The Hydrogen Atom Chapter 20

Transcription

1 4/4/17 Quantum mechanical treatment of the H atom: Model; The Hydrogen Atom Chapter 1 r -1 Electron moving aroundpositively charged nucleus in a Coulombic field from the nucleus. Potential energy term V(r) arises from this Coulombic field. The centrifugal potential due to angular motion of the electron also contributes to the potential energy term. Both terms are spherically symmetric. Quantum mechanical treatment of H & H -like atoms -e, m e Coulombic P.E. Quantum Mechanics predicts a stable system as opposed to the classical mechanical prediction of the demise of the atom. Ze, m N p Ze H = T V = µ 4πε r ħ Ze H = me x y z 4πεr mnme where µ = & mn >> me µ me m m N e H ψ = Eψ; eigenequation ħ Hψ = V ψ = Eψ me x y z H ħ ψ = ψ Vψ Eψ = me x y z me Ze ψ E ψ ( ) = x y z ħ 4πεr φ θ r x,y,z Strategy: changing the coordinate system to polar coordinates. ψ( x, y, z) ψ( r, θ, φ) Because r = x y z Separation of variables is not possible. Cartesian volume element dτ = dx dy dz Polar coordinate volume elementdτ= r sinθdθdφdr 1

2 4/4/17 r sinθdθdφdr _1%3A Atomic_Structure/Chapter_%3A_Atomic_Structure/Chapter_.5%3A_Atomic_Orbitals_and_Their_Energies Hamiltonian in Polar Coordinates (spherical symmetry): H atom Hamiltonian Time-independent Schrödinger Equation for H atom: KE PE KE Rigid rotor Hamiltonian (note) PE Wavefunction Separation of variables Rigid rotor Hamiltonian Defining R( r) Θ( θ) Φ ( φ) = RΘΦ Substituting for ψ(r,θ,φ); And using the rigid rotor Hamiltonian, ħ d dr 1 e ΘΦ r Rl ɵ R ER mr dr dr ΘΦ ΘΦ = ΘΦ mr 4πεr Note this form of the SE for H:

3 4/4/17 Recall that for rigid rotor(r constantand I = µr ): For an electron in an atom where r is a variable substituting for I = m e r in the following term of SE; E tot l l = H = ɵ I I For an electron in an atom where r is a variable substituting for I = m e r in the eigen equation above, ɵl I we get, ɵl 1 R ΘΦ = Rl ɵ ΘΦ I mr subscript e dropped. Using the result in the SE: Differential equation in R(r) is; ħ d dr 1 e ΘΦ r Rɵl ΘΦ R ER mr dr dr ΘΦ = ΘΦ mr 4πεr ħ d dr ħ l( l 1) e ΘΦ r R R ER mr dr dr ΘΦ ΘΦ = ΘΦ mr 4πεr Differential equation in R(r) result in; Solution of the above DE to find the radial part,r, of the eigen functions and their eigen values. Quantum numbers nand E n. Eigenvalues of SE are; free e R(r)solutions results in the radial part of the wave functions and the spherical harmonics results give the angular part of the wave function. The respective quantum numbers are n(principal), l(azimuthal) and m l (magnetic) such that; Bohr radius: a ε h = πme e Coulombic potential function Bound states 3

4 4/4/17 me Rydberg constant = 8ε hc More accurately; 4 e 3 Isotope peaks in atomic spectra predictable. Radial functions R nl ~ product of an exponential function, power of r and a polynomial of (r/a ) of order (n l -1). General form of Radial functions: Zr ( n l 1) ( n l) na Rnl = constant* polynomial r e n l 1 r r Polynomial = b br 1 b... bn l 1 a a Hydrogen wave functions (orbitals) are a product of the radial and angular functions (spherical harmonics). Real functions for m l = and complex functions if otherwise. Normalized functions are shown. Degeneracy (H atom and H like species only) = n Zr ( n l 1) ( n l) na Rnl = constant* polynomial r e actual solution linear combinations 4

5 4/4/17 Visualization of complex functions not possible m l. Therefore linear combinations wave functions can be constructed to obtain equivalent orbitals;from the available complete set of wave functions. ψnl = combinations of ψnlm, ψnl m for ml. orbital l l e.g. two porbitals ψ = ψ ψ ψ = ψ ψ px 1,, 1 1,, 1 px, pm, 1, pm, 1 ψ = ψ ψ ψ = ψ ψ py 1,, 1 1,, 1 px, pm, 1, p, m 1 At this point time independent SE provided us with eigen functions (orbitals) and eigen values (energies) for H atom that are independent of time. All eigen values are negative, more stable than the reference energy (reference energy = at n = ). n 1 (positive and nonzero) and energy can never approach negative infinity; i.e. the electron will not fall into the nucleus!! Visualization of orbitals Orbitals described by ψ s are not well defined shells, in keeping with the Heisenberg Principle. They, ψ s, are akin to probability functions. Probability of finding an electron in a volume element dτis; ψ*( r, θ, φ) ψ( r, θ, φ) dτ ψ 1 Spherical nodal surfaces - - #nodal surfaces (radial part) = n-l-1 not counting node at origin for l. #nodal surfaces (radial nodes) = n-l-1; not counting node at origin for l. #nodal surfaces (angular nodes) = l Total # nodal surfaces =n-1-1 5

6 4/4/17 - #nodal surfaces (angular part) = l #nodal surfaces (angular part) = l Radial Probability functions: P(r)dr Probability per unit volume = ψ*( r, θ, φ) ψ( r, θ, φ) Probability in volume dτ = ψ*( r, θ, φ) ψ( r, θ, φ) dτ Probability in nulcleus dτ = ψ*( r, θ, φ) ψ( r, θ, φ) dτ See Example Problem.4 = The electron distribution in general (for l ) is not spherically symmetric. But radial part is always spherically symmetric. To find the most probable position of finding electrons in an orbital would then be dictated by the radial function regardless of the spherical harmonics. Probability of finding the electron in a spherical shell at radius rof shell thickness dris defined as the Radial probability distribution function; P(r)dr. P(r)is a measure of the probability of electrons at a distance in a spherical shell of unit volume, rdistance away from the nucleus, for all angles θand φ, Example: =1 Radial Distribution Function P(r) is the probability function of choice to determine the most likely radius to find the electron in an orbital. =1 The Most Probable Radius Hydrogen Ground State Bohr Radius = a r/a 6

7 4/4/17 P(r)is the probability function of choice to determine the most likely radius to find the electron in an orbital. s is more dispersed than 1s. Bohr Radius = a r/a Wave functions are a manifestation of the wave particle duality, so is the non localization of orbitals. Spherical harmonic wavefunctions and their shapes: l = General form of Angular functions ml l Y = constant( sinθ and/or cosθ) e l imlφ l = 1 l = Complex functions - visualization not possible. 7

8 4/4/17 Under such situations alternate wave functions with same eigenvalues can be constructed from the complete set of wave functions available, i.e. properly constructedlinear combinations from the complete set of eigenfunctions. Note the sign of the wavefunction! Phase. p x = θ φ < 9 18 θ φ <9 θ 9 φ surface on x-z plane, for simplicity Note the signs of the wavefunction! 8

9 4/4/17 all p all d 9