Part C : Quantum Physics

Transcription

1 Part C : Quantum Pysics 1 Particle-wave duality 1.1 Te Bor model for te atom We begin our discussion of quantum pysics by discussing an early idea for atomic structure, te Bor model. Wile tis relies on a rater arbitrary assumptions (as we sall see), it does demonstrate te power of te idea of quantization and was te first model of te atom tat was truly predictive. In te model model, te atom consists of a massive nucleus wit carge +Ze surrounded by muc ligter electrons (carge e) tat are in circular orbits. We proceed using classical Newtonian arguments. Te equation of motion for te orbiting electron is m e v r = Ze 4πǫ 0 r, (1) wic tells us tat te angular momentum of te electron is Te energy of te electron is L m e vr = Ze m e r 4πǫ 0. () E = 1 m ev Ze 4πǫ 0 r, (3) wic, using te equation of motion, can be written in terms of just r: E = Ze 8πǫ 0 r (4) Bor stated tat, wenever electrons move one orbit to anoter, te cange in energy is released (or absorbed) as a packet of e/m radiation wit frequency proportional to te cange in energy (in modern language, we would say tat a poton is emitted or absorbed). But we know tat atoms ave well-defined emission and absorption lines, so it seems as if only very particular energy jumps are allowed. Wy? Bor asserted tat te angular momentum of te electron is quantized, L = n, (5) 1

2 were is te reduced Planck constant (/π) and n = 1,,3... Let us follow tis suggestion troug to its conclusion. Inserting equation into equation 5, we deduce tat only particular values of r are allowed, r n = 4πǫ 0 Ze m e n. (6) Tis in turn means tat only particular energy levels are allowed, E n = Z e 4 m e 3π ǫ 0 1 n (7) Suppose tat an electron jumps from a level wit n = n 1 to n = n. Te cange in energy is E n1 n = E n E n1 = Z e 4 ( m e 1 3π ǫ 0 1 ) (8) n 1 n If n < n 1, ten te electron loses energy (it falls deeper into te electrostatic potential) and a poton is emitted (emission line). If n 1 < n, te electron gains energy and it must ave absorbed a poton (absorption line). Te wavelengts of te corresponding emission/absorption lines are at c λ = E n1 n. (9) 1. Matter waves and particle-wave duality In 193, de Broglie built upon Einstein s ideas of potons to make a radical proposal tat matter particles ave wave-like properties, and tat te corresponding wavelengt is related to te particles momentum by λ = /p. Vectorially, we ave p = k were k is te wavenumber vector. Tis idea was verified by te detection of electron diffraction in 197. We can now understand te Bor angular momentum quantization in terms of de Broglie waves. Consider an electron in an atom, orbiting te nucleus at radius r. In order to fit, te wavelengt of te electron must be an integer number of multiples of te circumference of te orbit, i.e. πr = nλ. (10) But, since λ = /p, tis formula can be easily re-arranged to give rp =, (11) and we recognize te left-and side of tis as just te angular momentum L. Degeneracy pressure and compact stars Quantum pysics is crucial for an understanding of wite dwarfs and neutron stars. Te material in tese stars is rater ot and so all atoms are fully ionized. Tus we must examine te dynamics of free particles. Tis brings us to te classic discussion of...

3 .1 A particle in a box Consider a particles in a cubic box of side lengt L. Suppose tat te walls of te box are impenetrable. Te particle is described by a 3-d wave tat needs to fit inside te box. Tus, we need: n x λ x n y λ y n z λ z = L, (1) = L, (13) = L, (14) (15) were n x,n y,n z are positive integers. Using de Broglie s formula, tis gives a quantization of te particle s momentum components and ence a quantization of te energy, Some notes: p x = n x L, (16) p y = n y L, (17) p z = n z L, (18) E = p m = 8mL (n x +n y +n+z ) (non relativistic) (19) E = cp = c L (n x +n y +n+z ) 1/ (ultra relativistic). (0) 1. For any finite L, te allowed energy levels are quantized.. As L, te energy levels get closer togeter and (sort of) approac a continuum. 3. Tere is a minimum, non-zero, allowed energy! 4. Te state ofteparticlecanbelabeledbytetree quantumnumbers ; n x,n y,n z.. Fermions, Bosons and Pauli exclusion Tere are two basic types of particles in nature called Bosons and Fermions. Bosons (e.g. potons, gravitons, gluons) ave spin angular momenta tat are integer multiples of. For example, a poton as a spin of ; a graviton as a spin of. Tere is no restriction on te number of bosons tat can be in a given quantum state... indeed, bosons like being in te same quantum state! Fermions (e.g., electrons, quarks, protons, neutrons) ave spins tat are alf-integer multiples of. For examples, our familiar particles (electrons, protons, neutrons) all ave 3

4 1 1 spin. A given measurement of te spin will find it in one of two states, up (S = ) 1 or down (S = ). No more tan one fermion can occupy a given quantum state (including te spin as one of te quantum numbers) tis is te Pauli exclusion principle..3 Degenerate matter and te Fermi-energy Suppose tat we ave N spin-1/ fermions (e.g. electrons) in a box and tat tey are free and non-interacting. Tey must, owever, obey te Pauli exclusion principle. So, we can ave at most electrons in te (n x,n y,n z ) = (1,1,1) state, electrons in te (n x,n y,n z ) = (1,1,) state, electrons in te (n x,n y,n z ) = (1,,1) state etc. etc. If te system is very cold, te electrons will all attempt to occupy te lowest possible energylevels but, ofcourse, teyarenowallowedtoalloccupytelowest energylevel. Tey will stack up inorder of increasing E, wic means in order of increasing (n x +n y +n z )1/. In oter words, tey will stack up into a sperical ball in n-space and ence p space. Te maximum momentum acieved (i.e. te momentum of tis outer sell in momentum space) is p F = (3π n e ) 1/3, (1) were n e is te number density of te electrons. Tis is known as te Fermi-momentum. Te corresponding energy, E F = p F m e = m e (3π n e ) /3 (non-relativistic particles) () is known as te Fermi-energy..4 Degeneracy Pressure E F = p F c = c(3π n e ) 1/3 (relativistic particles), (3) Te Pauli exclusion principle means tat te distribution function of cold degenerate matter is particularly simple, f p (p) = A,constant for p < p F (4) f p (p) = 0 oterwise. (5) were f p (p)dp x dp y dp z is te probability of finding an electron in state wit momentum in range p x p x +dp x,p y p y +dp y,p z p z +dp z. We determine A by te normalization condition: f p (p)d 3 p = 1 A = π 3 3 (6) n e. From te distribution function, we can now calculate te pressure using our macinery of statistical mecanics. Assuming a non-relativistic relation between velocity and momentum (v x = p x /m e ), tis goes as: P = n e f(v)m e v x dv xdv y dv z (7) 4

5 = n e m p x f p(p)dp x dp y dp z (8) = (3π ) /3 5m e n 5/3 e (9) 5