Question

3. A particle is in a 1D box (infinite potential well) of dimension, a, situated symmetrically about the origin of the x-axis. A measurement of energy is made and the particle is found to have the ground state energy: 2ma The walls of the box are expanded instantaneously, doubling the well width symmetrically about the origin, leaving the particle in the same state. a) Sketch the initial potential well making it symmetric about x - 0 (note this is different to how we set up the problem in class). Sketch the wavefunction on the same x axes (of course noting that the y axis must now display the eigenstate). Note the functional form of this energy eigenstate in the position basis. b) Sketch the new potential well, again symmetric about x-0. Sketch the first and third energy eigenstates of the new well on these axes too. Now sketch the original wavefunction of the particle on these axes (perhaps in a different colour). Note the functional form of the new energy eigenstates in the position basis. c) Use the functional form of the appropriate wavefunctions and energy eigenstates to compute the probability that a subsequent energy measurement will yield the groundstate energy of the new well. (Hint: Prob - ^out in), what is your in) and what is your out)? and you'll need to transfer to the position basis to perform the computation.) d) What about the probability to find the first excited state (n -2) energy of the new well? Can any general statement be made about symmetry, that will play an important role in simplifying the computation here?

Answer 1

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