A.29 WKB so­lu­tion near the turn­ing points

Both the clas­si­cal and tun­nel­ing WKB ap­prox­i­ma­tions of ad­den­dum {A.28} fail near so-called turn­ing points where the clas­si­cal ki­netic en­ergy $E-V$ be­comes zero. This note ex­plains how the prob­lem can be fixed.

Fig­ure A.17: The Airy Ai and Bi func­tions that solve the Hamil­ton­ian eigen­value prob­lem for a lin­early vary­ing po­ten­tial en­ergy. Bi very quickly be­comes too large to plot for pos­i­tive val­ues of its ar­gu­ment.

The trick is to use a dif­fer­ent ap­prox­i­ma­tion near turn­ing points. In a small vicin­ity of a turn­ing point, it can nor­mally be as­sumed that the $x$-​de­riv­a­tive $V'$ of the po­ten­tial is about con­stant, so that the po­ten­tial varies lin­early with po­si­tion. Un­der that con­di­tion, the ex­act so­lu­tion of the Hamil­ton­ian eigen­value prob­lem is known to be a com­bi­na­tion of two spe­cial func­tions Ai and Bi that are called the Airy func­tions. These func­tions are shown in fig­ure A.17. The gen­eral so­lu­tion near a turn­ing point is:

$$\psi = C_{\rm {A}} {\rm Ai}(\overline{x}) + C_{\rm {B}} {\r... ...r^2}}\frac{V-E}{V'} \quad V' \equiv \frac{{\rm d}V}{{\rm d}x}$$

Note that $(V-E)$$\raisebox{.5pt}{$/$}$​$V'$ is the $x$-​po­si­tion mea­sured from the point where $V$ $\vphantom0\raisebox{1.5pt}{$=$}$ $E$, so that $\overline{x}$ is a lo­cal, stretched $x$-​co­or­di­nate.

The sec­ond step is to re­late this so­lu­tion to the nor­mal WKB ap­prox­i­ma­tions away from the turn­ing point. Now from a macro­scopic point of view, the WKB ap­prox­i­ma­tion fol­lows from the as­sump­tion that Planck’s con­stant $\hbar$ is very small. That im­plies that the va­lid­ity of the Airy func­tions nor­mally ex­tends to re­gion where $\vert\overline{x}\vert$ is rel­a­tively large. For ex­am­ple, if you fo­cus at­ten­tion on a point where $V-E$ is a fi­nite mul­ti­ple of $\hbar^{1/3}$, $V-E$ is small, so the value of $V'$ will de­vi­ate lit­tle from its value at the turn­ing point: the as­sump­tion of lin­early vary­ing po­ten­tial re­mains valid. Still, if $V-E$ is a fi­nite mul­ti­ple of $\hbar^{1/3}$, $\vert\overline{x}\vert$ will be pro­por­tional to 1/$\hbar^{1/3}$, and that is large. Such re­gions of large, but not too large, $\vert\overline{x}\vert$ are called “match­ing re­gions,” be­cause in them both the Airy func­tion so­lu­tion and the WKB so­lu­tion are valid. It is where the two meet and must agree.

Fig­ure A.18: Con­nec­tion for­mu­lae for a turn­ing point from nor­mal mo­tion to tun­nel­ing.

Fig­ure A.19: Con­nec­tion for­mu­lae for a turn­ing point from tun­nel­ing to nor­mal mo­tion.

It is graph­i­cally de­picted in fig­ures A.18 and A.19. Away from the turn­ing points, the clas­si­cal or tun­nel­ing WKB ap­prox­i­ma­tions ap­ply, de­pend­ing on whether the to­tal en­ergy is more than the po­ten­tial en­ergy or less. In the vicin­ity of the turn­ing points, the so­lu­tion is a com­bi­na­tion of the Airy func­tions. If you look up in a math­e­mat­i­cal hand­book like [1] how the Airy func­tions can be ap­prox­i­mated for large pos­i­tive re­spec­tively neg­a­tive $\overline{x}$, you find the ex­pres­sions listed in the bot­tom lines of the fig­ures. (Af­ter you rewrite what you find in ta­ble books in terms of use­ful quan­ti­ties, that is!)

The ex­pres­sions in the bot­tom lines must agree with what the clas­si­cal, re­spec­tively tun­nel­ing WKB ap­prox­i­ma­tion say about the match­ing re­gions. At one side of the turn­ing point, that re­lates the co­ef­fi­cients $C_{\rm {p}}$ and $C_{\rm {n}}$ of the tun­nel­ing ap­prox­i­ma­tion to the co­ef­fi­cients of $C_{\rm {A}}$ and $C_{\rm {B}}$ of the Airy func­tions. At the other side, it re­lates the co­ef­fi­cients $C_{\rm {f}}$ and $C_{\rm {b}}$ (or $C_{\rm {c}}$ and $C_{\rm {s}}$) of the clas­si­cal WKB ap­prox­i­ma­tion to $C_{\rm {A}}$ and $C_{\rm {B}}$. The net ef­fect of it all is to re­late, con­nect, the co­ef­fi­cients of the clas­si­cal WKB ap­prox­i­ma­tion to those of the tun­nel­ing one. That is why the for­mu­lae in fig­ures A.18 and A.19 are called the “con­nec­tion for­mu­lae.”

You may have noted the ap­pear­ance of an ad­di­tional con­stant $c$ in fig­ures A.18 and A.19. This nasty con­stant is de­fined as

$$c = \frac{\sqrt{\pi}}{(2m\vert V'\vert\hbar)^{1/6}} %$$ (A.215)

and shows up un­in­vited when you ap­prox­i­mate the Airy func­tion so­lu­tion for large $\vert\overline{x}\vert$. By clev­erly ab­sorb­ing it in a re­de­f­i­n­i­tion of the con­stants $C_{\rm {A}}$ and $C_{\rm {B}}$, fig­ures A.18 and A.19 achieve that you do not have to worry about it un­less you specif­i­cally need the ac­tual so­lu­tion at the turn­ing points.

As an ex­am­ple of how the con­nec­tion for­mu­lae are used, con­sider a right turn­ing point for the har­monic os­cil­la­tor or sim­i­lar. Near such a turn­ing point, the con­nec­tion for­mu­lae of fig­ure A.18 ap­ply. In the tun­nel­ing re­gion to­wards the right, the term $C_{\rm {p}}e^{\gamma}$ bet­ter be zero, be­cause it blows up at large $x$, and that would put the par­ti­cle at in­fin­ity for sure. So the con­stant $C_{\rm {p}}$ will have to be zero. Now the match­ing at the right side equates $C_{\rm {p}}$ to $C_{\rm {B}}e^{-\gamma_t}$ so $C_{\rm {B}}$ will have to be zero. That means that the so­lu­tion in the vicin­ity of the turn­ing point will have to be a pure Ai func­tion. Then the match­ing to­wards the left shows that the so­lu­tion in the clas­si­cal WKB re­gion must take the form of a sine that, when ex­trap­o­lated to the turn­ing point $\theta$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\theta_t$, stops short of reach­ing zero by an an­gu­lar amount $\pi$$\raisebox{.5pt}{$/$}$​4. Hence the as­ser­tion in ad­den­dum {A.28} that the an­gu­lar range of the clas­si­cal WKB so­lu­tion should be short­ened by $\pi$$\raisebox{.5pt}{$/$}$​4 for each end at which the par­ti­cle is trapped by a grad­u­ally in­creas­ing po­ten­tial in­stead of an im­pen­e­tra­ble wall.

Fig­ure A.20: WKB ap­prox­i­ma­tion of tun­nel­ing.

As an­other ex­am­ple, con­sider tun­nel­ing as dis­cussed in chap­ter 7.12 and 7.13. Fig­ure A.20 shows a sketch. The WKB ap­prox­i­ma­tion may be used if the bar­rier through which the par­ti­cle tun­nels is high and wide. In the far right re­gion, the en­ergy eigen­func­tion only in­volves a term $C^{\rm {r}}e^{{{\rm i}}\theta}$ with a for­ward wave speed. To sim­plify the analy­sis, the con­stant $C^{\rm {r}}$ can be taken to be one, be­cause it does not make a dif­fer­ence how the wave func­tion is nor­mal­ized. Also, the in­te­gra­tion con­stant in $\theta$ can be cho­sen such that $\theta$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\pi$$\raisebox{.5pt}{$/$}$​4 at turn­ing point 2; then the con­nec­tion for­mu­lae of fig­ure A.19 along with the Euler for­mula (2.5) show that the co­ef­fi­cients of the Airy func­tions at turn­ing point 2 are $C_{\rm {B}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 and $C_{\rm {A}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\rm i}$. Next, the in­te­gra­tion con­stant in $\gamma$ can be taken such that $\gamma$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 at turn­ing point 2; then the con­nec­tion for­mu­lae of fig­ure A.19 im­ply that $C^{\rm {m}}_{\rm {p}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\frac12{\rm i}$ and $C^{\rm {m}}_{\rm {n}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1.

Next con­sider the con­nec­tion for­mu­lae for turn­ing point 1 in fig­ure A.18. Note that $e^{-\gamma_1}$ can be writ­ten as $e^{\gamma_{12}}$, where $\gamma_{12}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\gamma_2-\gamma_1$, be­cause the in­te­gra­tion con­stant in $\gamma$ was cho­sen such that $\gamma_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. The ad­van­tage of us­ing $e^{\gamma_{12}}$ in­stead of $e^{-\gamma_1}$ is that it is in­de­pen­dent of the choice of in­te­gra­tion con­stant. Fur­ther­more, un­der the typ­i­cal con­di­tions that the WKB ap­prox­i­ma­tion ap­plies, for a high and wide bar­rier, $e^{\gamma_{12}}$ will be a very large num­ber. It is then seen from fig­ure A.18 that near turn­ing point 1, $C_{\rm {A}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $2e^{\gamma_{12}}$ which is large while $C_{\rm {B}}$ is small and will be ig­nored. And that then im­plies, us­ing the Euler for­mula to con­vert Ai’s sine into ex­po­nen­tials, that $\vert C^{\rm {l}}_{\rm {f}}\vert$ $\vphantom0\raisebox{1.5pt}{$=$}$ $e^{\gamma_{12}}$. As dis­cussed in chap­ter 7.13, the trans­mis­sion co­ef­fi­cient is given by

$$T = \frac{p_{\rm {c}}^{\rm {r}}}{p_{\rm {c}}^{\rm {l}}} \f... ...ert C^{\rm {l}}_{\rm {f}}/\sqrt{p_{\rm {c}}^{\rm {l}}}\vert^2}$$

and plug­ging in $C^{\rm {r}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 and $\vert C^{\rm {l}}_{\rm {f}}\vert$ $\vphantom0\raisebox{1.5pt}{$=$}$ $e^{\gamma_{12}}$, the trans­mis­sion co­ef­fi­cient is found to be $e^{-2\gamma_{12}}$.