7.13 Re­flec­tion and Trans­mis­sion Co­ef­fi­cients

Scat­ter­ing and tun­nel­ing can be de­scribed in terms of so-called re­flec­tion and trans­mis­sion co­ef­fi­cients. This sec­tion ex­plains the un­der­ly­ing ideas.

Fig­ure 7.22: Schematic of a scat­ter­ing po­ten­tial and the as­ymp­totic be­hav­ior of an ex­am­ple en­ergy eigen­func­tion for a wave packet com­ing in from the far left.

Con­sider an ar­bi­trary scat­ter­ing po­ten­tial like the one in fig­ure 7.22. To the far left and right, it is as­sumed that the po­ten­tial as­sumes a con­stant value. In such re­gions the en­ergy eigen­func­tions take the form

$$\psi_E = C_{\rm {f}} e^{{\rm i}p_{\rm {c}}x/\hbar} + C_{\rm {b}} e^{-{\rm i}p_{\rm {c}}x/\hbar}$$

where $p_{\rm {c}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sqrt{2m(E-V)}$ is the clas­si­cal mo­men­tum and $C_{\rm {f}}$ and $C_{\rm {b}}$ are con­stants. When eigen­func­tions of slightly dif­fer­ent en­er­gies are com­bined to­gether, the terms $C_{\rm {f}}e^{{{\rm i}}p_{\rm {c}}x/\hbar}$ pro­duce wave pack­ets that move for­wards in $x$, graph­i­cally from left to right, and the terms $C_{\rm {b}}e^{-{{\rm i}}p_{\rm {c}}x/\hbar}$ pro­duce pack­ets that move back­wards. So the sub­scripts in­di­cate the di­rec­tion of mo­tion.

This sec­tion is con­cerned with a sin­gle wave packet that comes in from the far left and is scat­tered by the non­triv­ial po­ten­tial in the cen­ter re­gion. To de­scribe this, the co­ef­fi­cient $C_{\rm {b}}$ must be zero in the far-right re­gion. If it was nonzero, it would pro­duce a sec­ond wave packet com­ing in from the far right.

In the far-left re­gion, the co­ef­fi­cient $C_{\rm {b}}$ is nor­mally not zero. In fact, the term $C^{\rm {l}}_{\rm {b}}e^{-{{\rm i}}p_{\rm {c}}x/\hbar}$ pro­duces the part of the in­com­ing wave packet that is re­flected back to­wards the far left. The rel­a­tive amount of the in­com­ing wave packet that is re­flected back is called the “re­flec­tion co­ef­fi­cient” $R$. It gives the prob­a­bil­ity that the par­ti­cle can be found to the left of the scat­ter­ing re­gion af­ter the in­ter­ac­tion with the scat­ter­ing po­ten­tial. It can be com­puted from the co­ef­fi­cients of the en­ergy eigen­func­tion in the left re­gion as, {A.32},

$$\fbox{$\displaystyle R = \frac{\vert C^{\rm{l}}_{\rm{b}}\vert^2}{\vert C^{\rm{l}}_{\rm{f}}\vert^2} $} %$$ (7.72)

Sim­i­larly, the rel­a­tive frac­tion of the wave packet that passes through the scat­ter­ing re­gion is called the “trans­mis­sion co­ef­fi­cient” $T$. It gives the prob­a­bil­ity that the par­ti­cle can be found at the other side of the scat­ter­ing re­gion af­ter­wards. It is most sim­ply com­puted as $T$ $\vphantom0\raisebox{1.5pt}{$=$}$ $1-R$: what­ever is not re­flected must pass through. Al­ter­na­tively, it can be com­puted as

$$\fbox{$\displaystyle T = \frac{p_{\rm{c}}^{\rm{r}}\vert C^... ...l}})} \quad p_{\rm{c}}^{\rm{r}}=\sqrt{2m(E-V_{\rm{r}})} $} %$$ (7.73)

where $p_{\rm {c}}^{\rm {l}}$ re­spec­tively $p_{\rm {c}}^{\rm {r}}$ are the val­ues of the clas­si­cal mo­men­tum in the far left and right re­gions.

Note that a co­her­ent wave packet re­quires a small amount of un­cer­tainty in en­ergy. Us­ing the eigen­func­tion at the nom­i­nal value of en­ergy in the above ex­pres­sions for the re­flec­tion and trans­mis­sion co­ef­fi­cients will in­volve a small er­ror. It can be made to go to zero by re­duc­ing the un­cer­tainty in en­ergy, but then the size of the wave packet will ex­pand cor­re­spond­ingly.

In the case of tun­nel­ing through a high and wide bar­rier, the WKB ap­prox­i­ma­tion may be used to de­rive a sim­pli­fied ex­pres­sion for the trans­mis­sion co­ef­fi­cient, {A.29}. It is

$$\fbox{$\displaystyle T \approx e^{-2\gamma_{12}} \qquad \... ...{ \rm d}x \quad \vert p_{\rm{c}}\vert = \sqrt{2m(V-E)} $} %$$ (7.74)

where $x_1$ and $x_2$ are the turn­ing points in fig­ure 7.22, in be­tween which the po­ten­tial en­ergy ex­ceeds the to­tal en­ergy of the par­ti­cle.

There­fore in the WKB ap­prox­i­ma­tion, it is just a mat­ter of do­ing a sim­ple in­te­gral to es­ti­mate what is the prob­a­bil­ity for a wave packet to pass through a bar­rier. One fa­mous ap­pli­ca­tion of that re­sult is for the al­pha de­cay of atomic nu­clei. In such de­cay a so-called al­pha par­ti­cle tun­nels out of the nu­cleus.

For sim­i­lar con­sid­er­a­tions in three-di­men­sion­al scat­ter­ing, see ad­den­dum {A.30}.

Key Points

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A trans­mis­sion co­ef­fi­cient gives the prob­a­bil­ity for a par­ti­cle to pass through an ob­sta­cle. A re­flec­tion co­ef­fi­cient gives the prob­a­bil­ity for it to be re­flected.

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A very sim­ple ex­pres­sion for these co­ef­fi­cients can be ob­tained in the WKB ap­prox­i­ma­tion.