1. Consider a wave function of Lorentzian form, i.e., 1 (x): = x² + a² a) What are < > and Ax? b) What are < p > and Ap? c) Compare with the uncertainty relation.

1) An electron is confined to a square box of length L, and the walls of that box are infinitely high. The zero-point energy (ZPE) is defined as the minimal energy that corresponds to the smallest quantum number n. What would be the length of the box L such that the ZPE of the electron located inside this box is equal to its rest mass energy mec2?

9B. a) Do particles of matter have wave properties? b). Who was the first physicist to give a convincing answer to the question? c). As the speed of a particle increases, does its associated wavelength increase or decrease? Your answer

5. A microscopic oscillator has its first and second excited states 0.05 eV and 0.10 eV above the ground-state energy. Calculate the Boltzmann factor for the ground state, first excited state, and second excited state, at room temperature.