Download 1 Niels Bohr`s semi-classical model (1913) 2 QM atomic shell model

Atomic shell model: single-particle wave functions and probability densities
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Niels Bohr’s semi-classical model (1913)
Starting from Rutherford’s atomic model (1908), Niels Bohr assumed that the electrons orbit the
atomic nucleus like the planets orbit the Sun. Using a combination of classical mechanics and
certain ad-hoc quantization conditions, Bohr’s 1913 model predicts circular electron orbits whose
radii rn are quantized. The radii of the electronic orbits increase with the square of the principal
quantum number n
a0
rn = n2 ( ) , (n = 1, 2, 3, ...)
Z
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). The quantized radii lead to quantized single-particle energies
which can be written in the following elegant form
me c2
En = −
2
2
Zα
n
2
α≈
1
137.036
QM atomic shell model, probability densities for electrons
If we neglect the spin-orbit interaction, relativistic correction and the two-body electron-electron
interaction, the Hamiltonian for an N-electron system has the structure
H(~r) =
N
X
h(~ri ) ,
i=1
where the single-particle Hamiltonian of the atomic shell model is given by
h(~r) =
−~2 2 Ze2
∇ −
.
2me
r
The single-particle Schrödinger equation is
h(~r)ψE (~r) = EψE (~r) .
Using spherical coordinates, the single-particle wave functions of the atomic shell-model have the
structure
ψn,`,m` (r, θ, φ) = Rn,` (r) Y`,m` (θ, φ) .
The radial wave functions Rn,` (r) are very similar to those of the hydrogen atom (see QM textbooks), except that one has to replace the Bohr radius a0 by
a0 →
a0
.
Z
As we will demonstrate below, the radial probability densities predicted by quantum mechanics
ρrad (r) = r2 [Rn,` (r)]2
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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.
atomic radial probability densities (n=2)
0.25
L=1
0.2
0.15
0.1
0.05
0
0
L=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).
are most closely related to the Bohr radii rn . The factor r2 in the last expression arises from the
volume element in spherical coordinates. We show plots of the 1-D radial probability densities for
the lowest quantum states with principle quantum numbers n = 1, 2, 3.
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. Quantum many2
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).
body theory shows that the electron density of an atom is the sum of the probability densities for
all occupied quantum states. 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 examine atomic electron densities later on which are calculated from the Hartree-Fock
equations which take the electron-electron interaction into account “on average” (via the one-body
mean field potential).
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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) .
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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 > .
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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 > .
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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 > .
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