Show that the energy of a simple harmonic oscillator in the n = 1 state is 3ℏω/2 by substituting the wave function ψ1 = axe-αx2/2 directly into the schroedinger equation, as broken down in the following steps. first, calculate dψ1/dx, using a, x, and α. dψ1/dx = second, calculate d2ψ1/dx2, using a, x, and α. d2ψ1/dx2 = third, calculate α2x2ψ1 - d2ψ1/dx2, using a, x, and α. α2x2ψ1 - d2ψ1/dx2 = fourth, calculate (α2x2ψ1 - d2ψ1/dx2)/ψ1, using a, x, and α. (α2x2ψ1 - d2ψ1/dx2)/ψ1 = finally, calculate e ≡ [(α2x2ψ1 - d2ψ1/dx2)/ψ1]ℏ2/(2m), using a, x, α, m, ω, and ℏ. e ≡

As part of an industrial process, air as an ideal gas at 10 bar, 400k expands at steady state through a valve to a pressure of 4 bar. the mass flow rate of air is 0.5 kg/s. the air then passes through a heat exchanger where it is cooled to a temperature of 295k with negligible change in pressure. the valve can be modeled as a throttling process, and kinetic and potential energy effects can be neglected. (a) for a control volume enclosing the valve and heat exchanger and enough of the local surroundings that the heat transfer occurs at the ambient temperature of 295 k, determine the rate of entropy production, in kw/k. (b) if the expansion valve were replaced by an adiabatic turbine operating isentropically, what would be the entropy production? compare the results of parts (a) and (b) and discuss.

What are the necessary conditions for fusion?

An object with initial temperature 130 ∘ f is submerged in large tank of water whose temperature is 50 ∘ f . find a formula for f ( t ) , the temperature of the object after t minutes, if the cooling constant is k = − 0.2 . remember newton's law of cooling (the rate of change of temperature with respect to time is equal to k times the difference between the temperature of the object and the surrounding temperature) ! : )