Question 3 of 7: Planck's Constant as the Quantum of Action

Question 3: Planck's Constant as the Quantum of Action (10 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Four physical relationships in which $h$ appears: photon energy, photon momentum, orbital angular momentum quantization, and the uncertainty principle.

Find. (a) each relation written explicitly in terms of $h$; (b) the SI dimensions of $h$, derived independently from two of the relations; (c) why $h$ is called the "quantum of action" in each case.

Approach. Write each defining relation, then read off $h$'s dimensions from two independent routes ($E=hf$ and $p=h/\lambda$) and confirm they agree; finally recognize that "action" (energy$\times$time $\equiv$ momentum$\times$length $\equiv$ angular momentum) is the common dimensional thread tying all four relations together.

1. Part (a) — the four relations. Photon energy: $E=hf$ (equivalently $E=\hbar\omega$). Photon linear momentum: $p=h/\lambda$ (equivalently $p=\hbar k$). Angular momentum of an orbiting electron (Bohr quantization): $L=n\dfrac{h}{2\pi}=n\hbar$, $n=1,2,3,\ldots$. Uncertainty principle: $\Delta x\,\Delta p \ge \dfrac{h}{4\pi}$ (equivalently $\Delta x\,\Delta p\ge\hbar/2$).
2. Part (b) — dimensions of $h$. From $E=hf$: $$[h] = \frac{[E]}{[f]} = \frac{\text{kg}\,\text{m}^2\,\text{s}^{-2}}{\text{s}^{-1}} = \text{kg}\,\text{m}^2\,\text{s}^{-1}$$ Independently, from $p=h/\lambda$: $$[h] = [p][\lambda] = (\text{kg}\,\text{m}\,\text{s}^{-1})(\text{m}) = \text{kg}\,\text{m}^2\,\text{s}^{-1}$$ $$\boxed{[h] = \text{kg}\,\text{m}^2\,\text{s}^{-1} = \text{J}\cdot\text{s}}$$ Both independent routes agree, which is itself a check that the four relations in part (a) are dimensionally consistent with one another.
3. Part (c) — why "quantum of action". In classical mechanics, "action" has exactly the dimension found in part (b): energy$\times$time, or equivalently momentum$\times$length, or equivalently angular momentum — all three are the same dimension, $\text{J}\cdot\text{s}$. Each of the four relations in (a) is a statement that some classical action-like quantity is quantized in units of $h$ (or $h/2\pi$): $E=hf$ says an oscillator/photon's energy comes in packets of $h$ per cycle; $p=h/\lambda$ says a wave's momentum comes in packets of $h$ per wavelength; $L=nh/2\pi$ says orbital angular momentum (itself dimensionally an action) is restricted to integer multiples of $h/2\pi$; and the uncertainty principle says the minimum "area" a quantum state can occupy in phase space ($\Delta x\,\Delta p$, also dimensionally an action) is bounded below by a quantity proportional to $h$. In every case, $h$ is the smallest indivisible unit in which nature lets an action-dimensioned quantity change — hence "quantum of action".