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Quantum Mechanics: the Hydrogen Atom

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Quantum Mechanics: the Hydrogen Atom Quantum Mechanics: The Hydrogen Atom 12th April 2008 I. The Hydrogen Atom In this next section, we will tie together the elements of the last several sections to arrive at a complete description of the hydrogen atom. This will culminate in the definition of the hydrogen-atom orbitals and associated energies. From these functions, taken as a complete basis, we will be able to construct approximations to more complex wave functions for more complex molecules. Thus, the work of the last few lectures has fundamentally been amied at establishing a foundation for more complex problems in terms of exact solutions for smaller, model problems. II. The Radial Function We will start by reiterating the Schrodinger equation in 3D spherical coordi- nates as (refer to any standard text to get the transformation from Cartesian to spherical coordinate reference systems). Here, we have not placed the con- straint of a constant distance separting the masses of the rigid rotor (refer to last lecture); furthermore, we will keep in the formulation the potential V (r; θ; φ) for generality. Thus, in spherical polar coordinates, H^ (r; θ; φ) (r; θ; φ) = E (r; θ; φ) becomes: −h¯2 1 @ @ 1 @ @ 1 @2 r2 + sinθ + + V (r; θ; φ) (r; θ; φ) = E (r; θ; φ) 2µ r2 @r @r r2sinθ @θ @θ r2sin2θ @φ2 " ! # Now, for the hydrogen atom, with one electron found in "orbits" (note the quotes!) around the nucleus of charge +1, we can include an electrostatic potential which is essentially the Coulomb potential between a positive and negative charge: −Ze2 V (jrj) = V (r) = 4πor It is important to note that the Coulomb potential as we have written it here is simply a function of the magnitude of the posi- tion vector between the 2 masses (i.e, between the electron and 1 nucleus). There is no angular dependence!. Recall: 1 @ @ 1 @2 L^2 = −h¯2 sinθ + r2sinθ @θ @θ r2sinθ @φ2 ! is the total angular momentum squared operator (function of θ and φ only!). Thus, we can rewrite the Schrodinger equation as: @ @ −h¯2 r2 + 2µr2 [V (r) − E] (r; θ; φ) + L^2 (r; θ; φ) = 0 @r @r This demonstrates that the Hamiltonian is separable since the terms in brackets are functions of r only, and the angular momentum operator is only a function of θ and φ. Thus, the wavefunction can be written in a form that lends to separation of variables. Recalling that the spherical harmonics are eigenfunctions of the angular momentum operator: m (r; θ; φ) = R(r)Yl (θ; φ) Separation of V ariables ^2 m 2 m L Yl (θ; φ) = h¯ l(l + 1)Yl (θ; φ) l = 0; 1; 2; ::: Accounting for separation of variables and the angular momentum resuls, the Schrodinger equation is transformed into the Radial equation for the Hydrogen atom: −h¯2 d dR(r) h¯2l(l + 1) r2 + V (r) − E R(r) = 0 µr2 dr dr µr2 2 " 2 # The solutions of the radial equation are the Hydrogen atom radial wave- functions, R(r). II. Solutions and Energies The general solutions of the radial equation are products of an exponential and a polynomial. The eigenvalues (energies) are: −Z2e2 −Zµe4 E = 2 = 2 2 2 n = 1; 2; 3; :: 8πoaon 8oh n The constant ao is known as the Bohr Radius: 2 2h2 a = o o πµe2 The Radial eigenfunctions are: 1 3 2 l+ 2 − (n − l − 1)! 2Z l Zr 2l+1 2Zr R (r) = − r e nao L nl 3 na n+l na "2n [(n + l)!] # o o The L2l+1 2Zr are the associated Laguerre functions. Those for n = 1 n+l nao and n = 2 are sho wn: 1 n = 1; l = 0; L1 = −1 Zr n = 2; l = 0; L1 = −2! 2 − 2 a o 3 n = 2; l = 1; L3 = −3! How to normalize: Spherical Harmonics: 2π π m m dφ dθ sinθ Yl ∗(θ; φ)Yl (θ; φ) = 1 Z0 Z0 Radial Wavefunctions: 1 2 dr r Rnl∗ (r)Rnl(r) = 1 Z0 The total hydrogen atom wavefunctions are: m (r; θ; φ) = RnlYl (θ; φ) Table 1. Nomenclature and Ranges of H-Atom quantum numbers Name Symbol Allowed Values principle quantum number n 1,2,3,.... angular momentum quantum number l 0,1,2,...,n-1 magnetic quantum number m 0,1; 2; 3; :::; l 3 Z2e2 ENERGIES: DEPEND on "n" E = 2 . 8π−oaon ANGULAR MOMENTUM: DEPEND on "l" jLj = h¯ l(l + 1) p Lz component : DEP ENDon"m" Lz = mh¯ − Total H atom wavefunctions are normalized and orthogonal: 2π π 1 2 dφ dθ sinθ drr nl∗ m(r; θ; φ) n∗0l0m0 (r; θ; φ) = δnn0 δll0 δmm0 Z0 Z0 Z0 Lowest total Hydrogen atom wavefunctions: n=1 and n=2 ( define σ ≡ Zr ) ao Table 2. Hydrogen Atom Wavefunctions n l m nlm Orbital Name 3 1 Z 2 σ 1 0 0 100 = e 1 pπ ao − s 3 − 1 Z 2 σ 2 0 0 200 = (2 − σ)e 2 2s p32π ao 3 − 1 Z 2 σ 1 0 210 = σe 2 cosθ 2p p32π ao z 3 −σ 1 Z 2 iφ 1 1 21 1 = σe 2 sinθe p64π ao Transforming to real functions via normalized linear combinations 3 2 −σ 1 Z 2 1 1 1 2px = σe sinθcosφ 2px = ( 21+1 + 21 1) p32π ao p2 − 3 2 −σ 1 Z 2 1 1 1 2py = σe sinθcosφ 2py = ( 21+1 − 21 1) p32π ao p2 − Thus, we have come to the point where we can connect what we already know from previous analysis: 4 Table 3. Quantum numbers n = 1; 2; 3; ::: l = 0; 1; 2; :::; n − 1 m = 0; 1; 2; :::; l Orbital 1 0 0 1s 2 0 0 2s 2 1 0 2pz 2 1 + 2px 2 1 - 2py What are the degeneracies of the Hydrogen atom energy levels? Recall they are dependent on the principle quantum number only. III. Spectroscopy of the Hydrogen Atom Transitions between the energy states (levels) of individual atoms give rise to characteristic atomic spectra. These spectra can be used as analytical tools to assess composition of matter. For instance, our knowledge of the atomic composition of the sun was in part aided by considering the spectra of the radiation from the sun. For the Hydrogen atom, early scientists observed that the emission spectra (generated by exciting hydrogen atoms from the ground to excited states), gave rise to specific lines; the spec- tra were NOT continuous. The understanding of the quantum mechanical nature of the hydrogen atom helps us understand how these lines arise. Series of lines in the hydrogen spectrum, named after the scien- tists who observed and characterized them, can be related to the energies associated with transsitions from the various energy lev- els of the hydrogen atom. The relation, simple enough as it is, turns out to accurately predict the spectral lines. The equation relating the wavelength (and thus energy via E = hν) associated with a transition from a state n1 to another tate n is given by: 1 1 1 = R − n1 = 2; Balmer Series λ Rydberg n2 n2 1 n1 = 1; Lyman Series n1 = 3; P aschen Series 5.
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