Symmetry of Spatial Wavefunctions

[cooev769]

I know about symmetry and antisymmetry and so on, but a thought that I had never considered just hit me. If we had two fermions in the triplet symmetric spin state and hence therefore an antisymmetric spatial state, for example a harmonic oscillator in the first excited state must be one in state n and one in k:

psi = 1/(sqr root 2)( n1k2 - n2k1 )

But this wavefunction is clearly different from

psi = 1/(sqr root 2)( n2k1 - n1k2 )

Clearly it is just the exchange operator applied to the wave function, i can see that the expectation values will always be the same, so can you just choose how you order these and in which order they go?

 

[DrDu]

The two functions are identical up to a phase factor -1. Wavefunctions differing only by a phase factor $\exp(i\phi)$ with phi independent of coordinates describe the same state as the phase factor cancels out when forming the expectation value of any operator.

 

[cooev769]

As I thought. But I'm not quite sure what a phase factor is or where it is coming from.

 

[DrDu]

A phase factor is simply a complex number of unit absolute value whence they can be expressed in polar coordinates as $r\exp{i\phi}$ with r=1. The angle phi is often called phase as we have a wave picture in mind. You can check that the time dependent Schrödinger equation changes the phase of an eigenstate with time. However, the phase of different eigenstates changes differently with time, so that a superposition of energy eigenstates will not be stationary.