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

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

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