Why is the ground state eigenfunction in a symmetric hamiltonian also symmetric?

[njoshi3]

Hi,
Why is that, the ground state eigenfunction in ANY quantum mechanical system is symmetric under inversion of co-ordinates?

 

[lethe]

Originally posted by njoshi3
Hi,
Why is that, the ground state eigenfunction in ANY quantum mechanical system is symmetric under inversion of co-ordinates?

well first of all, this isn t true. the groundstate is only symmetric if the hamiltonian is also symmetric.

so let s assume that you had asked this question about any quantum system with a symmetric hamiltonian.

when there is a symmetric hamiltonian, all eigenfuctions must be either even or odd. that is, they must be either symmetric or antisymmetric.

an antisymmetric wavefunction necessarily passes through zero. therefore it has a node, therefore it cannot be the ground state.

therefore the ground state must be symmetric.