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- 4. a) Describe the Variational Principle = e-ax? b) The variation method is applied to a harmonic oscillator using a trial function = e Minimize the energy as a function of the variable a. k The solution for energy is E = 24 8a Leave your answer in terms of h, k, and u. Compare your answer to the true energy of a harmonic oscillator, E = v+) ha, where v = 0,1,2, ... and w =Problem 1. Two State System Consider an atom with only two states: a ground state with energy 0, and an excited state with energy A. Determine the mean energy (e) and variance in energy (de). Sketch the mean energy versus A/k T.Q1: Find the first excited state of harmonic oscillator using the equation: P(x) = An(a*)(x), with E₂ = (n + hw
- The chemical potential of an ideal gas Use as ƏN E,V V S(E,V, N) = Nk| ln N 3 In 2 3Νπh? a. Derive the dependence of the chemical potential u on E, V, and N for an ideal classical gas. b. Use 3 E = -NkT. to determine µ(T, V, N)For a harmonic oscillator state |n⟩, determine a. <x(hat)^2> b. <p(hat)^2> c. ΔxΔpA simple harmonic oscillator is in the ground state with A-², where A=1/41/4 jm-1/2 and a = 1/2 m2, What is a (x²) and Ax? (Hint: [ [_xe-x²³ dx=0 and [_x²a-b²³dx==√m/6²³ ) Ax- μm m2 um
- A) A Photo by Kwines - Given the function =3xyz²î + 2xy³ĵ – x²yzk Find the divergence V. b. a. 3yz²î + 6xy²j - x²yk 3yz² + 6xy² - x²y c. 3xyz²+ 2xy³ – x²yz d. 85. A particle of mass m in a rectangular box with dimensions x, y, z has ground k² ( 1 1 + y? 1 state energy E(x, y, z) = where k is a physical constant. If + 8m x the volume of the box is fixed (say V =xyz ), find the values of x, y, and z that minimize the ground state energy.x + 5 x 5 Calculate the following values: a. q( – 3) : b. q(7) = q(5) = C.
- 4. Calculate the mean kinetic energy, , for a harmonic oscillator in the ground 1.3.5...(2n-1) O/2 , and 00 state. Given that Sx2n exp(-ax²) dx = 00 2nan L, exp(-ax²) dx = E)+/ . 00 (hv/4 or ½ Eo)0.|A particle is confined in a potential V(x)= -2x². A. The particle is at the second excited ground state. Please write down the wave function. (normalized) B. What the energy of the state. C. Write down the probability function.1. Show that the partition function for a harmonic oscillator in the x direction only is given by: exp(-ßhv) 1-exp(-ßhv) 9x =