Transfer-Matrix Methods

A transfer matrix propagates wave amplitudes across piecewise constant one-dimensional potentials. It is exact for each constant segment, making it a useful reference method. Its weakness is exponential conditioning when growing and decaying evanescent solutions coexist; a physically tiny transmission can be lost inside enormous intermediate numbers.

Required background. One-dimensional scattering explains reflection and transmission, while probability flux supplies normalization.

Amplitudes and matrix direction

In a region of constant potential and mass $\mu$, write $\psi(x)=A e^{ikx}+B e^{-ikx}$ using coordinates measured from the region’s left interface. Across a width $L$,

$$\begin{pmatrix}A_R\\B_R\end{pmatrix} =P(k,L)\begin{pmatrix}A_L\\B_L\end{pmatrix}, \qquad P=\begin{pmatrix}e^{ikL}&0\\0&e^{-ikL}\end{pmatrix}.$$

At an interface from momentum $k_a$ to $k_b$ with equal mass, continuity of $\psi,\psi'$ gives

$$I_{b\leftarrow a} =\frac12 \begin{pmatrix} 1+k_a/k_b&1-k_a/k_b\\ 1-k_a/k_b&1+k_a/k_b \end{pmatrix}.$$

The complete $M$ maps left amplitudes to right amplitudes. Matrix order is physical order with later operations multiplying on the left. Effective-mass discontinuities require the correct interface condition from the Hamiltonian, not this equal-mass formula by default.

Extracting amplitudes and flux

For incidence from the left with no incoming wave from the right,

$$\begin{pmatrix}t\\0\end{pmatrix} =M\begin{pmatrix}1\\r\end{pmatrix}, \qquad r=-\frac{M_{21}}{M_{22}}, \quad t=\frac{\det M}{M_{22}}.$$

In propagating exterior leads, $R=|r|^2$ and $T=(v_R/v_L)|t|^2$. For equal masses, $v=\hbar k/\mu$. Only equal exterior velocities allow $T=|t|^2$. The determinant telescopes to $k_L/k_R$ for this amplitude convention. Use that analytic ratio when evaluating $t$, rather than recomputing the determinant by subtracting products of large matrix entries. The opaque-barrier calculation shows how that subtraction can return zero transmission while $R+T=1$ still passes at machine precision. Use its logarithmic transmission formula when the probability is too small to represent; retain the distinction between an accurate single-barrier formula and a stable algorithm for a long stack.

For a real conservative potential, verify $R+T=1$. Arbitrary phase origins change amplitudes but must leave these probabilities invariant.

A square-barrier benchmark

For height $V_0>E>0$, width $L$, equal exterior leads, and $\kappa=\sqrt{2\mu(V_0-E)}/\hbar$,

$$T=\left[ 1+\frac{V_0^2\sinh^2(\kappa L)} {4E(V_0-E)} \right]^{-1}.$$

At $E=V_0/2$, the prefactor is one and $T=\operatorname{sech}^2(\kappa L)$. For $\kappa L=10$, $T\approx8.24\times10^{-9}$. This tests a regime where reflection is almost one but the small transmission still needs relative accuracy.

At a threshold $k_b=0$, the displayed interface formula is singular. Use the constant-region $(\psi,\psi')$ propagator instead. At $E=V$, the equation is $\psi''=0$, so its exact zero-momentum limit is

$$\begin{pmatrix}\psi(L)\\\psi'(L)\end{pmatrix} =\begin{pmatrix}1&L\\0&1\end{pmatrix} \begin{pmatrix}\psi(0)\\\psi'(0)\end{pmatrix}.$$

Why multiplication becomes unstable

For $k=i\kappa$, the propagation diagonal contains $e^{-\kappa L}$ and $e^{+\kappa L}$. Repeated products mix those scales and can destroy the decaying solution through roundoff even before overflow. Checking only $R\approx1$ may conceal a completely wrong transmission.

A scattering-matrix formulation relates incoming to outgoing amplitudes and composes segments through linear solves for internal reflections. It avoids carrying exponentially growing amplitudes through a long forbidden region. Log-derivative propagation and carefully stabilized transfer methods are other options.

For a smooth potential approximated by steps, reduce segment width and compare with direct matching. This spatial approximation is separate from transfer-product conditioning.

Exercises

What does a potential-free segment produce when both exterior leads are identical?

Solution

$M=P(k,L)$, so $r=0$ and $t=e^{ikL}$ with the declared local interface origins. Therefore $R=0,T=1$. A different global phase convention may give $t=1$, but any reflected flux signals a matrix-order or interface error.

References

• Descouvemont, Pierre, and Daniel Baye, “The R-matrix theory,” Reports on Progress in Physics 73, 036301 (2010), arXiv:1001.0678 — internal-region bases and channel matching.
• Taylor, John R., Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover Publications (2006; original edition 1972) — scattering normalization, amplitudes, and analytic structure.