Schrödinger Model
The Physics Hypertextbook™
© 1998-2008 by Glenn Elert -- A Work in Progress
All Rights Reserved -- Fair Use Encouraged
Discussion
dilemma
Electrons have three-dimensional extent, but the Bohr model assumes the
electron to be a one-dimensional standing wave wrapped around the nucleus.
major new idea
electron forms a three-dimensional standing wave around the nucleus, electron
clouds, restricted wavelengths (spherical harmonics)
Erwin Schrödinger (1887-1961) Austria, Abhandlungen zur Wellenmechanik. Wave equation for matter reminiscent of Maxwell's equations for electromagnetic waves. The story I heard is that Schrödinger went to Switzerland with two goals: to keep his mistress happy and to derive a wave equation for matter. How successful he was with the former is open to speculation.
full, time-dependent form
| iℏ | ∂ | Ψ(r, t) = − | ℏ2 | ∇2(r)Ψ(r, t) + V (r)Ψ(r, t) |
| ∂t | 2m |
can be separated into two halves
Ψ(r, t) = ψ(r) T(t)
spatial, time-independent half
| Eψ(r) = | ⎡ ⎣ |
− | ℏ2 | ∇2 + V(r) | ⎤ ⎦ |
ψ(r) |
| 2m |
four quantum numbers
- n -- energy quantization, linear momentum quantization? (K, L, M, N, O, … originally A, B, … )
principal quantum number - l -- angular momentum quantization (s, p, d, f, g, … sharp, principal, diffuse, fine/fundamental/faint, … )
(reduced? orbital?) azimuthal quantum number - m (ml) -- space quantization
magnetic quantum number - s (ms) -- spin quantization (up, down)
spin quantum number
 |
 |
 |
| | 2, 0, 0 > | | 2, 1, 0 > | | 2, 1, 1 > |
 |
 |
 |
| | 3, 0, 0 > | | 3, 1, 0 > | | 3, 1, 1 > |
 |
 |
 |
| | 3, 2, 0 > | | 3, 2, 1 > | | 3, 2, 2 > |
Some Probability Densities for Excited Electrons in a Hydrogen Atom
Wolfgang Pauli (1900-1958) Austria -- exclusion principle
The ground states of all elements follow the pattern of the excited states in the hydrogen atom. The structure of the periodic table, which was determined empirically, can be derived theoretically from first principles. Chemistry is the bastard child of physics.
The electrons around an atom are standing, probability waves. It's their interference, when atoms bond and form molecules, that determine molecular structures.
| This Table Needs to Be Checked for Accuracy | |||||
| n shells |
l subshell |
m orbital |
orbital name |
orbital shape |
|
|---|---|---|---|---|---|
| 1, 2, 3, 4, 5, 6, 7, 8 … | 0 | 0 | s | spherical | |
| 2, 3, 4, 5, 6, 7 … | 1 | +1 | p | x | dumbbell |
| 0 | p | y | dumbbell | ||
| −1 | p | z | dumbbell | ||
| 3, 4, 5, 6 … | 2 | +2 | d | xy | double dumbbell |
| +1 | d | xz | double dumbbell | ||
| 0 | d | z2 | dumbbell–torus | ||
| −1 | d | yz | double dumbbell | ||
| −2 | d | x2−y2 | double dumbbell | ||
| 4, 5 … | 3 | +3 | f | y(3x2−y2) | flat triple dumbbell |
| +2 | f | xyz | quadruple dumbbell | ||
| +1 | f | yz2 | triple dumbbell (double dumbbell-torus?) |
||
| 0 | f | z3 | dumbbell–double torus | ||
| −1 | f | xz2 | triple dumbbell (double dumbbell-torus?) |
||
| −2 | f | z(x2−y2) | quadruple dumbbell | ||
| −3 | f | x(3y2−x2) | flat triple dumbbell | ||
| 5 … | 4 | −3 to +3 | g | many-lobed | |
Dirac showed that there are no stable electron orbits for more than 137 electrons, therefore the last chemical element on the periodic table will be untriseptium (137Uts) also known informally as feynmanium (137Fy). It's full electron configuration would be something like …
1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 6s2 4f14 5d10 6p6 7s2 5f14 6d10 7p6 8s2 5g17
or is it …
1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 6s2 4f14 5d10 6p6 7s2 5f14 6d10 7p6 8s1 5g18
data tables
table-ionization.html
table-ionization.txt
table-energy-levels.html
Summary
- bullet
Problems
practice
- Write something.
- Answer it.
- Write something.
- Answer it.
- Write something.
- Answer it.
- Write something completely different.
- Answer it.
numerical
- problems
Resources
- general
- The Mechanical Universe and Beyond (video on demand, login required)
- The Atom, A history of the atom, from the ancient Greeks to the early 20th century, and a new challenge for the world of physics.
- Particles and Waves, Evidence that light can sometimes act like a particle leads to quantum mechanics, the new physics.
- From Atoms to Quarks, Electron waves attracted to the nucleus of an atom help account for the periodic table of the elements and ultimately lead to the search for quarks.
- The Mechanical Universe and Beyond (video on demand, login required)
- orbitals
- historical
- Nobel Prize in Physics 1933, Erwin Schrödinger and P.A.M. Dirac
- Nobel Prize in Physics 1945, Wolfgang Pauli
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