Tính đối xứng hoán vị, cặp spin và nguyên lý loại trừ Pauli
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| singlet/ˈsɪŋɡlɪt/ | The spin-zero two-particle state, antisymmetric under exchange. |
| exchange/eksˈtʃeɪndʒ/ | Swapping the labels of two identical particles. |
| triplet/ˈtrɪplɪt/ | The three spin-one two-particle states, symmetric under exchange. |
Two electrons, one orbital
- Two electrons can share a spatial orbital — but only with opposite spins in the singlet 单态. The rule is antisymmetry of the total state.
- GRE identical-particle questions test exchange 交换 symmetry, singlets and triplets 三重态, counting and exchange correlations.
Exchange symmetry
- Swap all labels (space and spin): fermions change sign, bosons do not.
- Orbitals $a \ne b$: $\Psi_\pm = \tfrac{1}{\sqrt2}[a(1)b(2) \pm b(1)a(2)]$. If $a = b$, only $a(1)a(2)$ exists — symmetric.
Singlet and triplet
- Singlet $\tfrac{1}{\sqrt2}(\uparrow\downarrow - \downarrow\uparrow)$, $S = 0$, antisymmetric. Triplet ($\uparrow\uparrow$, $\tfrac{1}{\sqrt2}(\uparrow\downarrow + \downarrow\uparrow)$, $\downarrow\downarrow$), $S = 1$, symmetric.
- Electrons: symmetric space × singlet, or antisymmetric space × triplet.
Allowed for two electrons?
Combine spatial and spin symmetry into an antisymmetric total state.
Two electrons both in orbital $a$ must be in
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With both electrons in orbital $a$ the spatial part $a(1)a(2)$ is symmetric, so the spin part must be antisymmetric: the singlet.
Counting
- Two orbitals × two spins = 4 complete states: $\binom42 = 6$ two-electron states — two double-occupation singlets, one open-shell singlet, three triplets.
- Three orbitals: $\binom62 = 15$. Spinless bosons in two orbitals: 3.
How many two-electron states can be built from three spatial orbitals (with spin)?
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Three orbitals with two spin states give 6 spin-orbitals; two electrons fill 2 of them: $\binom62 = 15$.
Two identical spinless bosons in three orbitals. How many states?
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Two identical bosons in 3 orbitals: $\binom{3+2-1}{2} = \binom42 = 6$.
How many of the six two-orbital states are triplets?
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All with one electron in each orbital.
Exchange correlation
- $\Psi_-(x, x) = 0$: antisymmetric spatial states keep particles apart, with no force involved.
Well of length $L$, $a = \sqrt{2/L}\sin(\pi x/L)$, $b = \sqrt{2/L}\sin(2\pi x/L)$. At $x_1 = L/4$, $x_2 = 3L/4$: $\Psi_- = -2/L$ (density $4/L^2$) and $\Psi_+ = 0$.
The vanishing of $\Psi_-$ at $x_1 = x_2$ comes from an added repulsive force.
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It comes from antisymmetry alone.
Recap
- Antisymmetric total state for electrons.
- Symmetric space ↔ singlet; antisymmetric space ↔ triplet.
- Count complete states, then choose.
- singlet: The spin-zero two-particle state, antisymmetric under exchange.
- exchange: Swapping the labels of two identical particles.
- triplet: The three spin-one two-particle states, symmetric under exchange.