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Lecture 22 Heisenberg Uncertainty Relations 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 Introduction The Nature of the Wave function Ψ • 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 wave associated with them with wavelength determined Ψ2 tells us the probability of finding the particle at a 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”. • Experiment: Davisson-Germer (1927) studies electron • Ψ is well-defined at every point in space and time scattering from crystals - see interference that corresponds exactly to the predicted de Broglie wavelength. • The Schrodinger equation: Master Equation of Quantum • But Ψ cannot be measured directly -Its square gives 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 Ψ2 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 Ψ2 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 electron doing repeated experiments: where ∆p and ∆x refer to the ∆p ∆x ≥ ~ h uncertainties in the measurement of momentum and position. (~ h means “roughly equal to h” -- Lower probability Higher probability will give exact factors later) to find electron to find electron far from nucleus near nucleus Since p = mv, this also means ∆v ∆x ≥ ~ h/m 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 the idea of de Broglie that particles also have wave but its wavelength λ and momentum p = h/ λ is not character well-defined , i.e., the wave does not correspond to a • What are properties of waves definite momentum or wavelength. • Waves are patterns that vary in space and time • A wave is not in only one place at a give time - it is “spread out” λ 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 How can one construct a Localized Wave? Localized Wave Packet Ψ(x) • An extended periodic wave is x • In order to have a wave localized in a region of a state of definite momentum space ∆x, it must have a spread of momenta ∆p • Note: This wave is not localized! • The smaller ∆x, the larger the range ∆p required • Problem: How to describe a localized wave? • Leads to the Heisenberg Uncertainty Principle: • Solution: Add other waves to form a “wave packet”. higher momentum ∆p ∆x ≥ ~h average momentum • Can understand from de Broglie’s Equation or lower momentum λ = h / p p λ = h • 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 more the wave spreads out in time. • The wave packet spreads out!. • Why isn’t that called an “uncertainty principle” and given philosophical hype? One particle has probability of being found in a range of • Because nothing is really “uncertain”: the wave is positions and velocities definitely spread out. If you measure where it is, you get the answer: “It is spread out.” Range of probable velocity around an average At a later time: • This is different in quantum mechanics where Range of probable me each particle is not spread out. Only the Ti positions spreads out in probability of where the particle will be found is time because of spread spread out. in velocities 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/2π = (1/2) h ∆E ∆t ≥ (1/2) h/2π = (1/2) h • The constant “h-bar” has approximately the value • In SI units: 2 ∆E ∆t ≥ 10 −34 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 • Examples: (See March Table 17-1) (atom size) conservation! • electron: m ~ 10-31 Kg, ∆x ~ 10 -10 m, ∆v ~ 10 7 m/s Can predict position in future for time ~ ∆x/∆v~ 10 -17 s • Particles can be in Virtual States for short times • 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”! 1020 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. • Quantum mechanics tells us only the probability ∆p ∆x ≥ (1/2) h ∆E ∆t ≥ (1/2) h 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 Decay one does not know the position accurately • Energy E can be measured to great accuracy - but Energy only if one measures over a long time - i.e., one does not know the time for an event accurately 3 Lecture 22 Heisenberg Uncertainty Relations Not everything is uncertain! II Not everything is uncertain! III • The Schrodinger wavefunction Ψ for a particle is • The most accurate clocks precisely defined for each quantum state. in the world are “atomic clocks” that produce light • The function Ψ2(r) , the probability to find the of frequency ν which is particle a distance r from the nucleus, is well- precisely defined beacuse defined. energies in atoms are quantized to definite values • The energies of quantum states of atoms are and E = h ν. extremely well-defined and measured to great • Time standard for the precision - often measured to accuracies of United States is a clock the 1/1,000,000,000 % = 10-12 uses Cesium atoms • But any one measurement will find an electron in Cs Light the atom at some particular point - the theory only atom predicts the probability of finding the electron at Uncertainty less than 2 x 10-15, which means it would neither gain any point nor lose a second in 20 million years! History of Atomic Clocks at NIST http://www.boulder.nist.gov/timefreq/cesium/atomichistory.htm Not everything is uncertain! IV Important Quantum Effects in Our World • Because the energy is so certain it means I Lasers something else must be very uncertain Usually light is emitted by an excited atom is • Example: the location of an electron in the atom. in a a random direction - light from many atoms Any one measurement will find an electron in the goes in all directions – direction and energy atom at some particular point - the theory only have uncertainty for light emitted from any one atom predicts the probability of finding the electron at any point – we cannot predict at which point Photons Excited Atoms Lower probability Higher probability to find electron to find electron far from nucleus near nucleus Nucleus What is special about a Laser?? Important Quantum Effects in Our World Important Quantum Effects in Our World I Lasers - continued I Lasers - continued Lasers work because of the quantum properties Since photons cannot be distinguished, which
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