1 Interstellar Medium (ISM) Week 2 March 26 (Thursday), 2020 updated 04/13, 08:56 선광일 (Kwangil Seon) KASI / UST 2 Atomic Structure, Spectroscopy 3 References • Books for atomic/molecular structure and spectroscopy - Astronomical Spectroscopy [Jonathan Tennyson] - Physics of the Interstellar and Intergalactic Medium [Bruce T. Draine] ⇒ see https://www.astro.princeton.edu/~draine/ for errata - Astrophysics of the Diffuse Universe [Michael A. Dopita & Ralph S. Sutherland] ⇒ many typos - Physics and Chemistry of the Interstellar Medium [Sun Kwok] - Atomic Spectrocopy and Radiative Processes [Egidio Landi Degl’Innocenti] 4 Hydrogen Atom: Schrödinger Equation • Momentum operator ℏ p = ∇ i • Hamiltonian operator p2 ℏ2 H = + V(r) = − ∇2 + V(r) 2m 2m • The time-dependent Schrödinger equation for a system with Hamiltonian H: ∂Ψ ∂Ψ (r, t) ℏ2 iℏ = HΨ iℏ = − ∇2Ψ (r, t) + V(r)Ψ (r, t) ∂t ∂t 2m The time and space parts of the wave function can be separated: Ψ (r, t) = ψ(r)eiEt/ℏ • Then, the time-independent Schrödinger equation is obtained: ℏ2 Hψ (r) = Eψ(r) ∇2ψ (r) + V(r)ψ (r) = Eψ(r) 2m 5 • Expectation value of an operator 3 F = ⇤F d x h<latexit sha1_base64="+p0ZDsGnVguDSgAa3yy1sewZ094=">AAACDnicbZDNSgMxFIUz/tb6N+rSTVAK4qLMWEEXKgWhuKxiq9CpJZPJtKGZzJDcEUvpE4jgq7hxoYhb127EhxB8BNPWhVoPBD7OvZebe/xEcA2O826NjU9MTk1nZrKzc/MLi/bSclXHqaKsQmMRq3OfaCa4ZBXgINh5ohiJfMHO/PZhv352yZTmsTyFTsLqEWlKHnJKwFgNO+cJFsJeyVO82YKDfY9L8BLNLzZxCfcBBxeFq4a97uSdgfAouN+wXnRPboK3z49yw371gpimEZNABdG65joJ1LtEAaeC9bJeqllCaJs0Wc2gJBHT9e7gnB7OGSfAYazMk4AH7s+JLom07kS+6YwItPTfWt/8r1ZLIdytd7lMUmCSDheFqcAQ4342OOCKURAdA4Qqbv6KaYsoQsEkmDUhuH9PHoXqVt4t5LeOTRrbaKgMWkVraAO5aAcV0REqowqi6BrdoQf0aN1a99aT9TxsHbO+Z1bQL1kvX4Ygn4c=</latexit> i Z • Wavefunction for a hydrogen-like atom: (r, ✓, φ)=R (r)Y (✓, φ) <latexit sha1_base64="omnm2UJR+hQQ9tJp3PqxmXxqUjc=">AAACH3icbZDNSwJBGMZn7cvsyz5uXZYkUBDZtagugdClo0Wa4coyO87q4OzsMvNuIIv/SZf+lS4diohu/jfNqgfTHhh4+D3vMPM+XsSZAssaG5mV1bX1jexmbmt7Z3cvv3/QVGEsCW2QkIey5WFFORO0AQw4bUWS4sDj9NEb3KT54zOVioXiAYYR7QS4J5jPCAaN3PyFEynmJoIHo6IsO9CngMtO1Gel6/sUa1p6cpM0ng/dfMGqWBOZy8aemULtyJ+o7uZ/nG5I4oAKIBwr1batCDoJlsAIp6OcEysaYTLAPdrWVuCAqk4y2W9knmrSNf1Q6iPAnND5GwkOlBoGnp4MMPTVYpbC/7J2DP5VJ2EiioEKMn3Ij7kJoZmWZXaZpAT4UBtMJNN/NUkfS0xAV5rTJdiLKy+bZrVin1Wqd7qNczRVFh2jE1RENrpENXSL6qiBCHpBb+gDfRqvxrvxZXxPRzPG7M4h+iNj/At5+KV5</latexit> nlm nl lm radial function: 2 1/2 2Z ~ ˚ 3 ⇢ = r, a0 2 =0.529A (Bohr radius), 2Z (n l 1)! ⇢/2 l 2l+1 na0 ⌘ mec R (r)= − − e− ⇢ L (⇢) nl na 2n (n + l)! 3 n+l L2l+1 = associated Laguerre polynomial − " 0 # <latexit sha1_base64="jWIE314tiehRkmRr6Us2tgPL1A4=">AAACgXicbVFNb9QwEHVSPsrytYUjF4uFqlVLlIQiQGilFi4ceigS21ZstpHjTDZWHTu1nYqV5f4Ofhc3/gzC+3GAlpGseXpvZmy/KVrOtInjX0G4duv2nbvr93r3Hzx89Li/8eRYy05RGFHJpTotiAbOBIwMMxxOWwWkKTicFOef5vrJJSjNpPhqZi1MGjIVrGKUGE/l/R+bmarlMKsUoTb95qwgeezU7tWVzxlcdOxyqWV1QdRZ6myTA/V5GEdv0veZge/GZgcHbonw1kdZK6xIyTq97XazrLd5mFuxw92ZTflO4oZ4VUm0lpQRAyU+JNMOlALcSj4TsmGEu7w/iKN4EfgmSFZggFZxlPd/ZqWkXQPCUO6Hj5O4NRNLlGGUg+tlnYaW0HMyhbGHgjSgJ3bhoMMvPVPiSip/hMEL9u8OSxqtZ03hKxtian1dm5P/08adqd5NLBNtZ0DQ5UVVx7GReL4OXDIF1PCZB4Qq5t+KaU284cYvredNSK5/+SY4TqPkdZR+2Rvs763sWEfP0HO0hRL0Fu2jz+gIjRBFv4MXwasgCtfC7TAO02VpGKx6nqJ/IvzwB/w/woQ=</latexit> n+l <latexit sha1_base64="yQNHlwr1tGt063ig4U+2GTQw7K4=">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</latexit> ✓ ◆ { } spherical harmonics 1/2 (m+ m )/2 (l m )! 2l +1 m imφ m Y (✓, φ)=( 1) | | P | |(cos ✓)e P | | = associated Legendre function − | | <latexit sha1_base64="33PsWIXiS2a42APBIst22wNE278=">AAACGXicbVDLSgMxFM34rPVVdekmWARXZUYLuhEENy5cVLCt0NYhk7lTg5nMkNwRy9jfcOOvuHGhiEtd+Tem7Sx8HQgczrknyT1BKoVB1/10pqZnZufmSwvlxaXlldXK2nrLJJnm0OSJTPRFwAxIoaCJAiVcpBpYHEhoB9fHI799A9qIRJ3jIIVezPpKRIIztJJfcRu+vMzv4rshPaRdhFvMKTMm4YIhhPQU+qBCDTTKFB8lhn6l6tbcMehf4hWkSgo0/Mp7N0x4FoNCLu3VHc9NsZczjYJLGJa7mYGU8WvWh46lisVgevl4syHdtkpIo0Tbo5CO1e+JnMXGDOLATsYMr8xvbyT+53UyjA56uVBphqD45KEokxQTOqqJhkIDRzmwhHEt7F8pv2KacbRllm0J3u+V/5LWbs3bq+2e1atH9aKOEtkkW2SHeGSfHJET0iBNwsk9eSTP5MV5cJ6cV+dtMjrlFJkN8gPOxxcjNqD5</latexit> lm − (l + m )! 4⇡ l l <latexit sha1_base64="BpgRYJu18ReYZaeTwWNo0m2uWNg=">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</latexit> | | • Quantum Number - n = 1, 2, 3, ⋯ : principal quantum number - l = 0, 1, 2, ⋯, n − 1 : orbital quantum number - m = − l, − l + 1, ⋯, 0, ⋯, l − 1, l : magnetic quantum number 6 28 The hydrogen atom 2.1 The Schr¨odinger equation 25 • Wavefunctions for a hydrogen-like atom Table 2.2 Radial hydrogenic wavefunctions Rn,l in terms of the variable ρ = Zr/(na0), which gives a scaling that varies with n. The Bohr radius a0 is defined in eqn 1.40. Table 2.1 Orbital angular momentum eigenfunctions. 3/2 Z ρ 1 R1,0 = 2e− Y0,0 = a0 4π ! " ! 3/2 Z ρ 3 R2,0 = 2(1 ρ)e− Y1,0 = cos θ 2a0 − 4π ! " ! 3/2 Z 2 ρ 3 iφ R2,1 = ρ e− Y1, 1 = sin θ e± 2a √ ± ∓ 8π ! 0 " 3 ! 3/2 Z 2 2 ρ 5 2 R3,0 = 2 1 2ρ + ρ e− Y2,0 = 3cos θ 1 3a0 − 3 16π − ! " ! " ! 3/2 " # Z 4√2 1 ρ 15 iφ R = ρ 1 ρ e− Y2, 1 = sin θ cos θ e± 3,1 ± 3a0 3 − 2 ∓ 8π ! " ! " ! 3/2 Z 2√2 2 ρ 15 2 2iφ − Y2, 2 = sin θ e± R3,2 = ρ e ± 3a0 3√5 32π ! " 2!π π ∞ 2 2 2 Normalisation: R r dr =1 Normalisation: Yl,m sin θ dθ dφ =1 n,l | | #0 $0 $0 Z Here, ρ ≡ r These show a general a feature of hydrogenicna0 wavefunctions, namely 12 12 that the radial functions for l =0haveafinitevalueattheorigin,i.e.repeated application of the lowering operator: This eigenfunction has magnetic quantum number l (l m)=m. the power series in ρ starts at the zeroth power. Thus electrons with − − l m l ilφ l =0(calleds-electrons)haveafiniteprobabilityofbeingfoundatthe Yl,m (l ) − sin θ e . (2.11) ∝ − position of the nucleus and this has important consequences in atomic Atomic Physics [Foot] physics. To understand the properties of atoms, it is important to know what Inserting E from eqn 2.20 into eqn 2.17 gives the scaled coordinate | | the wavefunctions look like. The angular distribution needs to be mul- Z r tiplied by the radial distribution, calculated in the next section, to give ρ = , (2.21) n a0 the square of the wavefunction as where the atomic number has been incorporated by the replacement 2 2 2 2 2 e /4π#0 Ze /4π#0 (as in Chapter 1). There are some important prop- ψ (r, θ, φ) = Rn,l (r) Yl,m (θ, φ) . (2.12) erties of→ the radial wavefunctions that require a general form of the| | | | solution and for future reference we state these results. The probability 2 This is the probability distribution of the electron, or e ψ can be in- density of electrons with l =0attheoriginis terpreted as the electronic charge distribution. Many atomic− | | properties, 3 2 1 Z however, depend mainly on the form of the angular distribution and ψn,l=0 (0) = . (2.22) 2 2 | | π na0 Fig. 2.1 shows some plots of Yl,m .ThefunctionY0,0 is spherically ! " | 2 | | | symmetric. The function Y has two lobes along the z-axis. The For electrons with l =0theexpectationvalueof1/r3 is | 1,0| " squared modulus of the other two eigenfunctions of l =1isproportional to sin23 θ.AsshowninFig.2.1(c),thereisacorrespondencebetween 1 ∞ 1 2 2 1 Z 3 = 3 Rn,l (r) r dr = 1 these distributions. (2.23) and the circular motion of the electron around the r 0 r l l + (l +1) na0 $ % # 2 !z-axis" that we found as the normal modes in the classical theory of the These results have been written in a& form' that is easyZeeman to remember; effect (in Chapter 1).13 This can be seen in Cartesian coordi- 13Stationary states in quantum 3 they must both depend on 1/a0 in order to have the correctnates where dimensions mechanics correspond to the time- and the dependence on Z follows from the scaling of the Schr¨odinger z averaged classical motion. In this Y1,0 , case both directions of circular mo- ∝ r tion about the x-axis give the same x +iy distribution. Y , (2.13) 1,1 ∝ r x iy Y1, 1 − . − ∝ r 7 • The spherical harmonics are eigenfunctions of the orbital angular momentum operator ℏ L2Y = l(l + 1) 2Y ,LY = m Y L = r × p = r × ∇ <latexit sha1_base64="q1zx6CT0ZG3kcmporaNQQccBWUY=">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</latexit> lm ~ lm z lm ~ lm i • The sizes of the angular moment and z-component are L = l(l + 1) , L = m <latexit