“Know how to solve every problem that has been solved.” “What I cannot create, I do not understand.” — Richard Feynman

The Particle in a Box by Hand — A dust grain does not feel quantization

Exercises

Introductory Quantum Mechanics › Unit 2 · Bound states in one dimension › The Particle in a Box by Hand › all problems

Practice Problem 7 of 10

A dust grain does not feel quantization

Problem

A 1

μ

g dust grain drifts at 1 mm/s in a 1 mm box. Which quantum number

n

describes it, and how large is the relative spacing between neighboring levels? Conclude something about why nobody noticed quantum mechanics before 1900.

Solution

Predict before reading on. Run the worked example’s formula backwards: from energy (or momentum $p = nh /2 L$ ) to $n$ . Predict the exponent of $n$ within a factor of a thousand before computing.

n = \frac{p \cdot 2 L}{h} = \frac{( 1 0 ^{- 9} ) ( 1 0 ^{- 3} ) \cdot 2 ( 1 0 ^{- 3} )}{6.6 \times 1 0 ^{- 34}} = 3.0 \times 1 0^{18}

\frac{E _{n + 1} - E _{n}}{E _{n}} = \frac{2 n + 1}{n ^{2}} \approx \frac{2}{n} = 6.6 \times 1 0^{- 19}

Nineteen orders of magnitude below any measurement. The spectrum is quantized in principle and perfectly continuous in practice — the correspondence principle as arithmetic. Quantum mechanics hid because $h$ is small compared to macroscopic actions $m vL$ , not because it is subtle.

Result

n \approx 3 \times 1 0^{18}, Δ E / E \approx 7 \times 1 0^{- 19}