Resonant Transmission

Resonant transmission occurs when a scattering structure that usually reflects waves becomes nearly transparent at special energies. The simplest wave-mechanics example is a double barrier: two finite barriers separated by a well-like region. At certain energies, the wave temporarily builds up between the barriers and then exits with high probability.

The effect is the quantum analog of a Fabry–Pérot cavity. Multiple reflected amplitudes interfere destructively on the reflected side and constructively on the transmitted side.

Double-Barrier Picture

A schematic double-barrier potential has the form

$$V(x)= \begin{cases} 0, & x\lt 0,\\ V_b, & 0\lt x\lt a,\\ 0, & a\lt x\lt a+L,\\ V_b, & a+L\lt x\lt 2a+L,\\ 0, & x\gt 2a+L. \end{cases}$$

For energies below the barrier top,

$$0\lt E\lt V_b,$$

each single barrier suppresses transmission. But the well region between the barriers can support standing-wave-like amplitudes. When the phase accumulated across the middle region is just right, the full double-barrier system can transmit strongly.

Quasi-Bound States

If the barriers were infinitely thick, the middle region would contain true bound states. With finite barriers, those states leak into the left and right continua. They are therefore quasi-bound states: they can trap probability for a while, but not forever.

A quasi-bound state appears in scattering as a resonance. Near the resonance energy, the probability density inside the middle region becomes large compared with the incoming density, and the transmission probability can peak sharply.

The word “quasi” matters. These states are not normalizable stationary bound states on the full line. They are long-lived scattering features.

Phase Condition

A rough resonance condition is that one round trip in the middle region returns with phase equal to an integer multiple of $2\pi$:

$$2kL+\phi_L+\phi_R=2\pi n.$$

Here $k=\sqrt{2mE}/\hbar$ is the wavenumber in the central well region, $L$ is the well width, and $\phi_L,\phi_R$ are reflection phase shifts from the left and right barriers as seen from inside the well.

If the barrier reflection phases are ignored as a first approximation, this becomes

$$kL\approx n\pi.$$

That is the same phase condition as a particle in a box of width $L$. Finite barriers shift and broaden the levels rather than leaving exact box eigenstates.

Transfer-Matrix View

The Transfer Matrix Method composes the two barriers and the middle propagation region into one matrix:

$$\mathbf v_R=M(E)\mathbf v_L.$$

For left incidence,

$$\mathbf v_L= \begin{pmatrix} 1\\ r \end{pmatrix}, \qquad \mathbf v_R= \begin{pmatrix} t\\ 0 \end{pmatrix}.$$

The transmission probability is

$$T(E)=\lvert t(E)\rvert^2$$

when the left and right asymptotic potentials are equal. Resonant transmission appears as sharp maxima of $T(E)$.

For a symmetric double barrier, the peak transmission can reach

$$T(E_{\mathrm{res}})=1$$

in the ideal conservative one-dimensional model. Asymmetry, absorption, inelastic channels, or decoherence can reduce the peak height.

Width and Lifetime

Near an isolated resonance, the transmission often has a Breit–Wigner-like form:

$$T(E) \approx \frac{\Gamma_L\Gamma_R} {(E-E_{\mathrm{res}})^2+\Gamma^2/4}, \qquad \Gamma=\Gamma_L+\Gamma_R.$$

Here $\Gamma_L$ and $\Gamma_R$ describe leakage through the left and right barriers, while $\Gamma$ is the total resonance width. At resonance,

$$T(E_{\mathrm{res}}) \approx \frac{4\Gamma_L\Gamma_R} {(\Gamma_L+\Gamma_R)^2}.$$

For symmetric barriers, $\Gamma_L=\Gamma_R$, so the peak can reach $1$. For strongly asymmetric barriers, one side leaks much faster than the other, and the peak is lower.

The lifetime scale of the quasi-bound state is roughly

$$\tau\sim\frac{\hbar}{\Gamma}.$$

A narrow resonance has a long lifetime; a broad resonance decays quickly. This width-lifetime relation is a recurring theme in scattering theory and spectroscopy.

Relation to Scattering Poles

In a more advanced scattering description, resonances correspond to poles of the analytically continued scattering amplitude at complex energies

$$E_{\mathrm{pole}} = E_{\mathrm{res}}-\frac{i}{2}\Gamma.$$

The negative imaginary part encodes decay. This pole language is not needed for the first double-barrier calculation, but it explains why resonances, lifetimes, and line shapes are tied together. The broader theory belongs in Resonances and Bound States and Scattering Poles.

Physical Interpretation

Resonant transmission is not the statement that tunneling has disappeared. Each barrier may still be individually opaque. The high transmission occurs because the whole structure supports a leaky standing wave whose outgoing amplitudes interfere coherently.

This is why small changes in energy, barrier spacing, or barrier height can change transmission dramatically. The middle region acts like an energy-selective cavity.

Physical settings where this idea appears include resonant tunneling diodes, semiconductor heterostructures, quantum wells coupled to leads, optical multilayers, and molecular tunneling problems. Those applications require additional material-specific modeling, but the one-dimensional double barrier captures the core mechanism.

Common Mistakes

• Assuming two barriers always suppress transmission more than one barrier.
• Confusing a quasi-bound resonance with a true bound state.
• Forgetting that transmission probabilities are current ratios.
• Assuming every resonance reaches $T=1$; asymmetric barriers need not.
• Treating the width $\Gamma$ as a numerical artifact rather than a physical leakage scale.
• Ignoring absorption or inelastic channels when checking whether $R+T=1$.

Exercises

1. Ignore reflection phase shifts and derive the approximate resonance condition for a middle well of width $L$.

Solution

A round trip across the well and back accumulates phase

$$2kL.$$

Constructive interference requires

$$2kL=2\pi n.$$

Therefore

$$kL=n\pi.$$

This is the same standing-wave condition as an infinite well, but finite barriers shift and broaden the resonances.

2. For the Breit–Wigner form, show that the peak transmission is $1$ when $\Gamma_L=\Gamma_R$.

Solution

At $E=E_{\mathrm{res}}$,

$$T(E_{\mathrm{res}}) = \frac{\Gamma_L\Gamma_R}{\Gamma^2/4} = \frac{4\Gamma_L\Gamma_R}{(\Gamma_L+\Gamma_R)^2}.$$

If $\Gamma_L=\Gamma_R=\gamma$, then

$$T(E_{\mathrm{res}}) = \frac{4\gamma^2}{(2\gamma)^2} =1.$$

3. A resonance has width $\Gamma=10^{-6}\,\mathrm{eV}$. Estimate the lifetime using $\tau\sim\hbar/\Gamma$ and $\hbar\approx6.58\times10^{-16}\,\mathrm{eV\,s}$.

Solution

Use

$$\tau\sim\frac{\hbar}{\Gamma}.$$

Then

$$\tau \sim \frac{6.58\times10^{-16}\,\mathrm{eV\,s}} {10^{-6}\,\mathrm{eV}} = 6.58\times10^{-10}\,\mathrm{s}.$$

So the lifetime scale is roughly $0.66\,\mathrm{ns}$.

Where This Is Used

• Transfer Matrix Method gives the matrix composition method for double barriers.
• Rectangular Barrier Tunneling gives the single-barrier baseline.
• Reflection and Transmission Coefficients defines current-based $R$ and $T$.
• Finite Square Well supplies the bound-state comparison for the central well region.
• Resonances develops the broader scattering-theory interpretation.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.
• S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, 1995.