[2] 2. (a) The expansion postulate allows us to write a general wave function, in terms of a linear combination of eigen

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2 2 A The Expansion Postulate Allows Us To Write A General Wave Function In Terms Of A Linear Combination Of Eigen 1 (96.75 KiB) Viewed 36 times

[2] 2. (a) The expansion postulate allows us to write a general wave function, in terms of a linear combination of eigenfunctions, or. Explain how this expansion can be used to make predictions about the possible outcomes of a measurement (b) Consider a particle of mass m in a region of zero potential between two impenetrable barriers at I = 0 and x = a. The particle has allowed energies E, and corresponding eigenstates on given by 7?hn En = and On (1) = sin 2ma? ("A"). At time t = 0, the wave function of the particle is n = = { 0<I<a/2 a/2 <r<a, -A [2] [4] where A= Draw a labelled sketch of this wave function. va (c) Determine an expression for the expansion coefficients, an that make up this wave function, and show that the first two non-zero coefficients are 22 22 02 and 06 3л TT [2] (] [2] ] (d) What is the probability that a measurement will determine the particle to be in either the state 02 or 06? (e) We notice that a = 12/3. We can therefore approximate (r) above by 1 Vapprox. (x)=B (02 +506 B ( Use your knowledge of the expansion postulate to calculate a value for the normalisation constant B for approx. (3). (f) For the wave function approx. (x) in part (e), write down the normalised wave function, Vapprox. (x, t) at some time t later. (9) Calculate the probability density for the wave function Vapprox. (x, t) from part (f), and show that it oscillates with time according to cos(AEt/h). You can leave your answer in terms of 2 and 06. (h) Determine the frequency at which this probability density oscillates. [2] . [4] [2] ]