A quantum particle has $V(x)=0$ for $-a/2<x<a/2$ and $V(x)=\infty$ outside, including the walls. Its ground-state wave function inside the well is proportional to
The ground state is even, has no interior node and must vanish at $x=\pm a/2$. The function $\cos(\pi x/a)$ has all these properties. The cosine with denominator $2a$ does not vanish at the walls, while the sine alternatives have a node at the centre or fail the boundary condition. Thus D.
More One dimensional potentials: step potential, finite rectangular well, tunnelling from a potential barrier, particle in 1,2,3-dimensional box, particle in single and double delta function potentials, 1,2,3 dimensional harmonic oscillator: concept of degeneracy questions
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