The time-dependent Schrödinger equation governs how a quantum state changes with time. In one spatial dimension it is
written as
Time-dependent Schrödinger equation.
\[
i\hbar\frac{\partial \psi(x,t)}{\partial t}
=
-\frac{\hbar^2}{2m}\frac{\partial^2 \psi(x,t)}{\partial x^2}
+V(x)\psi(x,t).
\]
Here, \(\psi(x,t)\) is the wave function, \(m\) is the particle mass, and \(V(x)\) is the potential. The quantity
\(|\psi(x,t)|^2\) gives the probability density for finding the particle at position \(x\) at time \(t\). Unlike a
classical point particle, a quantum packet can spread, interfere, and reflect from boundaries while still evolving
deterministically through the Schrödinger equation.
Gaussian wave packet in free space
A very common initial state is the Gaussian wave packet. It is localized around a center \(x_0\), has an initial width
\(\sigma_0\), and carries an average momentum \(p_0\). In free space, the packet’s center moves with the group
velocity
\[
v_g=\frac{p_0}{m}.
\]
Therefore the packet center is
\[
x_c(t)=x_0+v_g t.
\]
The most important physical effect is spreading. Even if the packet begins sharply localized, it contains a range of
momentum components. Those components evolve at different phase rates, so the packet broadens with time. A standard
result for the Gaussian width is
Free-particle spreading law.
\[
\sigma(t)=\sigma_0\sqrt{1+\left(\frac{\hbar t}{2m\sigma_0^2}\right)^2}.
\]
This formula shows that lighter particles spread faster than heavier ones, and narrower initial packets spread faster
than broader ones. That is why electron packets disperse much more quickly than proton packets under similar
conditions.
A simple model for the free-particle probability density is therefore
\[
|\psi(x,t)|^2
\propto
\exp\!\left[-\frac{(x-x_c(t))^2}{2\sigma(t)^2}\right].
\]
Infinite square well case
In an infinite square well, the particle is confined between two perfectly reflecting walls. The potential is zero
inside the box and effectively infinite outside, so the wave function must vanish at the boundaries. The stationary
states are
\[
\psi_n(x)=\sqrt{\frac{2}{L}}\sin\!\left(\frac{n\pi x}{L}\right),
\qquad
E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}.
\]
A localized packet inside the well is built from a superposition of many such states. Each stationary state evolves in
time with a phase factor \(e^{-iE_n t/\hbar}\). Because different energy levels acquire different phases, the total
packet changes shape with time. It reflects from the walls, spreads, and can partially reform through quantum
interference. In a simplified educational simulator, one often models the confinement qualitatively by reflecting the
packet center at the walls and showing the packet shape inside the box.
Sample interpretation
Suppose an electron begins as a Gaussian packet with nonzero momentum. In free space, the center drifts in the
direction of the momentum, while the width increases with time according to the spreading law above. If the same packet
is placed inside an infinite well, the motion is no longer unbounded. The packet is constrained by the walls and its
shape evolves under the combined effect of confinement and phase mixing.
The associated de Broglie wavelength of the packet’s average motion is
\[
\lambda=\frac{h}{p_0}.
\]
The momentum uncertainty associated with an initially localized packet is of order
\[
\Delta p \sim \frac{\hbar}{2\sigma_0},
\]
which is consistent with the Heisenberg uncertainty principle. This is another reason narrow packets spread more
rapidly: they require a broader momentum distribution.
Physical meaning
Wave-packet evolution connects the abstract Schrödinger equation to concrete quantum behavior. Free-particle spreading
illustrates how localization changes with time, while the infinite-well case shows how boundaries reshape evolution.
These ideas are important in quantum transport, electron microscopy, cold-atom dynamics, nanoscale devices, and basic
interpretations of measurement. At a more advanced level, one studies exact spectral decompositions, harmonic-oscillator
coherent states, and numerical propagation methods such as split-operator Fourier techniques.
| Concept |
Main relation |
Meaning |
| TDSE |
\(i\hbar \partial\psi/\partial t = -(\hbar^2/2m)\partial^2\psi/\partial x^2 + V\psi\) |
Fundamental equation of quantum time evolution |
| Group velocity |
\(v_g=p_0/m\) |
Motion of the packet center |
| Center position |
\(x_c(t)=x_0+v_g t\) |
Free-particle drift of the packet |
| Packet width |
\(\sigma(t)=\sigma_0\sqrt{1+(\hbar t/2m\sigma_0^2)^2}\) |
Spreading of a free Gaussian packet |
| Infinite-well states |
\(\psi_n(x)=\sqrt{2/L}\sin(n\pi x/L)\) |
Allowed standing-wave basis in the box |
| Infinite-well energies |
\(E_n=n^2\pi^2\hbar^2/(2mL^2)\) |
Quantized energies governing phase evolution |