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Obtain an approximate analytic cxpression for the energy level in a square well potential \((l=0)\) when \(V_{0} a^{2}\) is slightly greater than \(\pi^{2} \hbar^{2} / 8 m\)

A film of oil lies on wet pavement. The refractive index of the oil exceeds that of the water. The film has the minimum nonzero thickness such that it appears dark due to destructive interference when viewed in red light (wavelength \(=640.0 \mathrm{~nm}\) in vacuum). Assuming that the visible spectrum extends from 380 to \(750 \mathrm{~nm}\), for which visible wavelength(s) (in vacuum) will the film appear bright due to constructive interference?

It has been speculated that the present-day acceleration of the universe is due to the existence of a false vacuum, which will eventually decay. Suppose that the energy density of the false vacuum is \(\varepsilon_{\Lambda}=0.7 \varepsilon_{c, 0}=3600 \mathrm{MeV} \mathrm{m}^{-3}\), and that the current energy density of matter is \(\varepsilon_{m, 0}=0.3 \varepsilon_{c, 0}=1600 \mathrm{MeV} \mathrm{m}^{-3}\). What will be the value of the Hubble parameter once the false vacuum becomes strongly dominant? Suppose that the false vacuum is fated to decay instantaneously to radiation at a time \(t_{f}=50 t_{0}\). (Assume, for simplicity, that the radiation takes the form of blackbody photons.) To what temperature will the universe be reheated at \(t=t f ?\) What will the energy density of matter be at \(t=t_{f} ?\) At what time will the universe again be dominated by matter?

The polaron self-energy \(-\alpha \omega_{0}\), contains some kinetic energy of the electron and some potential energy. Use the Rayleigh-Schr枚dinger wave function to show these have the ratio of \(1: 3\) at \(p=0\).

Two identical, noninteracting spin-\frac{1 } particles of mass \(m\) are in the onedimensional harmonic oscillator for which the Hamiltonian is $$ \hat{H}=\frac{\hat{p}_{1 x}^{2}}{2 m}+\frac{1}{2} m \omega^{2} \hat{x}_{1}^{2}+\frac{\hat{p}_{2 x}^{2}}{2 m}+\frac{1}{2} m \omega^{2} \hat{x}_{2}^{2} $$ (a) Determine the ground-state and first-excited-state kets and corresponding energies when the two particles are in a total-spin-0 state. What are the lowest energy states and corresponding kets for the particles if they are in a totalspin-1 state? (b) Suppose the two particles interact with a potential energy of interaction $$ V\left(\left|x_{1}-x_{2}\right|\right)= \begin{cases}-V_{0} & \left|x_{1}-x_{2}\right|

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