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(Solved): Suppose we have an electron in a one dimensional potential well with thicknes ...

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Suppose we have an electron in a one dimensional potential well with thickness in the direction with infinitely high barriers on either side and zero potential at the bottom. Now suppose we add a perturbing Hamiltonian as a potential energy of the form , where is positive, non-zero real number, and corresponds to the middle of the well. The solutions to the unperturbed problem are the usual wavefunctions for the particle in such a box with infinitely high walls, where is the lowest energy state, is the second lowest, and so on. Consider whether each of the following six statements is true or false. Note: This is a compound true/false question. In the six statements below, exactly three are true and three are false. For each attempt you make to answer this question, select only the three correct statements. Be sure the other three boxes are not checked. You may want to keep a note of your answer attempts. First, consider the electron to be in the lowest eigenenergy state of the problem. In the resulting perturbed wavefunction, according to first order perturbation theory, there is no component of in the perturbed wavefunction (i.e., ). The first order correction to the wavefunction is an odd function. In the resulting perturbed wavefunction, according to first order perturbation theory, we can conclude solely from consideration of even and odd functions (i.e. parity arguments) that there is no component of in the perturbed wavefunction (that is, ). Now consider the electron to be in the second lowest eigenenergy state of the system . Presuming that there is a first order wavefunction correction to the second state (i.e., to the unperturbed state ), that wavefunction correction is an odd function. Now consider the electron to be in the third lowest eigenenergy state of the problem. In the first order perturbation correction to the wavefunction of this third energy eigenstate in the well, there will be a component of in the correction. Now consider the following, which applies for the electron in any specific energy eigenstate. To first order in perturbation theory, the energies of all the energy eigenstates are increased as the number is increased.

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