Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Physics 365 - Many-Particle Quantum Theory - Prof. Oberacker 1 Radial probability densities for electrons in an atom In our lecture notes (chapter 3, p. 10) we derived a simple expression for the ground state density of an N-electron atom if we neglect the electron-electron interaction. In this case the electron density of an atom is simply the sum of the probabitity densities (= absolute value squared of the wave functions) for all occupied single particle states. The spatial part of the electronic wave functions has the structure ψn,`,m` (r, θ, φ) = Rn,` (r) Y`,m` (θ, φ) . In Niels Bohr’s semi-classical atomic model, the radii rn of the spherical electronic orbits increase with the square of the principal quantum number n, i.e. rn = n2 ( a0 ), Z (n = 1, 2, 3, ...) where a0 = 0.55 Å is the Bohr radius of the hydrogen atom, and Z denotes the atomic number (= number of protons in the nucleus). In quantum mechanics, the radial probability densities r2 [Rn,` (r)]2 are most closely related to these classical radii. The factor r2 in the last expression arises from the volume element in spherical coordinates. Below we plot the 1-D radial probability densities for the lowest quantum states with principle quantum numbers n = 1, 2, 3. atomic radial probability density (n=1) L=0 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 r (units of Bohr radius / Z) Figure 1: For n = 1, the probability density peaks at 1 Bohr radius / Z. 1 atomic radial probability densities (n=2) 0.25 L=1 0.2 0.15 0.1 0.05 L=0 0 0 5 10 15 r (units of Bohr radius / Z) Figure 2: For n = 2, the probability densities peak at 3-5 Bohr radii /Z (for the two angular momentum substates). atomic radial probability densities (n=3) 0.12 L=2 0.08 0.04 L=1 L=0 0 0 5 10 15 20 25 r (units of Bohr radius / Z) Figure 3: For n = 3, the probability densities peak at 8-13 Bohr radii / Z (for the three angular momentum substates). From these plots we infer that the probability densities for n = 1, 2, 3 peak at different radial positions which roughly agree with the semi-classical Bohr radii rn given above. This suggests that the total density of an N-electron atom (which can be measured) might reveal the shell structure of the occupied orbitals! This is indeed the case. We will look at atomic electron densities later in Chapter 7 within the context of atomic Hartree-Fock calculations which take the electron-electron 2 interaction into account “on average” (via the one-body mean field potential). 2 Position probability densities in 3-D (‘atomic orbitals’) We calculate now the position probability densities in 3-D space. In cylindrical coordinates (z, r, φ) we have ρn,`,m` (z, r, φ) = [ψ ∗ ψ]n,`,m` (z, r, φ) . The probability densities are axially symmetric around the z-axis. In the contour plots below we show the quantities ρn,`,m` (z, r, φ = 0) . prob_density_linear 3 0.31 0.286 2 0.263 0.239 1 0.215 z 0.191 0.167 0 0.143 0.119 1 0.0955 0.0717 2 0.0478 0.0239 3 6.57e 05 0 1 2 r 3 Figure 4: Probability density ψ ∗ ψ for the state |n = 1, ` = 0, m` = 0 > . 3 prob_density_linear 11 10 0.00525 0.00242 0.00444 0.00222 0.00404 0.00202 0.00363 0.00182 0.00323 0.00162 0.00283 0 0.00262 0.00485 z z 11 10 prob_density_linear 0.00141 0 0.00242 0.00121 0.00202 0.00101 0.00162 0.000808 0.00121 0.000606 0.000808 0.000404 0.000404 10 11 0.000202 10 11 0 0 r 1011 0 0 r 1011 Figure 5: Probability densities ψ ∗ ψ for the state |n = 2, ` = 1 > . Left side: for the magnetic substate |m` = 0 > . Right side: for the magnetic substates |m` = ±1 > . 4 prob_density_linear 35 0.00033 30 0.000304 0.000279 20 0.000254 0.000228 10 z 0.000203 0.000178 0 0.000152 0.000127 10 0.000101 7.61e 05 20 5.07e 05 2.54e 05 30 35 0 0 10 r 20 30 35 Figure 6: Probability density ψ ∗ ψ for the state |n = 4, ` = 2, m` = 0 > . 5