Question

P7F.12 A particle confined within a spherical cavity is a starting point for the discussion of the electronic properties of spherical metal nanoparticles. Here, you are invited to show in a series of steps that the energy levels of an electron in a spherical cavity of radius Rare quantized and given by E = 'hº/8m.Rº. (a) The hamiltonian for a particle free to move inside a spherical cavity of radius a is V with V-18 Show that the Schrödinger equation is separable into radial and angular components. That is, begin by writing (1,0,0) = R(r)Y(0,0), where R(r) depends only on the distance of the particle from the centre of the sphere, and Y(0,) is a spherical harmonic. Then show that the Schrödinger equation can be separated into two equations, one for R(r), the radial equation, and the other for Y(0,6), the angular equation. (b) Consider the case I -0. Show by differentiation that the solution of the radial equation has the form R(P)(2a) sin(ntr/a) (c) Now go on to show (by acknowledging the appropriate boundary conditions) that the allowed energies are given by E.-'/8ma'. With substitution of m, for mand of R for a, this is the equation given above for the energy

Answer 1

Guieu hat, he hailtowan free pautile to meve shere of radius R io imaide a sn Sphetical cooidinates, r e 0.ms nod) e R YO, ut m +2mf Rerd)J = 0 Lin9CRl Y (9 + 3CROYO) T t Dividing we efuauin mith R Yo) T3RG Sin0 Ylo,4)rti0 sin'o Ye) de Oemg ro,b) SmE Ratial patt Pangulae pask

ib .. - Rir) ar fe -o 9 mE 2 2 Lut v r (r YR R & e pifferensatuiy ain e Suedige equarim cm be wntten as +2 me ur d2 0 kn2mEn 2 V A n Kn+Bces knA T) A knd +Beas Ky Y

Emce ,ab O, Rr) muat be inile, co, B mat be Rmola) A sn kn =a VE Srnce at So Rne O A Sn kna o a whee nl,,3 kna nTT Ehikn Kn 2 m Rno A Sm/nT ma ruau in Noemauzavin f fadial wae (A1 4TT 41 (AISm de 2 2nTTY 4T A) Tia Jafra Ro=ISin a En wao Calulaled uan beundaty condrtuon beet tnat ab amd En n T2 A ma

ne re meme aER En 2 me R 2 4T 2 & meR En 2 meR