Question

1. Recall that an energy eigenfunction of any central potential V(r) may be writtren as nemr, 0,) = Rre(r)Yem0,6). This problem explores the behavior of u in the vicinity of the origin r = 0. Recall that the function u(r) = rRre(r) satisfies the equation iz fu fil(l+1) +V(r) u = Eu, (1) 2m dr2 * 2mr2 where E is the energy eigenvalue. Note that Eq. (1) has the usual one-dimensional kinetic-plus- potential form with respect to the radial coordinater. The term hell + 1)/2mra is an additional part of the "effective potential energy" called the centrifugal potential. Now assume that when r is small, V(r) cr", where a > -1. In other words, even if the potential V(r) does blow up at the origin, it doesn't blow up too quickly - the fastest it can blow up is ro, which, of course, is just what the Coulomb potential does. Thus, the Coulomb potential is covered by our assumption, but not potentials that blow up faster than the Coulomb potential. Now let us assume that (> 0, so that the centrifugal potential does not vanish. Thus, near the origin, the centrifugal potential blows up like r-2, and so it dominates over V(r). So, near the origin, we are justified in neglecting V(r), i.e. we may assume V (r) = 0. (i) Make the ansatz that near the origin ur) Ar, and plug this in to Eq. (1). By keeping only the lowest powers of r, show that must equal either (+1 or - (ii) I assert that B =+1 is the only physically allowable value. Why is B = - not allowed for an energy eigenstate? (iii) We may now conclude that Rndr) cr near r=0. Sketch a graph showing r for l = 1,2,3. Here is the physical interpretation of what we have learned. As I increases, the strength of the centrifugal barrier increases, and it thus becomes harder for the wavefunction to tunnel through the barrier to the origin. Thus the larger , the more the wavefunction is suppressed near r = 0, as reflected by the behavior of rl in your sketch. (iv) Based on the preceding analysis, for which atomic state, a p-state (( = 1) or a d-state (l = 2), is it more likely that the electron will be found in the immediate vicinity of the nucleus?

Recall that an energy eigenfunction of any central potential V (r) may be writtren as ψn`m(r, θ, φ) = Rn`(r)Y`m(θ, φ). This problem explores the behavior of ψ in the vicinity of the origin r = 0. Recall that the function u(r) = rRn`(r) satisfies the equation

− ~ 2 2m d 2u dr2 + ~ 2 `(` + 1) 2mr2 + V (r) u = Eu, (1)

where E is the energy eigenvalue. Note that Eq. (1) has the usual one-dimensional kinetic-pluspotential form with respect to the radial coordinate r. The term ~ 2 `(` + 1)/2mr2 is an additional part of the “effective potential energy” called the centrifugal potential. Now assume that when r is small, V (r) ≈ crα , where α ≥ −1. In other words, even if the potential V (r) does blow up at the origin, it doesn’t blow up too quickly – the fastest it can blow up is r −1 , which, of course, is just what the Coulomb potential does. Thus, the Coulomb potential is covered by our assumption, but not potentials that blow up faster than the Coulomb potential. Now let us assume that ` > 0, so that the centrifugal potential does not vanish. Thus, near the origin, the centrifugal potential blows up like r −2 , and so it dominates over V (r). So, near the origin, we are justified in neglecting V (r), i.e. we may assume V (r) = 0.

(i) Make the ansatz that near the origin u(r) ≈ Arβ , and plug this in to Eq. (1). By keeping only the lowest powers of r, show that β must equal either ` + 1 or −`.

(ii) I assert that β = ` + 1 is the only physically allowable value. Why is β = −` not allowed for an energy eigenstate?

(iii) We may now conclude that Rn`(r) ≈ cr` near r = 0. Sketch a graph showing r ` for ` = 1, 2, 3. Here is the physical interpretation of what we have learned. As ` increases, the strength of the centrifugal barrier increases, and it thus becomes harder for the wavefunction to tunnel through the barrier to the origin. Thus the larger `, the more the wavefunction is suppressed near r = 0, as reflected by the behavior of r ` in your sketch.

(iv) Based on the preceding analysis, for which atomic state, a p-state (` = 1) or a d-state (` = 2), is it more likely that the electron will be found in the immediate vicinity of the nucleus?

Answer 1

u EU h24 - A -(10) - via) u Eu | - 49 x x po e = [ 41m) Vin) - Ponte] 2mn? As porno, the centrifugal termidominates, 2 an2 s Assume u= Now App dule Apppy; fu - Ap(p-1) g. 8-2 din

A(A-1) z lf4+1) a fb2-43 -1 (4+1) 20 a) Bz+#1+48(4+1) IV 14 4 1² +41 - 2 lt (al+1) - 2 ilo, pz (Atl) , EN then ua Arol which lelows up " If ß and en 3 en 2 ar Rre I 7

- larger the e, the more the wavefunction is suppressed neap 20. So ooo dal is the most likely state to found the election