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2. (14 pts) A quantum particle with unit mass (m = 1) is in a one dimensional box where the potential energy, V (x) = o for o < x < 2 and V (x) = 80 otherwise. Answer the following questions. (a) (6/14) Prove that on(x) = sin ( NTT ), n = 1,2, 2 is an eigenfunction of the Hamiltonian satisfying the boundary condition, determine its eigen- value, and the corresponding de Broglie wavelength. (b) (8/14) particle is in a state Given that the represented by the following wave function, = 1 TTX 2 + 13 2 what are the expectation values of the energy and position measurements?

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