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Science · Advanced Physics
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Write down the time dependent Schrödinger equation, and state its solution for the case where the Hamiltonian is not time dependent 1. (a) ) 2 marks) (i) A particle moves in a potential well such that V-0 for sa, and - otherwise. This system has energy eigenvalues En and corresponding normalised for n odd 2a pisa and un =0for |x/>a infor neven 2a At time t-0 this particle is described by the un-normalised wave function Normalise this wave function and show that the position probability density of the particle at any later time t is given by Given this expression, explain with the aid of suitable sketches how the probability of finding the particle at any position within the well will vary in time. Note: exact details are not reguired 4 marks b) The generalised uncertainty principle for two observables A and B, can be written as ABwh) represents the mean value of the commutator of operators A and B. (i) Show that this generalised statement reduces to the familiar statement of the uncertainty principle when dealing with a particles x-coordinate and its momentum in the x-direction. The uncertainty principle is a central element of the standard formalism of quantum theory that quantifies the unavoidable mutual disturbance associated with making observations of A and B. Explain briefly, in terms of the formalism, why this mutual disturbance is unavoidable. 2 marks) (ii) 2 marks)

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