Quantum Numbers in Atomic Physics
Quantum numbers describe the unique quantum state of an electron in an atom. They are derived from solutions to the Schrödinger equation and define atomic orbitals and electron properties.
The Four Quantum Numbers
| Quantum Number | Symbol | Possible Values | Physical Meaning |
|---|---|---|---|
| Principal | n | 1, 2, 3, ... | Energy level/shell (size of orbital) |
| Azimuthal (Angular Momentum) | l | 0 to n-1 | Subshell shape (s, p, d, f) |
| Magnetic | ml | -l to +l | Orbital orientation in space |
| Spin | ms | +½ or -½ | Electron spin direction |
Detailed Explanation
1. Principal Quantum Number (n)
- Determines the energy level and average distance from the nucleus
- Maximum electrons in shell = 2n2
- Higher n means higher energy and larger orbital size
2. Azimuthal Quantum Number (l)
- Defines the orbital shape and subshell type:
- s orbitals (l=0): spherical
- p orbitals (l=1): dumbbell-shaped
- d orbitals (l=2): cloverleaf-shaped
- f orbitals (l=3): complex shapes
3. Magnetic Quantum Number (ml)
- Specifies the spatial orientation of the orbital
- For l=1 (p orbitals), ml can be -1, 0, +1 → three p orbitals (px, py, pz)
4. Spin Quantum Number (ms)
- Represents the electron's intrinsic spin (↑ or ↓)
- Pauli Exclusion Principle: No two electrons can have identical sets of all four quantum numbers
Example: 3d2 electron configuration
For two electrons in the 3d subshell:
- First electron: n=3, l=2, ml=+2, ms=+½
- Second electron: n=3, l=2, ml=+1, ms=-½
Note: These follow Hund's Rule (maximize parallel spins) and the Pauli Exclusion Principle.
Visual Representation of Orbitals
Different combinations of quantum numbers produce different orbital shapes:
- s orbitals (l=0): 1 possible orientation (spherical)
- p orbitals (l=1): 3 possible orientations (px, py, pz)
- d orbitals (l=2): 5 possible orientations (dxy, dxz, dyz, dx²-y², dz²)
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