(a) The energy eigenstates of an infinite well (of width L) are well 9) = (x) = V (haj) | (pj); 2mL (b) The energy eigenstates of a harmonic oscillator are -19); j = 1,2,3... sin j(x+L/2), xe [-L/2, L/2] otherwise. The wave functions have been defined with the spatial origin (x = 0) fixed to the center of the well. (In the discussion section. x = 0 at the edge of the well. Choices of origin don't matter to physics.) From the behavior of the wave functions under flipping x -x, determine which of 1,) is parity-even and which parity-odd; confirm your calculation by the simpler exercise of staring at pictures of the wave functions. Hosen) = ho(n + 1/2)|n). The parity operator inverts position and momentum operators as (1) P&P-. Ppp- = -p. = (2) (3) The ground state 10) is parity-symmetric: P|0) = 10), as manifested by the Gaussian function being even in x x. De- termine Pp- and Pat P-1 in their simplest forms, then determine the parity eigenvalues P, in P|n) = P|n) for general n. applying that | n) ()"| 0) up to a normalization factor. Confirm your calculation by the simpler exercise of staring at pictures of the wave functions. (c) For which j = 1, 2, 3, ... and n = 0, 1, 2, ... is the inner product ( [n) = 0? You should assume that the center of the well coincides with the center of the oscillator, and you may apply that the integral (over x from-co to co) of an odd function vanishes. (d) Suppose the centers don't coincide. Answer question (c) again.