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Does anyone know how this works ? I have dψ = (x^2 -cy^2 )dy And my physics teacher says it is (or can be) a harmonic function (Δψ = 0) Can anyone explain ?

$$\mathrm{Does}\:\mathrm{anyone}\:\mathrm{know}\:\mathrm{how}\:\mathrm{this}\:\mathrm{works}\:? \\ $$$$ \\ $$$$\mathrm{I}\:\mathrm{have}\:{d}\psi\:=\:\left({x}^{\mathrm{2}} -{cy}^{\mathrm{2}} \right){dy} \\ $$$$ \\ $$$$\mathrm{And}\:\mathrm{my}\:\mathrm{physics}\:\mathrm{teacher}\:\mathrm{says}\:\mathrm{it}\:\mathrm{is}\:\left(\mathrm{or}\:\mathrm{can}\right. \\ $$$$\left.\mathrm{be}\right)\:\mathrm{a}\:\mathrm{harmonic}\:\mathrm{function}\:\left(\Delta\psi\:=\:\mathrm{0}\right) \\ $$$$\mathrm{Can}\:\mathrm{anyone}\:\mathrm{explain}\:? \\ $$

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Calculate the first order energy correction for the one dimentional non-degenerate an harmonic oscillator whose harmiltonian id written as; H^ =−(h^2 /(2m)) (d^2 /dx^2 ) +(1/2)kx^2 +(1/5)Υx^3 +(1/(12))βx^4

$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{first}\:\mathrm{order}\:\mathrm{energy}\:\mathrm{correction}\: \\ $$$$\mathrm{for}\:\mathrm{the}\:\mathrm{one}\:\mathrm{dimentional}\:\mathrm{non}-\mathrm{degenerate} \\ $$$$\mathrm{an}\:\mathrm{harmonic}\:\mathrm{oscillator}\:\mathrm{whose}\:\mathrm{harmiltonian} \\ $$$$\mathrm{id}\:\mathrm{written}\:\mathrm{as}; \\ $$$$\hat {\mathrm{H}}=−\frac{\mathrm{h}^{\mathrm{2}} }{\mathrm{2}{m}}\:\frac{\mathrm{d}^{\mathrm{2}} }{\mathrm{dx}^{\mathrm{2}} }\:+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{kx}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{5}}\Upsilon\mathrm{x}^{\mathrm{3}} \:+\frac{\mathrm{1}}{\mathrm{12}}\beta\mathrm{x}^{\mathrm{4}} \\ $$

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