[領域] (題目相關領域) 量子力學 [來源] (課本習題、考古題、參考書...) 課本 [題目] We remarked in Impact I 9.2 that the particle in a sphere is a reasonable starting point for the discussion of the electronic properties of spherical metal nanoparticles. Here,we justify eqn 9.54,which shows that the energy of an electron in a sphere is quantized. (a)The Hamiltonian for a particle free to move inside a sphere of radius R is ^ h 2 H = - _______ ▽ 2m Show that the Schrodinger equation is separable into radial and angular components.That is,begin by writing Ψ(γ,θ,ψ)=Ｘ(γ)Ｙ (θ,ψ),Where　Ｘ(γ)depends only on the distance of the particle away from the centre of the sphere, andＹ(θ,ψ)is a spherical harmonic.Then show that the Schrodinger equation can be separated into two equations,one forＸ,the radial equation,and the other for Ｙ,the angular equation: 2 2 2 2 2 -h /2m 【 d Ｘ(γ)／d γ + 2/γ*dＸ(γ)／ｄγ】+ι(ι+1)h /2mγ *X(γ) =EX(γ) 2 Λ Ｙ＝ -ι(ι+1)Ｙ Youmay wish to consult Further information 10.1 for additional help. (c)Consider the caseι=0. Show by differentiation that the solution of the radial equation has the from -1/2 Ｘ(γ)=(2πR) sin(nπγ/R) ／γ (e)Now go on to show that the allowed energies are given by: 2 2　　　2 Ｅn = n h ／８mＲ This result for the energy (which is eqn9.54 after substituting ｍe for ｍ) also applies when ι≠0 [瓶頸] (寫寫自己的想法，方便大家為你解答) 只知道大概的意思 不知道該怎麼解釋 請板上強者幫忙!! --

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