An electron is confined to move in the xy plane in a

Chapter 41, Problem 56

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An electron is confined to move in the xy plane in a rectangle whose dimensions are Lx and Ly. That is, the electron is trapped in a two-dimensional potential well having lengths of Lx and Ly. In this situation, the allowed energies of the electron depend on two quantum numbers nx and ny and are given by E 5 h 2 8me a nx 2 Lx 2 1 ny 2 Ly 2 b Using this information, we wish to find the wavelength of a photon needed to excite the electron from the ground state to the second excited state, assuming Lx 5 Ly 5 L. (a) Using the assumption on the lengths, write an expression for the allowed energies of the electron in terms of the quantum numbers nx and ny. (b) What values of nx and ny correspond to the ground state? (c) Find the energy of the ground state. (d) What are the possible values of nx and ny for the first excited state, that is, the next-highest state in terms of energy? (e) What are the possible values of nx and ny for the second excited state? (f) Using the values in part (e), what is the energy of the second excited state? (g) What is the energy difference between the ground state and the second excited state? (h) What is the wavelength of a photon that will cause the transition between the ground state and the second excited state?