Show that UtâÛ = e-iwtâ for H = wâtâ. (Hint: you may need to use the Baker-Campbell-Hausdorff formua and the Taylor expansion e" = En=o ".)
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- Suppose you have an ideal gas of fermions at room temperature (293 K). How large must E EF be for Fermi-Dirac and Maxwell-Boltzmann statistics to agree to within 1%? Do you think the agreement is within 1% for ideal gases under normal conditions?(a) Prove the "vertical angle hypothesis" (I. 15): opposing angles are congruent if two lines cut each other. (Hint: You'll need to use postulate 4 about right angles in this case.) ) (b) Complete the proof of the Exterior Angle Theorem using section (a): illustrate why beta < alphaWhich one of the following statements about the exchange energy of the few lowest excited states of helium, in which the two electrons are in different subshells, is incorrect? Select an answer and submit. For keyboard navigation, use the up/down arrow keys to select an answer. b с d e In the absence of the exchange energy, the degeneracy between 1s¹2s¹ and 1s¹2p¹ configurations would only be lifted by relativistic corrections. For a given configuration and L the exchange energy favours (ie the energy is lower for) S-1 compared with S=0 O For a given configuration and L the exchange energy favours states that are spatially anti-symmetric The exchange energy gives larger shifts in the levels than relativistic corrections L, S remain good quantum numbers in the presence of the exchange energy
- calculate the heat capacity of an Einstein solid in the low-temperature limit. Sketch the predicted heat capacity as a function of temperature. (Note: Measurements of heat capacities of actual solids at low temperatures do not confirm the prediction that you will make in this problem.Find the number density N/V for Bose-Einstein condensation to occur in helium at room temperature (293 K). Compare your answer with the number density for an ideal gas at room temperature at 1 atmosphere pressure.explain the Fermi Dirac distribution