Quantum Mechanics: Wave-Particle Duality, Uncertainty, and Applications, Slides of Physics

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Quantum Mechanics Small things are weird Docsity.com The Quantum Mechanics View • All matter (particles) has wave-like properties – so-called particle-wave duality • Particle-waves are described in a probabilistic manner – electron doesn’t whiz around the nucleus, it has a probability distribution describing where it might be found – allows for seemingly impossible “quantum tunneling” • Some properties come in dual packages: can’t know both simultaneously to arbitrary precision – called the Heisenberg Uncertainty Principle – not simply a matter of measurement precision – position/momentum and energy/time are example pairs • The act of “measurement” fundamentally alters the system – called entanglement: information exchange alters a particle’s state Docsity.com Problems, cont. • What caused spectra of atoms to contain discrete “lines” – it was apparent that only a small set of optical frequencies (wavelengths) could be emitted or absorbed by atoms • Each atom has a distinct “fingerprint” • Light only comes off at very specific wavelengths – or frequencies – or energies • Note that hydrogen (bottom), with only one electron and one proton, emits several wavelengths Docsity.com The victory of the weird theory • Without Quantum Mechanics, we could never have designed and built: – semiconductor devices • computers, cell phones, etc. – lasers • CD/DVD players, bar-code scanners, surgical applications – MRI (magnetic resonance imaging) technology – nuclear reactors – atomic clocks (e.g., GPS navigation) • Physicists didn’t embrace quantum mechanics because it was gnarly, novel, or weird – it’s simply that the #$!&@ thing worked so well Docsity.com Let’s start with photon energy • Light is quantized into packets called photons • Photons have associated: – frequency,  (nu) – wavelength,  ( = c) – speed, c (always) – energy: E = h • higher frequency photons  higher energy  more damaging – momentum: p = h/c • The constant, h, is Planck’s constant – has tiny value of: h = 6.6310-34 J·s Docsity.com The Uncertainty Principle • The process of measurement involves interaction – this interaction necessarily “touches” the subject – by “touch,” we could mean by a photon of light • The more precisely we want to know where something is, the “harder” we have to measure it – so we end up giving it a kick • So we must unavoidably alter the velocity of the particle under study – thus changing its momentum • If x is the position uncertainty, and p is the momentum uncertainty, then inevitably, xp  h/2 Docsity.com Example: Diffraction • Light emerging from a tiny hole or slit will diverge (diffract) • We know its position very well (in at least one dimension) – so we give up knowledge of momentum in that dimension—thus the spread large opening: greater position uncertainty results in smaller momentum uncertainty, which translates to less of a spread angle small opening: less position uncertainty results in larger momentum uncertainty, which translates to more of a spread angle angle  p/p  h/px  h/hx = /x Docsity.com Diffraction in Our Everyday World • Squint and things get fuzzy – opposite behavior from particle-based pinhole camera • Eye floaters – look at bright, uniform source through tiniest pinhole you can make—you’ll see slowly moving specks with rings around them—diffraction rings • Shadow between thumb and forefinger – appears to connect before actual touch • Streaked street-lights through windshield – point toward center of wiper arc: diffraction grating formed by micro-grooves in windshield from wipers – same as color/streaks off CD Docsity.com Wave or Particle? Neither; Both; take your pick • Non-intuitive combination of wavelike and particle-like • Appears to behave in wavelike manner. But with low intensity, see the interference pattern build up out of individual photons, arriving one at a time. • How does the photon know about “the other” slit? – Actually, it’s impossible to simultaneously observe interference and know which slit the photon came through – Photon “sees”, or “feels-out” both paths simultaneously! • Speak of wave-part describing probability distribution of where individual photons may land Docsity.com The hydrogen atom • When the mathematical machinery of quantum mechanics is turned to the hydrogen atom, the solutions yield energy levels in exact agreement with the optical spectrum – Emergent picture is one of probability distributions describing where electrons can be • Probability distributions are static – electron is not thought to whiz around atom: it’s in a “stationary state” of probability • Separate functions describe the radial and angular pattern – http://hyperphysics.phy-astr.gsu.edu/hbase/hydwf.html The energy levels of hydrogen match the observed spectra, and fall out of the mathematics of quantum mechanics Docsity.com The angular part of the story s p d f These plots describe the directions in which one is likely to find an electron. They are denoted with quantum numbers l and m, with l as the subscript and m as the superscript. The s state (l =0,m=0) is spherically symmetric: equal probability of finding in all directions. The p state can be most likely to find at the poles (and not at all at the equator) in the case of (1,0), and exactly the opposite situation in the (1,1) state. Docsity.com