Figures from Advanced Quantum Theory


Existence of bound states for spherical square well
1) v0 = 0.5: Fig. 5.1 . The potential is too shallow to support a bound state. (Recall: v0 = V0 2 M V02 / \hbar2, r0 being the extension of the well, V0 its depth and M the mass of the particle.)
2) v0 = 3: Fig. 5.2 . The potential can now (just) support one bound state.
3) v0 = 30: Fig. 5.3 . The potential can now support two bound states.
4) v0 = 1000: Fig. 5.4 . The potential can now support ten bound states. The first few solutions occur where the cot function is large and negative, i.e. for q r0 just below n pi, n being an integer. The last solutions occur near the zeros of the cot function, i.e. for q r0 near (n+1/2) pi.

Bound state wave functions for spherical square well
1) v0 = 3 or 30: Fig. 5.5 . The shallow potential only supports a loosely bound state, with very broad wave function. The deeper potential also supports a bound state whose wave function is peaked well within the potential. Note that |r R(r)|2 is the probability density to find the particle at distance r from the origin.
2) v0 = 30: Fig. 5.6 . There are two S-wave states and one P-wave state. The latter is intermediate in energy between the S-wave states. States with larger binding energy (more negative E) fall off faster at large distance.

S-wave scattering phase for spherical square well
1) Shallow well: Fig. 5.7 . If the potential is so shallow that it does not support a bound state, i.e. for v0 < pi2/4 = 2.47, the sine of the scattering phase remains below 1 in magnitude everywhere. Note that the maximum of the scattering cross section moves to smaller energies, i.e. smaller k, as v0 increases in this regime. At the critical value of v0 the first resonance appears, at k = 0. For yet larger v0 the maximum of the cross section moves to larger values of k again, until eventually a second resonance appears at k = 0.
Note also that the scattering phase becomes small at small k, except near the critical value of v0 where a new resonance appears. We saw in class that delta0 is proportional to k here. Similarly, the scattering phase becomes small again at large k, where delta0 = v0 / (2kr0) - v0 sin(2kr0) / (2kr0)2.
2) v0 = 1000: Fig. 5.8 . There are now ten resonances. Note that this corresponds exactly to the number of bound states for this value of v0, see Fig. 5.4 above!

S-wave unbound wave functions for spherical square well
Shallow well: Fig. 5.9 . The values of v0 are the same as in Fig. 5.7. k is chosen such that |sin(delta0)| takes its maximal value, i.e. such that the S-wave cross section is maximal. We see that this gives a universal result if v0 is below the critical value, i.e. if no bound state exists, reaching its maximum at r = r0. For larger v0 the maxima of the wave function outside of the potential, r > r0, are higher than the one inside the potential. Note that the wave functions are normalized such that the maximum of |rR0/r0| = 1.

P-wave scattering phase for spherical square well
1) Shallow well: Fig. 5.10 . For v0 < 3 pi2/4 = 7.4, the sine of the scattering phase remains below 1 in magnitude everywhere. The maximum of the scattering cross section again moves to smaller energies, i.e. smaller k, as v0 increases in this regime. At the critical value of v0 the first resonance appears, at kr0 = pi/2. Note that at this value of v0 the potential does not yet support a true bound state. However, the presence of a potential barrier in the effective potential already leads to a quasi-bound, resonance, state. For yet larger v0 the maximum of the cross section moves to larger values of k again, until eventually a second resonance appears at kr0 = pi/2.
Note also that the scattering phase is even more suppressed at small k than the S-wave scattering phase; we saw in class that delta1 is proportional to k3 here. The behavior at large k is very similar as in case of the S-wave, except that the sign of the modulated term is flipped, delta1 = v0 / (2kr0) + v0 sin(2kr0) / (2kr0)2.
2) Deeper well: Fig. 5.11 . v0 = 30 is well above the critical value for the first resonance, but still below the value where the second resonance appears. We see that the P-wave cross section has a couple of maxima well below the resonance; however, the cross section at these maxima is still quite small.
New resonances appear at v0 = (n2 - 1/4) pi2, whereas new true bound states appear at v0 = (n pi)2. We see that at the critical values where the second or third resonance appears, the behavior of the cross section near the first resonance, which always starts at kr0 = pi/2, is nearly universal.
Note also that the asymptotic behavior for large k sets in only for (kr0)2 >> v0. Larger values of v0 also support resonances at quite large values of kr0.

P-wave unbound wave functions for spherical square well
Shallow well: Fig. 5.12 . The values of v0 are the same as in Fig. 5.10. k is chosen such that |sin(delta1)| takes its maximal value, i.e. such that the P-wave cross section is maximal. The behavior of the wave functions is similar to that shown in Fig. 5.9 for the S-wave, except that the critical values of v0 have to be increased, as discussed above; moreover, at the critical value of v0 a new resonance appears for kr0 = pi/2, not at k = 0 as in case of the S-wave.