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I've always been curious -- how can two helium atoms enter the same quantum state when their constituent particles can't? What is happening to the spin-1/2 ele

by panic 8y ago

I've always been curious -- how can two helium atoms enter the same quantum state when their constituent particles can't? What is happening to the spin-1/2 electrons, for example?

cbkeller 8y ago
This may be a bit mangled from the last time I took a relevant class, but my recollection is that the "real" reason electrons can't share the same state is that they're what's called antisymmetric with respect to exchange. In general, most particles are fundamentally indistinguishable: if you swap two of them you haven't really changed anything. This leads to most particles being either symmetric (bosons, integer spin) or antisymmetric (fermions, half-integer spin) with respect to exchange. If symmetric then the wave function doesn't change at all; if antisymmetric then exchange reverses the sign of the wavefunction. If we have two electrons in the same place, and arbitrarily swap them (which we're allowed to do since they're identical), then we've reversed the sign on one of the wavefunctions. But since they're in the same place, the wave functions cancel. This would violate conservation of energy, so isn't allowed; instead, we just never let them occupy the same state in the same place, and call it the Pauli exclusion principle. While a single electron has half-integer spin, a He-4 nucleus has integer spin, so is symmetric to exchange, and it's wavefunction doesn't cancel if you have two in the same place. That was probably horribly oversimplified and unrigorous, but there's a bit of relevant discussion (in bra-ket notation that I can't really read) here: https://en.wikipedia.org/wiki/Pauli_exclusion_principle#Connection_to_quantum_state_symmetry https://en.wikipedia.org/wiki/Pauli_exclusion_principle#Conn...
scentoni 8y ago
If you exchange electron A with A', the wave function changes sign (antisymmetric). But if you exchange A with A' and B with B', nothing is changed (symmetric).

by panic 8y ago

I've always been curious -- how can two helium atoms enter the same quantum state when their constituent particles can't? What is happening to the spin-1/2 electrons, for example?

cbkeller 8y ago

This may be a bit mangled from the last time I took a relevant class, but my recollection is that the "real" reason electrons can't share the same state is that they're what's called antisymmetric with respect to exchange. In general, most particles are fundamentally indistinguishable: if you swap two of them you haven't really changed anything. This leads to most particles being either symmetric (bosons, integer spin) or antisymmetric (fermions, half-integer spin) with respect to exchange. If symmetric then the wave function doesn't change at all; if antisymmetric then exchange reverses the sign of the wavefunction. If we have two electrons in the same place, and arbitrarily swap them (which we're allowed to do since they're identical), then we've reversed the sign on one of the wavefunctions. But since they're in the same place, the wave functions cancel. This would violate conservation of energy, so isn't allowed; instead, we just never let them occupy the same state in the same place, and call it the Pauli exclusion principle. While a single electron has half-integer spin, a He-4 nucleus has integer spin, so is symmetric to exchange, and it's wavefunction doesn't cancel if you have two in the same place. That was probably horribly oversimplified and unrigorous, but there's a bit of relevant discussion (in bra-ket notation that I can't really read) here: https://en.wikipedia.org/wiki/Pauli_exclusion_principle#Connection_to_quantum_state_symmetry https://en.wikipedia.org/wiki/Pauli_exclusion_principle#Conn...

scentoni 8y ago

If you exchange electron A with A', the wave function changes sign (antisymmetric). But if you exchange A with A' and B with B', nothing is changed (symmetric).