Symmetric and Antisymmetric WF

[Cosmossos]

Hello,
Why do symmetric wave function has less energy than the anti symmetric wave function and how does it connect to the number of the nodes (why existence of a node point in the anti symmetric tells us that this is more energetic function?)

 

[Simon Bridge]

Depends on the system - you can get a symmetric and anti-symmetric solution for the same energy ... but the coupling with the spin states splits the energy-level. However, in harmonic or square potentials you will see clearly that the lowest energy state is always symmetric.

An even number of nodes means there is an antinode at the middle of the potential - so there is a relatively higher probability of finding the particle, in that state, close to the center, which makes it more "tight" - the particle spends more time hanging around the middle away from the edges. This makes it harder to pull away, therefore, lower energy.

That help?