Wave Packets and Scattering

Stationary scattering states are the natural way to compute reflection and transmission amplitudes, but physical particles and beams are localized wave packets. This page explains how the two descriptions fit together.

The core idea is simple: build an incoming packet as a superposition of right-moving stationary scattering states. Each wavenumber component scatters with its own amplitudes $r(k)$ and $t(k)$. At late times, the state separates into a reflected packet and a transmitted packet, and their norms give the physical probabilities.

Why Packets Are Needed

A stationary left-incident scattering state has the asymptotic form

$$\phi_k(x)\sim e^{ikx}+r(k)e^{-ikx} \quad (x\to-\infty),$$

and

$$\phi_k(x)\sim t(k)e^{ikx} \quad (x\to+\infty),$$

for a potential localized near the origin and equal asymptotic potentials on the two sides. This state is not square-normalizable on the full line. It is a generalized energy eigenstate, like a plane wave.

A localized incoming state is instead a packet:

$$\psi_{\mathrm{in}}(x,t) = \frac{1}{\sqrt{2\pi}} \int_0^\infty a(k)e^{i(kx-\omega(k)t)}\,dk,$$

with

$$\omega(k)=\frac{\hbar k^2}{2m},$$

and

$$\int_0^\infty \lvert a(k)\rvert^2\,dk=1.$$

The amplitude $a(k)$ is concentrated near some $k_0\gt 0$ for a right-moving packet launched from the left. The lower limit $0$ is a convenient way to say that the packet is built mainly from right-moving components.

Scattered Packet

After the packet reaches the scattering region, each component inherits the stationary amplitudes. Far from the scatterer and after the reflected and transmitted pieces have separated, the left-moving reflected packet is approximately

$$\psi_{\mathrm{ref}}(x,t) = \frac{1}{\sqrt{2\pi}} \int_0^\infty a(k)r(k)e^{-ikx-i\omega(k)t}\,dk,$$

on the left side, while the transmitted packet is approximately

$$\psi_{\mathrm{trans}}(x,t) = \frac{1}{\sqrt{2\pi}} \int_0^\infty a(k)t(k)e^{ikx-i\omega(k)t}\,dk,$$

on the right side.

These formulas should be read as asymptotic descriptions. During the collision, the incident, reflected, transmitted, and interaction-region parts overlap and cannot be cleanly assigned separate probabilities by eye.

Late-Time Probabilities

For equal asymptotic potentials on the left and right, the stationary coefficients satisfy

$$R(k)=\lvert r(k)\rvert^2, \qquad T(k)=\lvert t(k)\rvert^2.$$

The packet probabilities are the momentum-distribution averages

$$P_{\mathrm R} = \int_0^\infty \lvert a(k)\rvert^2 R(k)\,dk,$$

and

$$P_{\mathrm T} = \int_0^\infty \lvert a(k)\rvert^2 T(k)\,dk.$$

For a real conservative one-dimensional potential with one open reflected channel and one open transmitted channel,

$$R(k)+T(k)=1$$

for each $k$, so

$$P_{\mathrm R}+P_{\mathrm T}=1.$$

If the initial packet has appreciable overlap with bound states or trapped quasi-bound components, there may also be a localized leftover probability at intermediate times. In the standard scattering setup, the packet starts far from the localized potential with positive momentum, making bound-state overlap negligible.

Narrow-Packet Approximation

If $a(k)$ is sharply concentrated near $k_0$ and $T(k)$ varies slowly across the packet width, then

$$P_{\mathrm T} \approx T(k_0), \qquad P_{\mathrm R} \approx R(k_0).$$

This is why plane-wave transmission coefficients are so useful: a narrow packet behaves like a beam with a well-defined central energy.

The approximation can fail when:

• the packet has a broad momentum spread;
• $T(k)$ changes rapidly across the packet bandwidth;
• the packet straddles a threshold;
• the scatterer has a narrow resonance;
• the potential supports long-lived quasi-bound behavior that delays separation.

In those cases the measured transmission is an average over the packet spectrum, not simply the value at the central momentum.

Current Versus Packet Norm

Stationary scattering uses current ratios:

$$T(k)=\frac{j_{\mathrm{trans}}(k)}{j_{\mathrm{inc}}(k)}.$$

Packet scattering uses ordinary square norms after the outgoing pieces are separated:

$$P_{\mathrm T} = \int_{\text{right region}} \lvert\psi(x,t)\rvert^2\,dx$$

at sufficiently late time. These are consistent descriptions. The current ratio tells how each energy component scatters; the packet norm adds the contributions of the components present in the incoming state.

For unequal asymptotic potentials, $T(k)$ must already include the velocity factor. The packet formula still averages the physical transmission coefficient, but one must label components carefully because the transmitted wavenumber differs from the incident one.

Time-Dependent Picture

For a localized potential near the origin, choose two points $x_L\lt0\lt x_R$ outside the interaction region. A numerical or conceptual time-dependent scattering experiment tracks

$$P_L(t) = \int_{-\infty}^{x_L} \lvert\psi(x,t)\rvert^2\,dx,$$

the interaction-region probability

$$P_{\mathrm{int}}(t) = \int_{x_L}^{x_R} \lvert\psi(x,t)\rvert^2\,dx,$$

and

$$P_R(t) = \int_{x_R}^{\infty} \lvert\psi(x,t)\rvert^2\,dx.$$

Early on, nearly all probability is in the incoming packet on the left. During the collision, $P_{\mathrm{int}}(t)$ can be significant. At late times, if the outgoing packets have cleared the interaction region,

$$P_L(t)\to P_{\mathrm R}, \qquad P_R(t)\to P_{\mathrm T}, \qquad P_{\mathrm{int}}(t)\to0.$$

This is the clean operational meaning of reflected and transmitted packet probabilities.

Packet Distortion and Delay

The functions $r(k)$ and $t(k)$ are complex. Their magnitudes weight the packet spectrum, and their phases shift the outgoing packets. If

$$t(k)=\lvert t(k)\rvert e^{i\theta_t(k)},$$

then a narrow transmitted packet is shifted by the $k$-dependence of $\theta_t(k)$. In energy language, this is related to the phase time

$$\tau_t = \hbar\frac{d\theta_t}{dE} \bigg\rvert_{E_0}.$$

This estimate is useful but should not be turned into a universal statement about “the time spent inside the barrier.” Tunneling-time questions depend on the operational definition. The safe packet statement is that phases affect arrival times and packet shapes, while probabilities come from late-time norms or current ratios.

Numerical Scattering Checks

Wave-packet scattering is a useful numerical benchmark, but it has several traps. A reliable simulation should check:

• total norm conservation, unless absorbers are intentionally used;
• the packet starts far enough from the scatterer;
• the grid is large enough that outgoing packets do not reflect from boundaries;
• the momentum distribution is narrow enough for any comparison with a single $T(k_0)$;
• late-time region integrals are taken only after the packets separate;
• the final accounting includes reflected, transmitted, and interaction-region probability.

The barrier-scattering notebook listed in Numerical Notebooks Index uses this kind of probability accounting. The Wave-Packet Scattering Notebook develops the full split-step convergence study, including packet-spectrum, grid, time-step, box-size, and separation checks.

Common Mistakes

• Treating a plane-wave scattering state as a normalizable particle state.
• Reading transmission probability from the visual height of the transmitted packet.
• Comparing a broad packet to $T(k_0)$ without averaging over $\lvert a(k)\rvert^2$.
• Assigning reflected and transmitted probabilities while the packet still overlaps the barrier.
• Forgetting velocity factors when the left and right asymptotic wavenumbers differ.
• Interpreting phase delay as a unique tunneling traversal time without specifying a measurement protocol.
• Ignoring boundary reflections in finite-grid simulations.

Where This Is Used

• Wave Packets gives the general Fourier construction of normalizable packets.
• Gaussian Wave Packets gives the standard localized free packet.
• Wave Packets and Classical Trajectories explains why reflected and transmitted branches cannot be compressed into one trajectory.
• Group Velocity and Phase Velocity explains the motion of packet envelopes.
• Reflection and Transmission Coefficients defines the stationary current coefficients that are averaged here.
• Finite Potential Barrier and Rectangular Barrier Tunneling supply the standard stationary coefficients.
• Resonant Transmission is where narrow energy features make packet averaging essential.
• Wave Packet Spreading explains why incoming and outgoing packet widths change even away from the scatterer.

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.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.
• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.

Exercises

1. Suppose a normalized incoming packet has spectral weight $\lvert a(k)\rvert^2$ and the stationary transmission coefficient is $T(k)$. Why is the packet transmission probability an average over $k$?

Solution

Each incoming component scatters independently in the stationary basis. The component near $k$ carries probability weight $\lvert a(k)\rvert^2\,dk$ and transmits with probability $T(k)$. Adding the transmitted contributions gives

$$P_{\mathrm T} = \int_0^\infty \lvert a(k)\rvert^2T(k)\,dk.$$

This is the continuum version of summing probabilities over mutually orthogonal momentum components.

2. Show that a narrow packet gives $P_{\mathrm T}\approx T(k_0)$ when $T(k)$ varies slowly.

Solution

Expand $T(k)$ around the packet center:

$$T(k) =T(k_0)+T'(k_0)(k-k_0)+\cdots.$$

If $\lvert a(k)\rvert^2$ is normalized and centered at $k_0$, then

$$\int_0^\infty \lvert a(k)\rvert^2\,dk=1, \qquad \int_0^\infty (k-k_0)\lvert a(k)\rvert^2\,dk\approx0.$$

Keeping the leading term gives

$$P_{\mathrm T} = \int_0^\infty \lvert a(k)\rvert^2T(k)\,dk \approx T(k_0).$$

Higher terms matter when the packet is broad or $T(k)$ varies rapidly.

3. A packet has two narrow momentum components with weights $0.7$ and $0.3$. Their transmission coefficients are $0.2$ and $0.9$. Estimate the total transmission probability.

Solution

Approximate the spectral distribution by two discrete weights:

$$P_{\mathrm T} \approx 0.7(0.2)+0.3(0.9).$$

Thus

$$P_{\mathrm T} \approx 0.14+0.27 =0.41.$$

4. Why is it misleading to compute transmission from the height of the transmitted packet in a plot?

Solution

Probability is the integral of density over a region, not the maximum height of the density. A transmitted packet may be broader or narrower than the incident packet because of spreading and spectral filtering. The correct late-time probability is

$$P_{\mathrm T} = \int_{\text{right region}} \lvert\psi(x,t)\rvert^2\,dx,$$

after the transmitted packet has separated from the interaction region.