How “Stationary” Is a Particle in a Box? Momentum and Travel Distance from 1-Inch to Atomic Scales


How “Stationary” Is a Particle in a Box? Momentum and Travel Distance from 1-Inch to Atomic Scales

In the 1D infinite square well, the ground state is a standing wave. That means the expectation value of momentum is zero, yet the state can be written as an equal superposition of two traveling waves with definite momenta \pm p_1. The magnitude p_1 is well defined by the box width L.

Key idea. In the infinite well (with hard walls), the ground state is not a momentum eigenstate, but it carries a definite momentum magnitude p_1. Physically you can think of it as a balanced superposition of left- and right-moving components with +p_1 and -p_1.

1) Ground-State Momentum in a 1-Inch-Wide Box

Box width:

    \[ L = 1~\text{inch} = 2.54~\text{cm} = 0.0254~\text{m}. \]

For the n-th stationary state of a 1D infinite well, the magnitude of the momentum is

    \[ p_n = \frac{n\pi\hbar}{L}\,, \qquad \Rightarrow \qquad p_1 = \frac{\pi\hbar}{L}. \]

Using \hbar = 1.054\,571\,817\times10^{-34}\ \text{J·s} and L=0.0254~\text{m}:

    \[ p_1 \approx \frac{\pi\,(1.0546\times10^{-34})}{0.0254} \approx 1.30\times10^{-32}\ \text{kg·m/s}. \]

Quantity Value
Width L 0.0254~\text{m}
Ground-state momentum magnitude p_1 \approx 1.30\times10^{-32}\ \text{kg·m/s}

2) “Distance Traveled” Inside the 1-Inch Box

Because the ground state is a standing wave, the probability density is static. But if you decompose it into its two traveling components, each one behaves like a particle with momentum magnitude p_1 and speed v = p_1/m. The distance covered in time t would be d = vt = (p_1/m) t.

Electron

    \[ m_e = 9.109\times10^{-31}\ \text{kg}, \qquad v_e = \frac{p_1}{m_e} \approx \frac{1.30\times10^{-32}}{9.109\times10^{-31}} \approx 1.43\times10^{-2}\ \text{m/s}. \]

    \[ t_{\text{one\;traverse}}=\frac{L}{v_e} \approx \frac{0.0254}{1.43\times10^{-2}} \approx 1.77\ \text{s}. \]

Proton

    \[ m_p = 1.673\times10^{-27}\ \text{kg}, \qquad v_p = \frac{p_1}{m_p} \approx 7.80\times10^{-6}\ \text{m/s}, \qquad t_{\text{one\;traverse}} \approx \frac{0.0254}{7.80\times10^{-6}} \approx 3.26\times10^{3}\ \text{s} \approx 54\ \text{min}. \]

Particle Mass (kg) Speed v=p_1/m (m/s) Time to cross L=1“
Electron 9.109\times10^{-31} \approx 1.43\times10^{-2} (1.4 cm/s) \approx 1.77\ \text{s}
Proton 1.673\times10^{-27} \approx 7.8\times10^{-6} \approx 54\ \text{min}

Takeaway: With a macroscopic well, the quantized momentum is tiny. Even an electron moves only about a centimeter per second in this ground-state momentum scale.

3) Atomic-Scale Box: L = 1~\text{\AA} = 10^{-10}\ \text{m}

Now the same formulas, but with a microscopic width.

Ground-state momentum

    \[ L = 1\times10^{-10}\ \text{m}, \qquad p_1=\frac{\pi\hbar}{L} \approx \frac{\pi\,(1.0546\times10^{-34})}{10^{-10}} \approx 3.31\times10^{-24}\ \text{kg·m/s}. \]

Electron

    \[ v_e=\frac{p_1}{m_e}\approx 3.64\times10^{6}\ \text{m/s},\qquad t_e=\frac{L}{v_e}\approx 2.75\times10^{-17}\ \text{s}\ (\text{~27.5 attoseconds}). \]

    \[ K_e=\frac{p_1^2}{2m_e}\approx 6.02\times10^{-18}\ \text{J}\ \approx 37.6\ \text{eV}. \]

Proton

    \[ v_p=\frac{p_1}{m_p}\approx 1.98\times10^{3}\ \text{m/s},\qquad t_p=\frac{L}{v_p}\approx 5.05\times10^{-14}\ \text{s}\ (\text{~50.5 fs}), \]

    \[ K_p=\frac{p_1^2}{2m_p}\approx 3.28\times10^{-21}\ \text{J}\ \approx 2.05\times10^{-2}\ \text{eV}. \]

Particle p_1 (kg·m/s) Speed (m/s) Traverse time for L=1\ \text{\AA} Kinetic energy (eV)
Electron 3.31\times10^{-24} \approx 3.64\times10^{6} \approx 2.75\times10^{-17}\ \text{s} \approx 37.6
Proton 3.31\times10^{-24} \approx 1.98\times10^{3} \approx 5.05\times10^{-14}\ \text{s} \approx 2.05\times10^{-2}

4) Why the Huge Difference?

  • The momentum scale is set by p \sim \hbar/L. Shrinking L by a factor of 2.54\times10^{8} (from 1 inch to 1 Å) increases p by the same factor.
  • Electron speeds at atomic scales naturally land in the 10^{6}\ \text{m/s} range, and characteristic times are attoseconds to femtoseconds—matching intuition from atomic physics.
  • At macroscopic L, quantization steps are so fine that the ground-state momentum (and kinetic energy) are minuscule.

5) The Standing-Wave Subtlety

The ground state wavefunction in the infinite well is

    \[ \psi_1(x)=\sqrt{\frac{2}{L}}\sin\!\left(\frac{\pi x}{L}\right),\qquad 0<x<L, \]

which is a standing wave with \langle p\rangle=0. Yet it decomposes into traveling waves carrying momenta \pm p_1. So while there is no unique classical trajectory for the stationary state, the momentum magnitude scale that governs any localized wavepacket built from the ground state is precisely

    \[ p_1=\frac{\pi\hbar}{L}. \]

Bottom Line

  • 1-inch box: p_1\approx 1.30\times10^{-32}\ \text{kg·m/s}. An electron’s speed would be \sim 1.4\ \text{cm/s}, taking \sim 1.8\ \text{s} to cross once.
  • 1 Å box: p_1\approx 3.31\times10^{-24}\ \text{kg·m/s}. An electron’s speed is \sim 3.6\times10^{6}\ \text{m/s}, crossing in \sim 28\ \text{as} with kinetic energy \sim 38\ \text{eV}.

Same quantum rule, wildly different scales—because everything scales as 1/L.