An energy eigenstate of a time-independent Hamiltonian. Its wavefunction has the form `ψ_E(x,t) = φ_E(x)e^(-iEt/ħ)` with time-independent probability density.

A localized quantum state built as a superposition of many momentum or energy eigenstates. It moves and usually spreads over time.

Reflection and transmission amplitudes (r, t)

Complex coefficients in scattering states describing the amplitudes of reflected and transmitted waves relative to the incoming wave.

The property of time evolution that preserves inner products and total probability: U(t) is unitary, so `U†U = I`.

A localized quantum state formed by superposing many plane waves or energy eigenstates, which moves and often spreads in time.

What is Scattering state?

A solution of the stationary Schrödinger equation that describes an incoming wave plus reflected and transmitted components.

What is Stationary state?

An energy eigenstate of a time-independent Hamiltonian whose probability density is independent of time.

Time Evolution, Dynamics, and Scattering Basics — Quantum Mechanics Mastery: From Foundations to Modern Frontiers

Time Evolution: From States to Motion

From States to Motion

We now ask: how do quantum states actually change in time, and what happens when particles collide and scatter? The key tool is the time-dependent Schrödinger equation (TDSE).

The TDSE

For a particle in 1D with Hamiltonian `H = -(ħ²/2m) d²/dx² + V(x)`, the TDSE is `iħ ∂ψ/∂t = H ψ`. This is the fundamental law of nonrelativistic quantum dynamics.

Core Ideas

We will explore: unitary time evolution, the time-evolution operator `U(t)`, stationary states vs moving wavepackets, and basic 1D scattering with reflection and transmission.

Focus on Use

The goal is not just to know formulas, but to use them: evolve simple states in time and compute measurable reflection and transmission probabilities.

The Time-Dependent Schrödinger Equation and Unitarity

TDSE in State Form

The TDSE in abstract form is `iħ ∂|ψ(t)⟩/∂t = H|ψ(t)⟩`. The Hamiltonian H is the energy operator that drives time evolution.

What is Unitarity?

Unitarity means time evolution preserves inner products: if `⟨ψ(t0)|ψ(t0)⟩ = 1`, then `⟨ψ(t)|ψ(t)⟩ = 1` for all t. Total probability stays 1.

Continuity Equation

In position space, `ρ = |ψ|²` and `j = (ħ/m) Im(ψ* ∂ψ/∂x)` obey `∂ρ/∂t + ∂j/∂x = 0`, expressing local probability conservation.

Physical Meaning

This continuity equation is like fluid or charge conservation: probability does not disappear, it flows from one region to another.

Time-Evolution Operator: Solving the TDSE

Defining U(t)

For time-independent H, write `|ψ(t)⟩ = U(t)|ψ(0)⟩`. Then `iħ dU/dt = H U` with `U(0)=I`. The solution is `U(t) = e^(-iHt/ħ)`.

Key Properties of U(t)

The evolution operator is unitary: `U†U = I`, and has the group property: `U(t1 + t2) = U(t1)U(t2)`. Time evolution is reversible.

Energy Eigenbasis

Physical Insight

Time evolution does not mix energy eigenstates (for time-independent H). It only rotates their phases, leading to interference in observables.

Example: Time Evolution of a Two-Level System

Two-Level Hamiltonian

For a spin-1/2 in a static B field along z: `H = -(γħB/2) σz`. Eigenstates are `|+⟩` and `|-⟩` with energies `E+` and `E_-`.

Initial Superposition

Take `|ψ(0)⟩ = (1/√2)(|+⟩ + |-⟩)`. This state is not an eigenstate of H; its time evolution will show interference effects.

Time Evolution

Larmor Precession

The changing relative phase makes `⟨σ_x⟩(t) = cos(ωt)` with `ω=γB`. This is Larmor precession, the basis of magnetic resonance and qubit control.

Stationary States vs Moving Wavepackets

Stationary States

Stationary states satisfy `H|E⟩ = E|E⟩`. In position space `ψE(x,t) = φE(x) e^(-iEt/ħ)`, so `|ψE|² = |φE|²` is time-independent.

What Moves?

In a stationary state, only the global phase changes. The probability density does not move, so there is no localized particle-like behavior.

Wavepackets

A wavepacket is a superposition of many energy or momentum eigenstates, e.g., a Gaussian localized around `x0` with mean momentum `p0`.

Packet Motion and Spreading

Wavepacket centers move roughly classically, while the packet spreads over time. In scattering, a packet splits into reflected and transmitted parts.

Basics of 1D Scattering: Reflection and Transmission

Scattering Setup

In 1D scattering, a particle approaches a localized potential V(x) from the left. It can be reflected back or transmitted through the region.

Two Pictures

We use stationary scattering states (energy eigenstates) to compute amplitudes, and wavepackets to visualize a packet splitting into reflected and transmitted parts.

Asymptotic Form

For incidence from the left: as x→-∞, `ψ ≈ e^(ikx) + r e^(-ikx)`; as x→+∞, `ψ ≈ t e^(ik'x)`. Here r and t are reflection and transmission amplitudes.

From Amplitudes to Probabilities

Reflection probability R is roughly `|r|²`, transmission T is `|t|² (k'/k)`. For real, time-independent potentials in 1D, R + T = 1.

Worked Example: Scattering from a Potential Step

Potential Step Setup

Take `V(x)=0` for x<0 and `V(x)=V0` for x≥0. A particle with energy E>V0 comes from the left with wave numbers k (left) and k' (right).

Waveforms in Each Region

Region I: `ψI = e^(ikx) + r e^(-ikx)`; Region II: `ψII = t e^(ik'x)`. r and t are unknown amplitudes to be found.

Matching at x=0

Continuity of ψ and dψ/dx at x=0 gives: 1+r=t and k(1−r)=k't. Solving yields `r=(k−k')/(k+k')` and `t=2k/(k+k')`.

R and T

Reflection: `R=|r|²=((k−k')/(k+k'))²`. Transmission: `T=(k'/k)|t|²=4kk'/(k+k')²`. Verify that R+T=1, expressing probability conservation.

Thought Exercise: Wavepacket View of the Step

Imagine a Gaussian wavepacket centered far to the left of a potential step, moving to the right with average momentum `p0` such that its mean energy `E > V0`.

Mentally simulate what happens:

Before the step:

The packet moves right with group velocity `v_g = p0/m`.
Its shape is roughly Gaussian, maybe slowly spreading.

During interaction with the step:

The front part of the packet reaches the step first.
At the step, part of the wave is reflected, part transmitted.

Long after the interaction:

You see two separated packets:
A reflected packet moving left in region I.
A transmitted packet moving right in region II.

Questions to reflect on (no need to compute):

How would the width of the initial momentum distribution affect the sharpness of the reflected/transmitted packets?
How would the picture change if `E` were only slightly above `V0` versus much larger than `V0`?
In real experiments with electrons hitting a metal-semiconductor junction, which part of this picture corresponds to reflection at an interface?

Numerical Taste: Evolving a Free Gaussian Wavepacket (Python)

You can get intuition for time evolution by numerically evolving a Gaussian wavepacket for a free particle (V=0) in 1D.

Below is a minimal Python example using `numpy` and `matplotlib`. It does not use the most advanced numerical methods, but it gives a flavor of how time evolution works via Fourier transforms.

```python

import numpy as np

import matplotlib.pyplot as plt

Physical and numerical parameters

ħ = 1.0

m = 1.0

Nx = 1024

L = 200.0

x = np.linspace(-L/2, L/2, Nx, endpoint=False)

dx = x[1] - x[0]

Momentum grid (Fourier conjugate of x)

dk = 2 * np.pi / L

k = dk np.fft.fftfreq(Nx) Nx

Initial Gaussian wavepacket parameters

x0 = -40.0 # initial center

p0 = 2.0 # average momentum

sigma = 5.0 # spatial width

Initial wavefunction in x-space

norm = (1.0 / (2 np.pi sigma2))0.25

psi_x = norm np.exp(-(x - x0)2 / (4 sigma2)) np.exp(1j p0 * x / ħ)

Time evolution parameters

dt = 0.1

steps = 400

Precompute free-particle energy E(k) = ħ²k² / (2m)

E_k = (ħ2 * k2) / (2 * m)

plt.ion()

fig, ax = plt.subplots()

line, = ax.plot(x, np.abs(psi_x)2)

ax.set_xlim(-L/2, L/2)

ax.setylim(0, np.max(np.abs(psix)2) * 1.2)

ax.set_xlabel('x')

ax.set_ylabel('|psi(x,t)|^2')

psik = np.fft.fft(psix)

for n in range(steps):

Evolve in momentum space: multiply by phase factor

phase = np.exp(-1j E_k dt / ħ)

psi_k *= phase

Transform back to x-space

psix = np.fft.ifft(psik)

if n % 20 == 0:

line.setydata(np.abs(psix)2)

ax.set_title(f't = {n*dt:.1f}')

plt.pause(0.01)

plt.ioff()

plt.show()

```

What you should observe:

The packet moves to the right with approximately constant shape at short times.
Over longer times, the packet spreads because different `k` components travel at different speeds.

You can modify this code to add a potential barrier and see reflection and transmission numerically (commonly done using split-operator or Crank–Nicolson methods in research and teaching codes as of 2026).

Check Understanding: Time Evolution and Scattering

Test your understanding of key ideas from this module.

For a particle scattering from a real, time-independent 1D potential, which statement is correct?

The reflection and transmission amplitudes r and t satisfy |r|² + |t|² = 1, independent of wave numbers.
Probability conservation is expressed by a continuity equation involving the probability density and current, leading to R + T = 1 when currents are used.
Stationary scattering states have time-dependent probability densities that oscillate between left and right of the barrier.

Show Answer

Answer: B) Probability conservation is expressed by a continuity equation involving the probability density and current, leading to R + T = 1 when currents are used.

In 1D scattering, probability conservation is encoded in the continuity equation for density and current. For a real, time-independent potential this leads to R + T = 1 when R and T are defined using probability currents. The simple relation |r|² + |t|² = 1 holds only when k' = k; in general T includes a factor k'/k. Stationary states have time-independent densities.

Review Key Terms

Flip through these flashcards to reinforce the main concepts from this module.

Time-dependent Schrödinger equation (TDSE): The fundamental equation of nonrelativistic quantum dynamics: `iħ ∂|ψ(t)⟩/∂t = H|ψ(t)⟩`, determining how quantum states evolve in time.
Time-evolution operator U(t): For time-independent H, `U(t) = e^(-iHt/ħ)` with `|ψ(t)⟩ = U(t)|ψ(0)⟩`. U(t) is unitary and preserves total probability.
Stationary state: An energy eigenstate of a time-independent Hamiltonian. Its wavefunction has the form `ψ_E(x,t) = φ_E(x)e^(-iEt/ħ)` with time-independent probability density.
Wavepacket: A localized quantum state built as a superposition of many momentum or energy eigenstates. It moves and usually spreads over time.
Reflection and transmission amplitudes (r, t): Complex coefficients in scattering states describing the amplitudes of reflected and transmitted waves relative to the incoming wave.
Reflection and transmission probabilities (R, T): Measurable quantities derived from amplitudes and currents. In 1D with wave numbers k (left) and k' (right): R = |r|², T = (k'/k)|t|², with R + T = 1 for real, time-independent potentials.
Probability current j(x,t): For a particle in 1D, `j = (ħ/m) Im(ψ* ∂ψ/∂x)`. Together with ρ = |ψ|² it satisfies the continuity equation `∂ρ/∂t + ∂j/∂x = 0`.

Key Terms

Unitarity: The property of time evolution that preserves inner products and total probability: U(t) is unitary, so `U†U = I`.
Wavepacket: A localized quantum state formed by superposing many plane waves or energy eigenstates, which moves and often spreads in time.
Scattering state: A solution of the stationary Schrödinger equation that describes an incoming wave plus reflected and transmitted components.
Stationary state: An energy eigenstate of a time-independent Hamiltonian whose probability density is independent of time.
Larmor precession: Precession of a magnetic moment (such as a spin-1/2 particle) around an external magnetic field at the Larmor frequency ω = γB.
Probability current: A quantity j(x,t) that, together with the probability density ρ(x,t), satisfies the continuity equation expressing probability conservation.
Time-evolution operator: An operator U(t) that maps an initial state to its state at time t. For time-independent H, `U(t) = e^(-iHt/ħ)`.
Reflection coefficient (R): The probability (or current fraction) that a particle is reflected by a potential, often R = |r|² in 1D.
Transmission coefficient (T): The probability (or current fraction) that a particle passes through a potential; in 1D, T often includes a factor of k'/k.
Time-dependent Schrödinger equation (TDSE): The core equation of nonrelativistic quantum mechanics describing time evolution: `iħ ∂|ψ(t)⟩/∂t = H|ψ(t)⟩`.