[題目] 量子力學

[almgb (almgb)]

[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

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