Chapter 1. Introduction
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1 I. Classical Physics Chater 1. Introduction Classical Mechanics (Newton): It redicts the motion of classical articles with elegance and accuracy. d F ma, mv F: force a: acceleration : momentum q: osition dq m Particle travels in a trajectory q(t), (t) with well-defined energy: E dv ( q) + V ( q), F -. m dq Translation Zero force (F 0), initial momentum 0 so d 0, 0. 0 E m 1
2 Rotation Angular momentum J Iω ω: angular velocity I mr : moment of inertia dj F T F T : torque force. For constant torque, J ( t) J + F t, 0 J ( J0 + FT t) E. I I Vibration (Harmonic oscillator) Hook s law: F -kq, k: force constant q: dislacement T Potential energy:
3 1 V ( q) Fdq ( kq) dq kq Newton s equation d q m F kq Solutions: q Asinωt, ω ( k/m) 1/ dq m mωacosωt E + V m ( mωacosωt) m mω A mω A 1 k( Asinωt) [(cosωt) + (sinωt) ] ka + When ωt 0, q 0 and max mωa When ωt π/, 0 and q q max A 3
4 Classical Electrodynamics (Maxwell): The four equations secify roerties of electromagnetic wave. Light is an electromagnetic wave with a seed of light with oscillating electric and magnetic fields. λν c λ: wavelength (cm) ν: frequency (Hz s -1 ) c: seed of light ( cm/s) γ-ray x-ray UV visible IR radio wave λ ν II. Failures of classical hysics So in classical world, there is either wave or article. The two never mix. But this was about the change. Several exerimental observations around 1900 rovided critical tests of the classical theory and gave birth to the new quantum mechanics. Blackbody radiation: Blackbody is an object that absorbs and emits at all frequencies. A good examle is a cavity with a inhole. A ractical examle is a heated metal or the sun. Observations: 4
5 indeendent of material radiation has continuous sectrum and a eak sectral eak changes with temerature Wien's dislacement law (emirical): c λ max, c 5T 1.44 cm K Examle: Calc. T of the sun (λ max 480 nm) c nm K T 6000K 5 λ 5 480nm max 7 Rayleigh-Jeans law (classical electrodynamics theory) ρ 8πkT 4 λ ρ: energy density k: Boltzmann constant ( J/K) Assumtion: blackbody consists of a collection of oscillators with arbitrary energies. Fits the exerimental curve well at long wavelengths, But when λ 0, ρ (ultraviolet catastrohe). Planck's distribution (1900, henomenological): 5
6 ρ 8πhc λ e 1 5 hc / λkt 1 Assumtion: energies of oscillators are quantized! E nhν nhc/λ h : Planck's constant ( Js) When λ 0, ρ 0. 8πkT When hc / λ kt << 1, ρ 4. ( e x 1 + x) λ The hotoelectric effect (1887 Hertz) Observations: No e- if ν < ν t even at high light intensity. Above ν t, # of e - increases with light intensity. Emit e- if ν > ν t even at low light intensity. Kinetic energy of e- linearly roortional to ν. Classically, e- emission roortional to light intensity, not ν. Einstein s theory (1905, quantization of radiation field) 6
7 1 υ m e hν Φ Work function (E min for e- escae) Φ hν t. Light is made of articles called hotons, each with E hν! Electron diffraction (197, Davisson and Germer) Diffraction of e- beam on crystal surfaces Classically, e- cannot interfere because it is a article. Conclusions: i. electron is wave-like (interference). ii. its λ is comarable to lattice size. de Broglie wave (193): For hoton: so E mc hν hc mc λ ( λν c ) h λ ( mc) 7
8 Matter-wave dualism: hoton and other quantum objects have both wave and article nature. Classical object Quantum object Light : electromagnetic wave hoton and electromagnetic wave Electron: article electron and matter wave Examle: Calculate de Broglie wavelength for an e- accelerated from rest thru a otential difference of 1.5 kv Kinetic energy and momentum m e υ / eδφ or υ (eδφ/m e ) 1/ m υ e (m e eδφ) 1/ de Broglie wavelength λ h/ 1/ h / (m e eδφ) [( Js kg)( C)( V )] m nm Crystal lattice const. is arox 0.1 nm, so e- can be diffracted / 8
9 Atomic sectrum In 1909, Rutherford established the lanetary model for atoms. However, this model redicts an unstable atom since accelerate charges (electrons) emit radiation according to classical theory, thus they lose energy and will eventually collase. The classical model also redicts continuous sectra. Observations: Stable atoms Line sectrum Bohr s atom model (1911) Assume electron moves in an orbital associated with quantized energy. Transitions between orbitals corresond to absortion or emission hν E n - E m but it did not exlain why. Summary: classical theory is inadequate in describing microscoic world quantum objects have both wave and article nature quantum objects often have discrete energies 9
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