Particle in a Box: Odd/Even Parity Explained

[Amith2006]

Homework Statement

Consider a particle in a 1-D box with impenetrable walls at x=0 and x=L. Then the eigen function for this problem is of the form,
$$\Psi(x) = \sqrt{2/L}sin(n\pi\ x/L)$$
This is always of odd parity. The question which is troubling me is that, in this case though the eigen function seems to be of odd parity mathematically, when see their eigen states, they are alternately symmetric and anti-symmetric i.e odd and even parity.
On the contrary, if the origin is shifted to the centre of the box in which case the particle is confined between -L/2 to +L/2, then there are 2 sets of eigen functions. One set of functions is of odd parity and other set of functions is of even parity. Since both are of same width, how can there exist such a difference? Could someone point out the flaw in my understanding?

Homework Equations

The Attempt at a Solution

 

[badphysicist]

For the first situation with walls at x=0 and x=L, the solution you gave is correct and it has odd parity with respect to the origin. However, that equation only gives the particle's wave function from 0 to L. The functions have alternating odd and even parity with respect to x=L/2 though. I hope that helps.

 

[lanedance]

the results are equivalent, but as L/2 is the natural axis of symmetry of the system, that's the one I'd condier parity in