Which one of the following physical quantities has the same dimensions as Planck constant h in dimensional analysis?

Introduction / Context:
Planck constant h is a fundamental constant in quantum mechanics. It appears in relations such as E = h * f for photons and p = h / lambda for de Broglie waves. Understanding the dimensions of h helps in interpreting these equations and recognising which physical quantities are of the same dimensional type. The question asks you to identify which listed quantity has the same dimensions as Planck constant. This is a classic dimensional analysis problem that reinforces the connection between quantum constants and mechanical quantities like action and angular momentum.

Given Data / Assumptions:
• Planck constant h has known dimensions that can be derived from photon energy formulas. • The options include linear momentum, angular momentum, force and potential energy. • We use base dimensions mass, length and time. • We assume standard definitions of each mechanical quantity listed.

Concept / Approach:
From the photon formula E = h * f, where E is energy and f is frequency, we can determine the dimensions of h. Energy has dimensions of mass times length squared divided by time squared, and frequency has dimensions of one over time. Therefore h has dimensions of energy divided by frequency, which simplifies to mass times length squared divided by time. Angular momentum also has dimensions mass times length squared divided by time, since it is defined as moment of momentum (position vector cross linear momentum). Linear momentum, force and potential energy each have different dimensions: momentum is mass times length divided by time, force is mass times length divided by time squared, and energy is mass times length squared divided by time squared. Hence, angular momentum is the quantity whose dimensions match those of Planck constant.

Step-by-Step Solution:
Step 1: Start from the relation E = h * f and write the dimensions of energy E as M * L^2 * T^-2. Step 2: Write the dimensions of frequency f as T^-1. Step 3: Solve for the dimensions of h: [h] = [E] / [f] = (M * L^2 * T^-2) / (T^-1) = M * L^2 * T^-1. Step 4: Compare this with the dimensions of angular momentum L, which is r cross p, so dimensions are (L) * (M * L * T^-1) = M * L^2 * T^-1. Step 5: Note that linear momentum p has dimensions M * L * T^-1, which do not match. Step 6: Note that force and potential energy both involve T^-2, not T^-1, so they do not match either.

Verification / Alternative check:
An alternative way is to observe that in quantum mechanics, Planck constant h and its reduced form h slash 2 pi often appear multiplying phase angles, and are associated with action, which is the time integral of the Lagrangian. Action has the same dimensions as angular momentum (mass times length squared divided by time). This connection is why h is sometimes described as a quantum of action. In atomic physics, orbital angular momentum of electrons is quantised in units of h slash 2 pi, again emphasising that h and angular momentum share the same dimensional nature. These theoretical considerations support the dimensional analysis result.

Why Other Options Are Wrong:
Option A, linear momentum, has one less factor of length in its dimensions compared with h; it is M * L * T^-1, not M * L^2 * T^-1. Option C, force, has dimensions M * L * T^-2, with an extra factor of T^-1 compared with h. Option D, potential energy, shares the same dimensions as energy, M * L^2 * T^-2, again differing by an extra T^-1 factor. These differences show that momentum, force and energy are not dimensionally equivalent to Planck constant, which instead matches angular momentum and action.

Common Pitfalls:
A frequent mistake is to assume that because h appears in energy and momentum formulas, it must have the same dimensions as energy or momentum. Without doing explicit dimensional analysis, this intuition can easily mislead. Another pitfall is to treat h as dimensionless because it often appears in exponentials or phase factors; in fact, expressions such as S / h inside exponentials are dimensionless because S and h both share the same dimensions. Always write down base dimensions systematically when comparing quantities to avoid such confusion.

Final Answer:
The correct choice is Angular momentum, because Planck constant h has dimensions M * L^2 * T^-1, which are the same as those of angular momentum and action, not those of linear momentum, force or potential energy.