8.2: Infinite Spherical Potential Well
( \newcommand{\kernel}{\mathrm{null}\,}\)
Consider a particle of mass m and energy E>0 moving in the following simple central potential:
V(r)={0for 0≤r≤a∞otherwise.
Clearly, the wavefunction ψ is only non-zero in the region 0≤r≤a. 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
we deduce (see the previous section) that the radial function Rn,l(r) satisfies
d2Rn,ldr2+2rdRn,ldr+[k2−l(l+1)r2]Rn,l=0 in the region 0≤r≤a, where
k2=2mEℏ2.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)=sinzz2−coszz,y0(z)=−coszz,y1(z)=−coszz2−sinzz.
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,lℏ22ma2. 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.
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 n≠n′. 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)