Lecture 22 Heisenberg Uncertainty Relations Does God play Dice? Announcements Heisenberg’s Uncertainy Principle • Schedule: • Last Time: Matter waves : de Broglie, Schrodinger’s Equation March (Ch 16), Lightman Ch. 4 • Today: Does God play Dice? Probablity Interpretation, Uncertainty Principles ∆ p ∆ x ≥ h/2 March (Ch 17) Lightman Ch 4 • Next time: Measurement and Reality - Does observation determine reality? - Meaning of two-slit experiment - Schrodinger’s Cat March (Ch 18), Lightman Ch 4 ∆ E ∆ t ≥ h/2 • Essay/Report • Last time: Short statement of subject your essay due h = “hbar” • Monday, December 8: Essay due = h/2 π The Nature of the Wave function Ψ Introduction • Last Time: Matter Waves • Max Born proposed: • Theory: de Broglie (1924) proposes matter waves Ψ is a probability amplitude wave! • assumes all “particles” (e.g. electrons) also have a Ψ 2 tells us the probability of finding the particle at a wave associated with them with wavelength determined by its momentum, λ = h/p. given place at a given time. • Bohr’s quantization follows because the electron in an atom is described by a “standing electron wave”. Ψ is well-defined at every point in space and time • Experiment: Davisson-Germer (1927) studies electron • scattering from crystals - see interference that corresponds exactly to the predicted de Broglie wavelength. • But Ψ cannot be measured directly - Its square gives • The Schrodinger equation: Master Equation of Quantum Mechanics: like Newton’s equation F=ma in classical the probability of finding a particle at any point in mechanics. space and time • But what waving? • Today: Probability is intrinsic to Quantum Mechanics; Heisenberg Uncertainty Principle Probability interpretation for Ψ 2 The Uncertainty Principle • The location of an electron is not determined by Ψ . The probability of finding it is high where Ψ 2 is large, and small where Ψ 2 is small. • Werner Heisenberg proposed that the basic ideas on quantum mechanics could be understood in terms • Example: A hydrogen atom is one electron around a of an Uncertainty Principle nucleus. Positions where one might find the where ∆ p and ∆ x refer to the electron doing repeated experiments: ∆ p ∆ x ≥ ~ h uncertainties in the measurement of momentum and position. (~ h means “roughly equal to h” -- will give exact factors later) Lower probability Higher probability to find electron to find electron far from nucleus near nucleus ∆ v ∆ x ≥ ~ h/m Since p = mv, this also means Nucleus (Neglecting relativistic effects - OK for v << c) 1
Lecture 22 Heisenberg Uncertainty Relations Uncertainty Principle and Matter Waves The Nature of a Wave - continued • The uncertainty principle can be understood from • Example of wave with well-defined position in space but its wavelength λ and momentum p = h/ λ is not the idea of de Broglie that particles also have wave character well-defined , i.e., the wave does not correspond to a definite momentum or wavelength. • What are properties of waves • Waves are patterns that vary in space and time • A wave is not in only one place at a give time - it is “spread λ out” 0 Most probable position • Example of wave with well-defined wavelength λ and Position x momentum p = h/ λ , but is spread over all space, i.e., its position is not well-defined Localized Wave Packet How can one construct a Localized Wave? Ψ( x) • An extended periodic wave is • In order to have a wave localized in a region of x a state of definite momentum space ∆ x, it must have a spread of momenta ∆ p • The smaller ∆ x, the larger the range ∆ p required • Note: This wave is not localized! • Problem: How to describe a localized wave? • Leads to the Heisenberg Uncertainty Principle: • Solution: Add other waves to form a “wave packet”. ∆ p ∆ x ≥ ~h higher momentum • Can understand from de Broglie’s Equation average momentum λ = h / p p λ = h or lower momentum • The minimum range ∆ x is of order the wavelength λ which requires a range of momenta ∆ p at least Wave packet = Sum as large as ∆ p ∆ x ≥ ~h Is This Like a Classical Wave? Time Evolution of the Wave Packet • Suppose one measures the position and velocity of • Yes --- And No! a particle at one time - each has some uncertainty • A classical wave also spreads out. The more • What happens at later times? localized the region in which the wave is confined, • The wave packet spreads out!. the more the wave spreads out in time. • Why isn’t that called an “uncertainty principle” O n e p a r t i c l e h a s p r o b a b i l i t y and given philosophical hype? o f b e i n g f o u n d i n a r a n g e o f • Because nothing is really “uncertain”: the wave is p o s i t i o n s a n d v e l o c i t i e s definitely spread out. If you measure where it is, you get the answer: “It is spread out.” Range of probable velocity around an A t a l a t e r t i m e : average • This is different in quantum mechanics where Time R a n g e o f p r o b a b l e each particle is not spread out. Only the p o s i t i o n s s p r e a d s o u t i n o f s p r e a d probability of where the particle will be found is t i m e b e c a u s e i n v e l o c i t i e s spread out. 2
Lecture 22 Heisenberg Uncertainty Relations Examples of Uncertainty Principle Uncertainty Principle in Energy & Time • The more exact form of the uncertainty principle is • Similar ideas lead to uncertainty in time and energy ∆ p ∆ x ≥ (1/2) h/2 π = (1/2) h ∆ E ∆ t ≥ (1/2) h/2 π = (1/2) h • • The constant “h-bar” has approximately the value 2 ∆ E ∆ t ≥ 10 −34 In SI units: h = 10 -34 Joule seconds • In quantum mechanics energy is conserved over So in SI units: 2m ∆ x ∆ v ≥ 10 −34 long times just as in classical mechanics • But for short times particles can violate energy (atom size) • Examples: (See March Table 17-1) conservation! • electron: m ~ 10 -31 Kg, ∆ x ~ 10 -10 m, ∆ v ~ 10 7 m/s • Particles can be in Virtual States for short times Can predict position in future for time ~ ∆ x / ∆ v ~ 10 -17 s • pin-head m ~ 10 -5 Kg, ∆ x ~ 10 -4 m, ∆ v ~ 10 -25 m/s • Things that are impossible in classical mechanical Can predict position in future for time ~ ∆ x / ∆ v ~ 10 21 s are only improbable in quantum mechanics! (greater than age of universe!) Quantum Tunneling Example of Quantum Tunneling • In classical mechanics an object can never get • The decay of a nucleus is the escape of particles over a barrier (e.g. a hill) if if does not have bound inside a barrier enough energy • The rate for escape can be very small. • In quantum mechanics there is some probablility • Particles in the nucleus “attempt to escape” for the object to “tunnel through the hill”! 10 20 times per second, but may succeed in • The particle below has energy less than the energy escaping only once in many years! needed to get over the barrier tunneling tunneling Radioactive Decay Energy Energy Example of Probability Intrinsic to Not everything is uncertain! I Quantum Mechanics • The uncertainly principle only says that the product • Even if the quantum state (wavefunction) of the of the uncertainties in two quantities must exceed a nucleus is completely well-defined with no minimum value uncertainty, one cannot predict when a nucleus will decay. ∆ p ∆ x ≥ (1/2) h ∆ E ∆ t ≥ (1/2) h • Quantum mechanics tells us only the probability per unit time that any nucleus will decay. • Demonstration with Geiger Counter • Momentum p can be measured to great accuracy - tunneling but only if one measures over a large region - i.e., Radioactive one does not know the position accurately Decay Energy • Energy E can be measured to great accuracy - but only if one measures over a long time - i.e., one does not know the time for an event accurately 3
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