energy-Momentum
62 Another Example:
63 In the Classical Limit?
64 Classical Limit : v << c
65 Classical Limit : v << c << 1
66 Classical Limit : v << c ~ 1 << 1
67 In the Classical Limit First case: v = 0, g = 1 ~1 ~0 ~1 ~0 ~1 ~0 ~1 ~0 x ~ x t ~ t
68 In the Classical Limit General case: v << c, g ~ 1 ~1 ~1 ~0 ~1 ~1 ~0 x ~ x vt t ~ t OK
69 What Follows: Lorentz transformations of distance and time intervals The Twin Paradox, revisited Velocity Transformation (1.7) HOMEWORK: Lorentz Transformations and 4-vectors (1.11) The Doppler Effect (1.6)
70 x1 x2
71 Lorentz Transformation of Distances and Time Intervals IMPORTANT: space and time distances mix together
72 This time using Lorentz Transformations
73 Lorentz Transformation of Distances and Time Intervals 0 = 0 0
74 Lorentz Transformation of Distances and Time Intervals -2 m = 0 = -2 m 0-2 m 0
75
76 SUMMARY Einstein s Postulates of Relativity: Michelson- Morley Experiment NO AETHER! Consequences of Einstein s Postulates: 1. Relative Simultaneity 2. Time Dilation 3. Length Contraction EXPERIMENT
77 Next: Velocity Transformation (1.7) Is causality absolute? Energy and mass HOMEWORK: Lorentz Transformations and 4-vectors (1.11) The Doppler Effect (1.6)
78 Velocity Transformation? u u
79 Velocity Transformation u u u v + u
80
81 What about y and z coordinates? (x - direction of motion)
82 Lorentz Transformations NOTE that coordinates orthogonal to the direction of motion stay the same
83 NO (Lorentz) Length Contraction in directions other than along the direction of relative motion Lorentz Transformations NOTE that coordinates orthogonal to the direction of motion stay the same
84 Lorentz Transformations Relativistic velocity Transformations BUT ALL Components of the Velocity Vector Transform!
85 Lorentz Transformations Relativistic velocity Transformations BUT ALL Components of the Velocity Vector Transform! WHY?
86 Lorentz Transformations Relativistic velocity Transformations BUT ALL Components of the Velocity Vector Transform! WHY?
87 Because: The time transforms independently of the direction of motion, coordinates do not, and velocity combines both
88 Velocity Transformation u u u v + u
89
90 dt / dt u
91
92
93 d d
94
95 Parallel to the Direction of Relative motion Orthogonal to the Direction of Relative motion
96 Classical Limit? (both u and v << c) O.K.
97 u and/or v ~ c? Please try to figure out yourself
98 ux = c ux = c uy = 0 uz = 0
99 Is there Absolute Causality? Might cause precede effect in one reference frame but effect precede cause in different reference frame(s)? e.g. can someone see you first die, and then see you get born? Benjamin Button
100 Let s assume that the order of events is changed in some reference frame S t > 0 t' 0 Is that Possible? t and t are the time intervals between the same two events observed in S and S, respectively
101 = = 1 ' ' 2 2 t x c v t t t x c v t v v g g Using Lorentz transformations. 0 ' 0 > t t if then
102 = = 1 ' ' 2 2 t x c v t t t x c v t v v g g Using Lorentz transformations. 0 ' 0 > t t if then t v c x t x c v >
103 = = 1 ' ' 2 2 t x c v t t t x c v t v v g g Using Lorentz transformations. 0 ' 0 > t t if then t v c x t x c v > Impossible
104 Is there Absolute Causality? Might cause precede effect in one reference frame but effect precede cause in different reference frame(s)? e.g. can someone see you first die, and then see you get born?
105 Viewed by Carl: - Bob moves towards right at v = c in his 10 m long spaceship. He wears a clock that blinks every 10 seconds. - Anna stands next to Carl in her 20 m long spaceship, and doesn t move. She wears a clock that blinks every 10 seconds, too. a. What is the length of Bob s spaceship, when measured by Anna? b. What is the length of Anna s spaceship when measured by Bob? c. What is the time interval between two blinks of Bob s clock, when measured by Carl? d. What is the time interval between two blinks of Anna s clock, when measured by Bob? e. What is the time interval between two blinks of Anna s clock, when measured by Carl?
106 Light Cone
107 New topics: 1. Relativistic Dynamics: 2 E = mc 2. General Theory of Relativity skipped END of Relativistic Physics (Next time Quantum Physics)
108 Another way of expressing the Lorentz Transformation Equations 4-vectors x x + y y + z z xx + yy + zz The LENGTH is not INVARIANT, i.e. it is not conserved under Lorentz transformations
109 4-dimensional LENGTH Is there anything that IS INVARIANT under Lorentz Transformations? x x + y y + z z xx + yy + zz x x + y y + z z (ct )(ct ) = xx + yy + zz (ct)(ct)
110 Another way of expressing the Lorentz Transformation Equations 4-vectors
111
112 Another Example: Energy-Momentum
113 Another Example:
114 ' ' ' ' c p c p c p E c p c p c p E c m INVARIANT c p c p c p E z y x z y x z y x = = =
115 ' ' ' ' c p c p c p E c p c p c p E c m INVARIANT c p c p c p E z y x z y x z y x = = = ) ( ) ( ) ( mc c p E mc c p E mc c p E = = = E = TOTAL ENERGY Revolutionary Concept
116 What about 2 E = mc? E = p 2 c 2 (mc 2 ) 2
117 E = INTERNAL ENERGY (when p = 0) 2 E = mc E = TOTAL ENERGY E = p 2 c? 2 (mc 2 ) 2
118 E = INTERNAL ENERGY 2 E INTERNAL = mc E = TOTAL ENERGY E = p 2 c 2 (mc 2 ) 2
119 Expressions for (total) Energy and Momentum of a particle of mass m, moving at velocity u
120 Classical Limit 1 p mu 1 NEW E mu 2 mc 2 2 FAMILIAR kinetic energy
121 Kinetic Energy = KE
122 Energy Matter
123 Energy Matter Our Experiment: Muons are CREATED From the Energy of Cosmic Rays
124
125 Energy Matter Atomic Bomb (Chapter 10): Energy is CREATED From the Mass of Nuclei (Internal energy is transformed into kinetic energy)
126 g u
127 g u
128 General Theory of Relativity
129 Acceleration Profound Link Gravitational Force
130 Cannon Free-falling Physicist Dog
131 Cannonball Trajectory:?
132 Cannonball Trajectory: Relative to the Earth = Parabola Relative to the Falling Physicist = Straight Line
133 Einstein s Principle of Equivalence In any small, freely falling reference frame anywhere in our real, gravity-endowed Universe, the laws of physics must be the same as they are in an inertial reference frame in an idealized, gravity-free universe.
134 Special Theory of Relativity The two postulates: BUT: earth Accellerating frames
135 Special Theory of Relativity: General Theory of Relativity: locally
136 Special Theory of Relativity: General Theory of Relativity: Deals also with Accelerating - LOCALLY INERTIAL FRAMES Deals exclusively with globally INERTIAL FRAMES - v = constant locally
137
138
139 Curved Space-time: (Intercontinental flights)
140 Experimental verification of General Relativity: Precise Measurements of Orbits
141 QUESTIONS
142 Lecture Today Electromagnetic Waves behaving like Particles (Chapter 3 (or 2-old book)) 1. Black Body Radiation (Max Planck 1900) 2. The Photoelectric Effect (Albert Einstein 1905) 3. The Production of X-Rays (Wilhelm Roentgen 1901) 4. The Compton Effect (Arthur Compton 1927) 5. Particle-Antiparticle Pair Production (Carl Anderson 1932) 6. Discussion: a Wave or a Particle (?) > Duality
143 (More in Appendix C) Black Body Radiation (Max Planck 1900)
144 The Planck s Black-Body Radiation Law: The Energy (E) in the electromagnetic radiation at a given frequency (f) may take on values restricted to E = n h f where: n = an integer (1, 2, 3, ) h = a constant ( Planck Constant )
145 Experimental Fact: E = nhf BUT, why should the energy of an Electromagnetic wave be Quantized? (n= integer) No explanation until 1905 AWave is a Continuous Phenomenon
146 Experimental Fact: E = nhf BUT Why should the energy of an Electromagnetic wave be Quantized? (n= integer) No Explanation until 1905 Albert Einstein The Photoelectric Effect A wave is a Continuous Phenomenon
147 metal The Photoelectric Effect (Albert Einstein)
148 metal The Photoelectric Effect (Albert Einstein 1905)
149 (More in Appendix C) Black Body Radiation (Max Planck 1900)
150 The Photoelectric Effect (Albert Einstein 1905) metal Phenomenon observed long time before Einstein, and something very strange was observed:
151 The Photoelectric Effect (Albert Einstein 1905) Even With Very strong light of low frequency metal NO electrons ejected
152 The Photoelectric Effect (Albert Einstein 1905) Even With Very strong light of low frequency metal NO electrons ejected
153 The Photoelectric Effect (Albert Einstein 1905) Even With Very strong light of low frequency metal NO electrons ejected
154 The Photoelectric Effect (Albert Einstein 1905) Even With Very-Very weak light intensity, but of high enough frequency Electrons ejected
155 The Photoelectric Effect (Albert Einstein 1905) Even With Very strong light of low frequency metal NO electrons ejected
156 Albert Einstein 1905
157 Planck s Law ( E = n h f ) Photoelectric Effect (Threshold frequency) Albert Einstein proposed: The light is behaving as a collection of particles called photons each of them having energy E = h f
158 The Photoelectric Effect (Albert Einstein 1905) Even With Very-Very weak light intensity, but of high enough frequency E E photon beam = hf = nhf Electrons ejected What happens is that 1 PHOTON ejects 1 ELECTRON
159 Photoelectric Effect E photon = hf KE max = hf
160 Also known at that time: To free an electron from the metal, one has to pay a certain amount of energy the Work Function
161 QUIZ 3: a. At what speed will the kinetic energy of a particle of mass m = g (gram) be equal to its internal energy? b. What is the ratio of that kinetic energy and the kinetic energy of the same particle moving at v = 100 km/h?
162 E = INTERNAL ENERGY 2 E INTERNAL = mc E = TOTAL ENERGY E = p 2 c 2 (mc 2 ) 2
163 Expressions for (total) Energy and Momentum of a particle of mass m, moving at velocity u
164 Classical Limit 1 p mu 1 NEW E mu 2 mc 2 2 FAMILIAR kinetic energy
165 Kinetic Energy = KE
166 Photon s Mass = 0 E = INTERNAL ENERGY 2 E INTERNAL = mc E = TOTAL ENERGY E = p 2 c 2 (mc 2 ) 2
167 Photoelectric Effect E photon = hf KE max = hf
168 Electromagnetic Waves behaving like Particles (Chapter 2) 1. Black Body Radiation (Max Planck 1900) 2. The Photoelectric Effect (Albert Einstein 1905) 3. The Production of X-Rays (Wilhelm Roentgen 1901) 4. The Compton Effect (Arthur Compton 1927) 5. Particle-Antiparticle Pair Production (Carl Anderson 1932) 6. Discussion: a Wave or a Particle (?) > Duality
169 The Production of X-Rays (Wilhelm Roentgen 1901) (The reverse of the Photoelectric Effect) CLASSICAL physics: Radiation covers entire spectrum Bremsstrahlung
170 SURPRISE: Experiments indicate a cutoff wavelength: Frequency f, Energy E=hf 1 photon 1 electron (?) 1 electron 1 photon (?)
171 The Production of X-Rays (Wilhelm Roentgen 1901) (The reverse of the Photoelectric Effect) CLASSICAL physics: Radiation covers entire spectrum Bremsstrahlung
172 Experiments indicate a cutoff wavelength: Frequency f, Energy E=hf INDEED:
173 Frequency f INDEED: 1 electron 1 photon
174 E = INTERNAL ENERGY 2 E INTERNAL = mc E = TOTAL ENERGY E = p 2 c 2 (mc 2 ) 2
175 Photon s Mass = 0 E = INTERNAL ENERGY 2 E INTERNAL = mc E = TOTAL ENERGY E = p 2 c 2 (mc 2 ) 2
176 Photoelectric Effect E photon = hf KE max = hf
177 The Compton effect (Arthur Compton 1927) Hypothesis: Experiment?
178 momentum energy
179
180 Energy and Momentum Conservation
181
182 The Compton Effect Photons carry momentum like particles and scatter individually with other particles
183 QUESTIONS?
184 Electromagnetic Waves behaving like Particles (Chapter 3, or 2-old book) 1. Black Body Radiation (Max Planck 1900) 2. The Photoelectric Effect (Albert Einstein 1905) 3. The Production of X-Rays (Wilhelm Roentgen 1901) 4. The Compton Effect (Arthur Compton 1927) 5. Particle-Antiparticle Pair Production (Carl Anderson 1932) 6. Discussion: a Wave or a Particle (?) > Duality
185 Particle-Antiparticle Pair Creation
186 Bubble Chamber
187
188 Previous Lectures Electromagnetic Waves behaving like Particles (Chapter 3, or 2 in old book) PHOTONS 1. Black Body Radiation (Max Planck 1900) 2. The Photoelectric Effect (Albert Einstein 1905) 3. The Production of X-Rays (Wilhelm Roentgen 1901) 4. The Compton Effect (Arthur Compton 1927) 5. Particle-Antiparticle Pair Production (Carl Anderson 1932) 6. Discussion: a Wave or a Particle (?) > Duality
189 QUESTIONS?
190 Electromagnetic Waves behaving like Particles (Chapter 2) PHOTONS Black Body Radiation The Photoelectric Effect The Production of X-Rays The Compton Effect PHOTONS E = hf PHOTONS p = hf/c = h/l Particle-Antiparticle Pair Production
191 Double-slit Diffraction Experiment light
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193
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195
196
197 INDIVIDUAL PHOTON HITS
198 Although diffraction of light is a wave phenomenon, there is no smooth distribution of light in the diffraction pattern, but the pattern is rather formed of many individual hits of particles the photons
199 A single photon DOES NOT get disintegrated in the Diffraction process to make a smooth diffraction pattern
200 QUESTIONS?
201 Electromagnetic waves (light) Particles (photons)? Waves Massive Particles
202
203
204 Very strange
205 Very strange Like a Wave
206 electrons? Waves Massive Particles
207
208
209 X-ray photons or electrons
210 electrons Waves Massive Particles
211
212 Properties of Matter Waves De Broglie (1929) Wavelength l = h p Frequency f = E h
213 ~1905 ~1930
214 Properties of Matter Waves Amplitude = function of (x,t) ( x, t ) Probability Amplitude
215 Amplitude = function of (x,t) ( x, t ) Probability Amplitude The probability density to find a particle at coordinate x, at time t ( x, t) 2
216 The probability to find a particle in an interval x P = dx ( x, t) 2 Integrate over x
217 x t P dx ( x) 2 =
218 l = h p Why l is so Small? Because h is very-very Small
219 QUESTIONS?
220 DEFINITIONS k 2p Wave Number l 2 p = 2 T pf Angular Frequency h 2p
221 k 2p 2 p = 2 pf h l T 2p p h = = l k E = hf =
222 The probability to find a particle in an interval x P = dx ( x) 2 Integrate over x How does the Probability Wave Move? Equation of Motion for y?
223 The Free-Particle Schrodinger Wave Equation y ( x, t ) Probability Wave Function * y ( x, t) y( x, t) = y( x, t) 2 Probability Density y ( x, t ) a complex function
224 ), ( ), ( ), ( 2 1 t x i t x t x y y y Probability Density =? ), ( ), ( ), ( 2 1 * t x i t x t x y y y Complex Conjugate 1 i 1 2 = i
225 ), ( ), ( ), ( 2 1 t x i t x t x y y y Probability Density ), ( ), ( ), ( 2 1 * t x i t x t x y y y 1 = i = = ), ( ), ( ), ( * 2 t x t x t x y y y = = y y y y y y y y i i ii y y y =y ), ( ), ( ), ( t x t x t x y y y =
226
227 The Plane Wave
228 Is the Plane Wave a solution of the Schrodinger Equation?
229 Is the Plane Wave a solution of the Schrodinger Equation?
230 Is the Plane Wave a solution of the Schrodinger Equation?
231 Is the Plane Wave a solution of the Schrodinger Equation?
232 Is the Plane Wave a solution of the Schrodinger Equation? 2 ( mv) E = = 2m mv 2 2
233 The Magnitude of a Plane Wave Constant in space and time! Constant Probability Density
234 Kinetic energy Plane wave = constant (x,t)
235 The Uncertainty Principle
236 The Uncertainty Relations in 3 Dimensions
237
238
239 The Uncertainty Relations and the Fourier Transform Any wave may be expressed mathematically as a superposition of plane waves of different wavelengths and amplitudes
240 A Single-Slit Spectral Content
241
242 Gaussian Wave form 1 x p = h x 2 bar = minimum uncertainty
243 Bound States - Chapter 3 Some Simple Cases The End 1. A Review of Classical Bound States (4.4) 2. The Schrodinger Equation for Interacting Particles (4.1) 3. Stationary States (4.2) 4. Well-Behaved Functions and Normalization (4.3) 5. Case I: Particle in a Box Infinite Potential Well (4.5) 6. Expectation Values, Uncertainties and Operators (4.6) 7. Case II: The Finite Potential Well (4.7) 8. Case III: The Simple Harmonic Oscillator (4.8)
244 The Schrodinger Equation for Interacting Particles A Particle Interacting With What? F=mg
245 The Schrodinger Equation for Interacting Particles A Particle Interacting With What? Simplification: The Concept of Potential (replaces all individual particle-particle interactions with a single smooth potential)
246 The Schrodinger Equation for Interacting Particles For free particles Try to add potential energy U(x)
247
248 The Schrodinger Equation for Interacting Particles and for Stationary Potentials U = U ( x) U U ( t)
249 Key Assumption: Factorization of the wave function Spatial Part Temporal Part What happens with the Schrodinger equation?
250
251 t and x are independent
252 Temporal part Total wave function
253 Temporal part Total wave function The probability density is time-independent Stationary States
254 The spatial part of y(x,t) The time-independent Schrodinger equation: NOTE: y(x) is Real, Spatial part but y(x,t) is Complex, because (t)=e -i t
255 Well-behaved wave functions Normalization of y(x,t) Smoothness of y(x,t)
256 Normalization of y(x,t) The particle must be somewhere in the universe at any time (the total probability should be = 1)
257 Smoothness of y(x,t) 1. Continuity of y(x,t) 2. Continuity of (dy(x)/dx)
258 Smoothness of y(x,t) y (x) x Discontinuity in y(x)
259 Smoothness of y(x,t) y (x) x Discontinuity in y(x) ~ k = 1 y ( ) ( x ) e 2p y ikx dx Extremely large k (or short l) > > Infinite Momentum impossible
260 Summary Temporal part Normalization of y(x,t) Smoothness of y(x,t) Total wave function Spatial part
261 Case I: Particle in a box Infinite Potential well
262 Case I: Particle in a box Infinite Potential well
263 1. Continuity of y(x,t) 2. Continuity of (dy(x)/dx)
264
265 General solution for region I
266 positive
267
268 x y F = D = 0
269
270 C = G = 0
271 A =? B =? From Smoothness:
272
273
274 Only certain Energy Levels are Allowed
275 Smoothness of y(x,t) Normalization of y(x,t):
276
277 Infinite Potential Well: E 1 0
278 Bound States Some Simple Cases (Chapter 5 The End (old book 4)) A Review of Classical Bound States (4.4) The Schrodinger Equation for Interacting Particles (4.1) Stationary States (4.2) Well-Behaved Functions and Normalization (4.3) Case I: Particle in a Box Infinite Potential Well (4.5) 1. Case II: The Finite Potential Well (4.7) 2. Case III: The Simple Harmonic Oscillator (4.8) 3. Read at Home: Expectation Values, Uncertainties and Operators (4.6)
279 Finite Potential Well: Infinite Potential Well:
280 Case 2: The Finite Potential Well A, B, C, G, E n =?
281 Requirement: 1. Continuity of y(x,t) 2. Continuity of (dy(x)/dx)
282
283
284 continuity at x = 0 continuity at x = L
285 continuity at x = 0 C A =, B = k C continuity at x = L
286 /(- )
287 /(- ) Transcendental Equation impossible to solve analytically
288 Transcendental Equation impossible to solve analytically, but NUMERICAL Solution is possible (e.g. graphing solution )
289 Graphing solution
290 Graphing solution
291 Graphing solution Parabola ~ k 2
292 Graphing solution Parabola ~ k 2
293 Graphing solution Solutions left=right Parabola ~ k 2
294 Graphing solution Solutions left=right
295 Graphing solution Solutions left=right
296 Graphing solution U U
297
298 Finite Potential Well Infinite Potential Well: Finite probability - penetration
299 Finite Potential Well Infinite Potential Well:
300 Case 3: The Simple Harmonic Oscillator
301 The Importance of the Harmonic Oscillator
302
303
304 Unbound States (Chapter 6 (old book 5))
305 Unbound States (Chapter 6 (old 5)) 5.1 Obstacles and home 5.2 Decay and other applications 5.3 Particle-wave propagation 5.4 The Classical home
306 Obstacles and Tunneling
307 Potential Wall
308 E > U Potential Wall
309 Obstacles Potential and Tunneling Wall
310 Obstacles Potential and Tunneling Wall
311 < 0
312 incident transmitted reflected incident reflected
313 incident reflected transmitted
314 90 protons Decay
315 Decay Only?
316 Decay
317 Hydrogen Atom (Chapter 7 (old book - 6)) 6.1 The Schrodinger Equation in 3 dimensions 6.2 The 3D Infinite Potential well 6.3 Toward the Hydrogen Atom 6.4 Central Forces
318 Schrodinger Equation in 3 Dimensions KE operator In 3-D
319 Schrodinger Equation in 3 Dimensions
320 Probability Density in 3 Dimensions
321 Stationary States f(r) f(t)
322 Stationary States temporal part Timeindependent Sch. equation
323 Stationary States in a 3-D Box
324 Stationary States in a 3-D Box Factorization
325 f(x) f(y) f(z) f(x,y,z)
326
327
328
329 Stationary States in a 3-D Box The Solution
330 Stationary States in a 3-D Box Solution
331 example : L 1, L = 2, L = x = y z 3
332 Stationary States in a 3-D Box Degeneracy
333
334
335
336 Toward the Hydrogen Atom The Problem: proton electron Energy loss due to Bremsstrahlung (because of centripetal acceleration)
337 Toward the Hydrogen Atom electron Stationary state? proton
338 Towards the Hydrogen Atom
339 Toward the Hydrogen Atom Spherical Polar Coordinate system
340 Toward the Hydrogen Atom
341 Toward the Hydrogen Atom: Schrodinger equation in Spherical Polar Coordinates
342
343 Summary: Hydrogen Atom
344 The meaning of l Angular part of Schrodinger Eq. Units of angular momentum L ~ Magnitude of Angular Momentum
345 Quantized so far: The projection of angular momentum to z- axis m l = MAGNETIC QUANTUM NUMBER The magnitude of (orbital) angular momentum l = ORBITAL QUANTUM NUMBER
346
347 The Uncertainty Principle
348 Angular Momentum Quantization
349 Angular Momentum Quantization
350 Energy levels of a hydrogen atom: homework
351 Ground State Radial Distribution of the Electron Probability Density in a Hydrogen Atom
352 Bohr Radius How Small is Small?
353 Quantum numbers accidental degeneracy Because of 1/r
354 Traditional naming scheme
355
356 Lecture Today
357 L GroundStat e llecture ( l 1) = 0Today s = h p L s = l( l 1) = 0 L p = l( l 1) = 2
358 L GroundStat e = l( l 1) = 0 Ground State: The Electron is NOT Orbiting around the proton Classical Physics: The Electron is Orbiting around the Proton
359 Spectral Lines
360 New Topic Spin and Atomic Physics The concept of SPIN The Stern-Gerlach Experiment Toward the Exclusion Principle for Fermions
361 Orbiting in Classical Physics Magnetic Dipole Moment
362 Surprise: Real Experimental Result Ground State -> l = 0 -> L = 0 -> F = 0 (???)
363 SPIN Like for L: s the quantum number of SPIN Intrinsic property of a particle
364 For an electron: s = 1/2
365 Spin Orientation
366 All Elementary Particles: either Bosons or Fermions Bosons Fermions
367 The Pauli Exclusion Principle for Fermions Sz = 1 2 S z = 1 2
368 The Exclusion Principle for Fermions Sz = 1 2 S z 1 = 2
369 The Exclusion Principle for Fermions Sz = 1 2 S z 1 = 2 S z = 1 2
370 The Exclusion Principle for Fermions Sz = 1 2 S z 1 = 2 S z = 1 2
371 Let s create multi-electron atoms Our Toolkit: The Schrodinger Equation Quantization of Energy Levels The Exclusion Principle Progressive Occupation of Energy Levels
372 Hydrogen H
373 Helium He
374 Lithium Li (3 electrons)
375 Energy Levels Hydrogen H, ground state
376 He 2-electrons Energy Levels
377 Energy Levels Neon (Ne) 10-electrons
378 Neon Ne
379 Energy Levels Fluorine: 9-electrons
380 Fluorine F Like Ne with one electron (and proton) less
381 Fluorine F Like Ne with one electron (and proton) less A fluorine atom would gladly accept one more electron, to look more like Ne That is why it is chemically very reactive
382 What matters for Chemical Properties is the state of the most loose electrons
383 What maters for Chemical Properties is the state of the most loose electrons The other, more strongly bound electrons are merely passive placeholders
384 The traditional naming scheme Principal quantum number n The number of electrons in that subshell
385
386 Noble gasses
387 Noble gasses He + 1 electron Li: 1s 2 2s 1
388 VALENCE
389 First Ionization Energy
390 Bonding of Atoms Chapter 10 (old book - 9) 9.1 When atoms come together 9.5 Energy Bands 9.6 Conductors, Insulators, Semiconductors
391 2 Atoms Come Together Electrons become shared
392 N Atoms Come Together Formation of energy Bands
393
394 Molecules Kovalent Bond
395
396 Molecules Ionic Bond Extremely Loose
397 Molecules Ionic Bond Extremely Loose Xe Ne Electrostatic Attraction: Cs(+) F(-)
398 Cs: First Ionization Energy
399 Form Energy Levels to Energy Bands 4- atom crystal
400
401 N- atomic States Formation of ~continuous Bands
402 Band Structure Central role in condensed matter physics!
403 Form Energy Levels to Energy Bands 4- atom crystal
404 Fermi-Dirac Distribution
405 Insulators, Conductors, Semiconductors Semiconductors insulators with a small energy gap (Eg < 2 ev)
406 conductor Band filling
407 conductor Band filling semiconductor Energy gapno electrons!
408 Electron-hole creation and conduction Current conducted by electrons and holes
409 Semiconductor Doping Doping elements: As, Al, Ga, B,
410
411 Semiconductor Doping n-type p-type Doping elements: As and B
412 Semiconductor Doping Intrinsic semiconductor e.g. Si Extrinsic (doped) semiconductor
413 n-type p-type
414 Semiconductor Devices Equilibrium depletion zone
415 p-n Junction Bias
416 p-n Junction Bias and the DIODE Current can flow through the diode in only one direction
417 n-p-n structure and the TRANSISTOR Signal amplification Very weak (low doping)
418 The Light Emitting DIODE (LED) Recombination: Electron-hole annihilation photon
419
420 LASER
421 LASER
422 The Laser Diode MASSIVE Recombination: LED Very high doping level needed, and REFLECTION Mirrors needed Laser
423 The Genealogy of Forces Unified Theories Physics & Cosmology
424 The forces [MeV] L
425 Bosons: Fermions [MeV] L
426 How the elementary particles interact Exchange of mediators (bosons)
427 The range of a force and the mass of the mediator Zero Mass Infinite Range Finite Mass Finite Range
428 The range of the force and the mass of the mediator Photon - the mediator of the Electromagnetic force Mass = 0 Range = infinite (~1/r potential)
429 The range of a force and the mass of the mediator Finite Mass Finite Range Zero Mass Infinite Range m c x range mc mc c t c x mc E t mc E t E m
430 Relativistic Treatment is Mandatory Because particles may be created and annihilated (internal energy plays a role; mc 2 ) Schrodinger Equation NON-relativistic Relativistic Quantum Mechanics
431 Schrodinger Equation NON-relativistic
432 Relativistic Quantum Mechanics: The Klein-Gordon Wave Equation The Dirac Equation (for Fermions) The concept of SPIN follows naturally
433 The Atomic Nucleus All have spin s = ½ Fermions Fermi-Dirac Statistics
434 Let s create multi-nucleon nuclei Our Toolkit: The Schrodinger Equation Quantization of Energy Levels The Exclusion Principle Progressive Occupation of Energy Levels
435 The Helium Nucleus 2 protons + 2 neutrons
436 The Hydrogen Isotopes Heavy Water: D 2 O
437 Rutherford Experiment
438 Rutherford Experiment Back-scattering Electrostatic repulsion
439 Rutherford Back-scattering and The measurement of the Size of a Nucleus r r = 0 No return - the nuclei collide Kinetic Energy = Potential Energy at Closest Approach r
440 The Radius of a Nucleus From Rutherford back-scattering, and other types of measurements, the radius of a nucleus: Atomic number 1.2 fm (femtometer) 1 fm = m
441 The Volume of a Nucleus Spherical Shape Assumed (an excellent approximation) The volume is proportional to the number of nucleons (A)
442 The Density of Nuclear Matter The density is constant, and: independent of A ~ everywhere inside a nucleus
443 Constant Density of Nuclear Matter Nucleons are closely packed in an ~incompressible liquid-like droplet
444 Binding Why would protons and neutrons stay together in a nucleus? Strong Force
445 Forces The real Strong Force acts among the constituents of a NUCLEON quarks and gluons
446 The potential well due to the (residual) strong force Nucleon-nucleon distance Nucleons cannot penetrate each other Finite-range interaction
447 Building a nucleus: Deuteron Deuteron is barely bound!
448 Building a nucleus: Deuteron Note: no Coulomb repulsion Deuteron is barely bound!
449 Building a nucleus: No Coulomb repulsion
450 Building a nucleus: With Coulomb repulsion
451 Building a nucleus: with Coulomb repulsion Nucleons are Fermions!
452 Building a nucleus: with Coulomb repulsion Without Coulomb interaction
453 Binding Energy per Nucleon
454 Building a nucleus: Shell Structure Nucleons are Fermions!
455 Building a nucleus: Shell Structure Nucleons are Fermions!
456 Binding energy per (the least bound) electron in an atom (~10 ev) Binding energy per nucleon in a nucleus (~ MeV)
457 Building a nucleus: Binding Energy
458 Building a nucleus: Large Nuclei N > Z
459 Building a nucleus: the Curve of Stability N > Z
460 Away from the Curve of Stability (?)
461 Radioactivity Alpha He nucleus Beta - Electron Gamma Photon Fission breakup into two nuclei
462 Beta-decay first evidence for the existence of Neutrinos One more particle in a decay!
463 Spontaneous Fission
464 Neutron-induced Fission
465 Fission
466 Neutron-induced Fission Chain Reaction
467 Example: n + U Applications: Nuclear Bombs and Nuclear Power
468 Controlled Fission (Chain) Reaction Reactors Power reactors Research Reactors Converters
469
470
471
472
473
474 Fission Explosives U-235 and Pu-239 Exponential increase in energy release of a supercritical assembly Without control Nuclear ( atomic ) Bomb
475 Fusion Fusion Nucleons like to be together
476 Fusion the creation of light elements in Stars He + He + He 12 C
477 The composition of Nucleons 3 Quarks (u = up, d = down )
478 Many Interesting Experiments Search for Proton Decay Cosmic Rays of unexpectedly high energies (10 20 ev) SUSY (Super Symmetric Particles), annihilation in space etc. Distribution of the Cosmic Background Radiation The mass of the neutrinos What is Dark Matter? Dark Energy?
479 The End Thank You! It was a great pleasure
480
481
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