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Physics LibreTexts

8.2: Infinite Spherical Potential Well

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Consider a particle of mass m and energy E>0 moving in the following simple central potential:

V(r)={0for 0raotherwise.

Clearly, the wavefunction ψ is only non-zero in the region 0ra. Within this region, it is subject to the physical boundary conditions that it be well behaved (i.e., square-integrable) at r=0, and that it be zero at r=a. (See Section [s5.2].) Writing the wavefunction in the standard form

ψ(r,θ,ϕ)=Rn,l(r)Yl,m(θ,ϕ),

we deduce (see the previous section) that the radial function Rn,l(r) satisfies

d2Rn,ldr2+2rdRn,ldr+[k2l(l+1)r2]Rn,l=0 in the region 0ra, where

k2=2mE2.Defining the scaled radial variable z=kr, the previous differential equation can

be transformed into the standard form

\[\frac{d^{\,2} R_{n,l}}{dz^{\,2}} + \frac{2}{z}\frac{dR_{n,l}}{dz} + \left[1 - \frac{l\,(l+1

)}{z^{\,2}}\right] R_{n,l} = 0.\]

The two independent solutions to this well-known second-order differential equation are called spherical Bessel functions, and can be written

jl(z)=zl(1zddz)l(sinzz),yl(z)=zl(1zddz)l(coszz).

Thus, the first few spherical Bessel functions take the form j0(z)=sinzz,j1(z)=sinzz2coszz,y0(z)=coszz,y1(z)=coszz2sinzz.

These functions are also plotted in Figure [sph]. It can be seen that the spherical Bessel functions are oscillatory in nature, passing through zero many times. However, the yl(z) functions are badly behaved (i.e., they are not square integrable) at z=0, whereas the jl(z) functions are well behaved everywhere. It follows from our boundary condition at r=0 that the yl(z) are unphysical, and that the radial wavefunction Rn,l(r) is thus proportional to jl(kr) only. In order to satisfy the boundary condition at r=a [i.e., Rn,l(a)=0], the value of k must be chosen such that z=ka corresponds to one of the zeros of jl(z). Let us denote the nth zero of jl(z) as zn,l. It follows that

ka=zn,l, for n=1,2,3,. Hence, from Equation ([e9.29]), the allowed energy levels are En,l=z2n,l22ma2. The first few values of zn,l are listed in Table [tsph]. It can be seen that zn,l is an increasing function of both n and l.

The first few zeros of the spherical Bessel function jl(z).
n=1 n=2 n=3 n=4
l=0 3.142 6.283 9.425 12.566
[0.5ex] l=1 4.493 7.725 10.904 14.066
[0.5ex] l=2 5.763 9.095 12.323 15.515
[0.5ex] l=3 6.988 10.417 13.698 16.924
[0.5ex] l=4 8.183 11.705 15.040 18.301

We are now in a position to interpret the three quantum numbers— n, l, and m—which determine the form of the wavefunction specified in Equation ([e9.27]). As is clear from Chapter [sorb], the azimuthal quantum number m determines the number of nodes in the wavefunction as the azimuthal angle ϕ varies between 0 and 2π. Thus, m=0 corresponds to no nodes, m=1 to a single node, m=2 to two nodes, et cetera. Likewise, the polar quantum number l determines the number of nodes in the wavefunction as the polar angle θ varies between 0 and π. Again, l=0 corresponds to no nodes, l=1 to a single node, et cetera. Finally, the radial quantum number n determines the number of nodes in the wavefunction as the radial variable r varies between 0 and a (not counting any nodes at r=0 or r=a). Thus, n=1 corresponds to no nodes, n=2 to a single node, n=3 to two nodes, et cetera. Note that, for the case of an infinite potential well, the only restrictions on the values that the various quantum numbers can take are that n must be a positive integer, l must be a non-negative integer, and m must be an integer lying between l and l. Note, further, that the allowed energy levels ([e9.39]) only depend on the values of the quantum numbers n and l. Finally, it is easily demonstrated that the spherical Bessel functions are mutually orthogonal: that is, a0jl(zn,lr/a)jl(zn,lr/a)r2dr=0 when nn. Given that the Yl,m(θ,ϕ) are mutually orthogonal (see Chapter [sorb]), this ensures that wavefunctions ([e9.27]) corresponding to distinct sets of values of the quantum numbers n, l, and m are mutually orthogonal.

Contributors and Attributions

  • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)


This page titled 8.2: Infinite Spherical Potential Well is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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