The wave function corresponding to energy level is
where and is the th Hermite polynomial. These wavefunctions represent all possible solutions to the harmonic oscillator potential that go to when or equivalent goes to infinity. Therefore, these wavefunctions represent all possuble solutions to the harmonic oscillator potential on a domain that is infinite in at least one direction (like in this problem). Now, Hermite polynomials are even functions if is even and odd functions if is odd. Therefore, only those where is odd satisfy the boundary condition . Therefore, only these are solutions for the potential in this problem. Therefore, answer (E) is correct.
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