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A Review of Quantum Mechanics A Review of Quantum Mechanics Chun Wa Wong Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547∗ (Dated: September 21, 2006) Review notes prepared for students of an undergraduate course in quantum mechanics at UCLA, Fall, 2005 – Spring, 2006. Copyright c 2006 Chun Wa Wong 2 2 I. The Rise of Quantum Mechanics (eG = e /4πǫ0 in Gaussian units). Hence U 1.1 Physics & quantum physics: E = T + U = = T. (5) 2 − Physics = objective description of recurrent physi- cal phenomena Bohr’s two quantum postulates (L quantization and Quantum physics = “quantized” systems with both quantum jump on photon emission): wave and particle properties L = mva = n~, n =1, 2, ..., − 1.2 Scientific discoveries: hν = Eni Enf . (6) Facts Theory/Postulates Facts → → For ground state: 1.3 Planck’s black body radiation: U e2 E = 1 = G = 13.6 eV Failure of classical physics: 1 2 − 2a − ~2 1 = T1 = 2 , Emissivity = uc, λmaxT = const, − −2ma 4 ~2 ˚ energy density = u = ρν E , a = 2 =0.53A (Bohr radius), (7) h i meG E exp = kBT, (1) h i 6 For excited states: (ρ = density of states in frequency space). ν E r = n2a, E = 1 . (8) Planck hypothesis of energy quantization (En = n n n2 nhν) explains observation: 1.6 Compton scattering: nx hν ∞ ne hν E = n=0 − = , (2) Energy-momentum conservation for a photon of lab h i ∞ e nx ehν/kB T 1 Pn=0 − − energy E = hc/λ scattered from an electron (of mass m) initially at rest in the lab (before=LHS, P where x = hν/(kBT ). after=RHS; 1=photon, 2= recoiling e): 1.4 Einstein’s photoelectric effect: p = p1 + p2, Photoelectrons ejected from a metal by absorbed E + mc2 = E + m2c4 + p2c2. (9) light of frequency ν have maximum energy: 1 2 q Photon scattered into lab angle θ has longer wave- E = hν W (work function) (3) − length λ1: that depends on the frequency ν of light and not λ λ = λ (1 cos θ), where 1 − e − its intensity. h hc λe = = 2 =2.4 pm (10) 1.5 Bohr’s atomic model: mc mc Electron of mass m in a classically stable circular is the electron Compton wavelength. orbit of radius r = a and velocity v around the 1.7 de Broglie’s matter wave: atomic nucleus: Massive matter and massless light satisfy the same mv2 e2 U energy-momentum and momentum-wavelength re- = G , or T = , (4) r r2 − 2 lations E2 = p2c2 + m2c4, h p = ~k = . (11) ∗Electronic address: [email protected] λ 2 Thus moving matter is postulated to be a wave with LDEs (linear DEs) with constant coefficients a motional de Broglie wavelength λ = h/p, leading have exponential solutions: to the diffraction maxima for both matter and light d of eikx = ikeikx, dx 2-sided formula: 2d sin θ = nλ, d iωt iωt e− = iωe− . (14) 1-sided formula: D sin θ = nλ. dt − These DE are called eigenvalue equations be- II. Waves and quantum waves cause the constant ik is called an eigenvalue of the ikx Classical and quantum waves have identical mathemat- differential operator d/dx, while the solution e is ical properties. called its eigenfunction belonging to its eigenvalue ik. 2.1 Waves in physics: The 1DWEq (12) has four distinct (i.e., linearly Waves rise and fall, travel, have coherence in space independent) eigenfunctions: and in time, interfere, and diffract. i(kx ωt) Ψ(x, t)= e± ± . (15) Two classes of waves: (a) Inertial waves in massive media: ocean waves, Energy in both classical and quantum waves is car- sound waves, vibrations of violin strings; ried by the intensity or energy flux (b) Noninertial waves in vacuum: EM waves, I Ψ 2. (16) matter waves. ∝ | | Its quadratic dependence on Ψ is responsible for the 2.2 Mathematical description of waves: interference between two traveling waves Ψi = iθi The 1-dimensional wave equation (1DWEq) Aie with real Ai: 2 2 2 2 2 ∂ 1 ∂ Ψ1 +Ψ2 = Ψ1 + Ψ2 + 2Re(Ψ∗Ψ2) Ψ(x, t)=0 (12) | | | | | | 1 ∂x2 − v2 ∂t2 = A2 + A2 +2A A cos(∆s), 1 2 1 2 ∆ω is an equation of state that gives a wave function where ∆s = ∆k x t . (17) − ∆k Ψ(x, t) describing an unfolding event in spacetime. The wave equation is a partial differential equation In the limit ∆k 0, (PDE) because it depends on more than one vari- → able, here the two variables x and t. dω = v = group velocity In contrast, the solution x(t) of an equation of mo- dk g ω tion such as a Newton equation describes how a = v = = wave velocity. (18) single point x(t) on an object evolves in time. Sim- 6 k ilarly, a snapshot of an object is not necessarily a wave. A wave function Ψ(x, t) is needed to describe Examples: Certain waves satisfy the relation ω = p a wave’s coherent structure in both space and time. Ak . Then vg/v = p. Specific examples are p = 2 for transverse vibrations of a thick bar, p = 3/2 However, a wave (any wave) can have particle prop- for short ripples moving under surface tension, and erties if a point x can be defined on it such that one p =1/2 for long waves on deep sea. can describe how its position changes in time. The result (17) is similar in structure to the squared The wave equation (12) describes linear waves sat- length of a sum of two vectors in space that inter- isfying the superposition principle that a sum sect at an angle θ: of solutions is also a solution : 2 2 2 A1 + A2 = A1 + A2 +2A1A2 cos θ, (19) Ψ = a1Ψ1 + a2Ψ2. (13) | | Two waves of the same amplitude traveling in op- Traveling waves: posite directions interfere form a standing wave: (a) f(x vt)= a wave traveling to the right (+x), − Ψ = A[cos(kx ωt) + cos(kx + ωt)] (b) g(x + vt)= a wave traveling to the left (-x), − = 2A cos(kx)cos(ωt) (20) (c) the composite wave a1f(x vt)+ a2g(x + vt) is also a wave. − with factorized space and time dependences. 3 2.3 Matter-wave quantization: is the normalized probablity density of finding the Light wave of momentum p and energy E = pc is wave in “k space”. (We shall not consider more described by the wave function: complicated properties that also depend on the time t or the energy E.) i(kx ωt) i[(p/~)x (E/~)t] e − = e − . (21) Uncertainty relations: A wave of any kind sat- de Broglie and Schr¨odinger suggested that the RHS isfies the uncertainty relation expression holds also for matter waves so that: 1 ∆x∆k , where ~ ≥ 2 ∂ i(p/~)x i(p/~)x i(p/~)x e =pe ˆ = pe , (∆x)2 = (x x¯)2 , x¯ = x , i ∂x h − i h i (∆k)2 = (k k¯)2 , k¯ = k . (28) ∂ i(E/~)t i(E/~)t i(E/~)t i~ e− = Heˆ − = Ee− .(22) h − i h i ∂t The expression for matter waves is usually written The differential operators that appear are called as the Heisenberg uncertainty principle: the momentum operatorp ˆ and the Hamiltonian (or ~ ˆ ∆x∆p . (29) energy) operator H, respectively. ≥ 2 Commutation relations: The rules of differen- tial calculus dictates that a differential operator Spreading wave packets: A Gaussian wave does not commute with its own variable: packet has the spectral amplitude ∂ ∂ ∂ αk2 1 x = 1+ x , or , x = 1; g(k)= e− , with ∆k = . (30) ∂x ∂x ∂x 2√α ∂ ∂ ∂ It gives rise to a wave function at time t =0 of t = 1+ t , or ,t =1. (23) ∂t ∂t ∂t x2/(4α) f(x, t = 0) e− , with ∆x = √α. (31) These results give the fundamental commuta- ∝ tors of wave mechanics Hence the wave packet at time t = 0 has the mini- ~ mal uncertainty product of [ˆp, x]= , [H,tˆ ]= i~. (24) i 1 ∆x∆k = . (32) 2 2.4 Wave packet and the uncertainty principle: For t = 0, however, the uncertainty product could A wave packet is a superposition of waves of the 6 type (21) be greater than the minimal value of 1/2. This hap- pens if the system is dispersive, meaning a nonzero 1 ∞ i[kx ω(k)t] second Taylor coefficient β in the following expan- f(x, t)= g(k)e − dk, (25) √2π sion: Z−∞ dω 1 d2ω with a linear superposition function g(k) sometimes ω(k) ω(k = 0)+ k + k2 called a spectral amplitude. ≈ dk 2 dk2 0 0 Expectation value (= mean value) of a property 1 = ω(0) + kv + k 2β. (33) A(x, t) of the wave: g 2 ∞ The resulting approximate wave function at any A(x, t) ρ (x, t)A(x, t)dx, where h i ≡ P time t can then be evaluated to the closed form Z−∞ 2 x2/(4α ) f(x, t) f(x, t) e− t t , with ρP (x, t) = | | (26) ∝ ∞ f(x, t) 2dx iβ | | xt = x vgt, αt = α + t. (34) −∞ − 2 is the normalizedR probablity density of finding the wave at the spacetime point (x, t). Thus the position x of the wave moves in time with the group velocity The expectation value of a time-independent prop- erty B(k) of the wave can be calculated by using dω v = . (35) the spectral amplitude g(k): g dk 0 ∞ Its position uncertainty increases in time to B(k) ρ˜P (k)B(k)dk, where h i ≡ Z−∞ 1/2 2 2 g(k) αt βt ρ˜P (k) = | | (27) ∆x(t)= | | = √α 1+ .
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