Scaling and Units

Introduction

In our previous lesson, we explored how dimensionless ratios, such as the Reynolds number, quantify the competition between physical effects and define distinct regimes of behavior. This naturally raises a powerful question: once we identify the dominant physics, can we simplify our mathematical description to reflect this, clearing away the non-essential details?

This lesson introduces the heuristic that formalizes this simplification: the use of natural units. You will learn how to select a unit system based on the intrinsic scales of the problem you are studying. By doing so, you can often set the most important physical constants of your problem—like $c$ or $ℏ$ —to unity. This practice does more than just tidy up equations; it strips away the "crud" of arbitrary human-centric units, revealing the essential physical relationships and making the underlying structure of a theory transparent. For a theoretical physicist, especially one navigating the complexities of quantum field theory and condensed matter, mastering this technique is not just a convenience but a fundamental way of thinking.

From Arbitrary Conventions to Intrinsic Scales

The units we typically use in introductory physics—meters, kilograms, seconds (MKS)—are deeply ingrained, but they are fundamentally arbitrary. They are anthropocentric conveniences, not quantities dictated by nature. As you delve deeper into theoretical physics, it becomes clear that these arbitrary choices can obscure the fundamental relationships you seek to understand.

A more profound approach is to recognize that every physical problem suggests its own set of intrinsic, or "natural," scales. For studying planetary motion, the Earth-Sun distance (the AU) is a more natural unit of length than the meter. In atomic physics, the mass of the electron is a natural unit of mass. The core idea of natural units is to build your measurement system from these intrinsic scales.

[PDF] Department of Physics Notes on Dimensional Analysis and Units

This document from CSU Chico's physics department provides an excellent philosophical and practical grounding for the concept of natural units. It argues that units are merely an intermediate, sociological function and that any well-posed problem determines its own intrinsic scales.

Please read the following two sections: In the Introduction (Section I), focus on the core argument presented in the last two paragraphs. Start from the main thesis about the role of units. The author makes the striking claim that units have "no 'physical content' whatever" and that a well-expressed relationship should be independent of them. Next, jump to Section IV, "NATURAL UNITS." Read the introductory paragraphs on pages 6 and 12. This section explains why we use natural units: to make numerical values close to unity (which is ideal for computation) and, more importantly, to make the "essential physical relationships stand out much more clearly."

This perspective shift is crucial. We are not just changing units; we are re-framing the problem to align with its own physics. The goal is to express relationships within the setting of the problem alone, a process the author calls "self-referencing."

The Mechanics of Natural Units

So, how do we practically switch to a natural unit system? The fundamental goal is to make the important constants and parameters of our problem have a numerical value of 1.

Let's consider a simple example: a falling body described by the conservation of energy, where $m$ , $g$ , and the total energy $E$ are the defining constants.

\frac{1}{2} m (\frac{d x}{d t})^{2} + m g x = E

We want to find natural units of length $[L]$ , time $[T]$ , and mass $[M]$ for this problem. The "Notes on Dimensional Analysis and Units" you just read outlines a rigorous, step-by-step method to do this.

[PDF] Department of Physics Notes on Dimensional Analysis and Units

This part of the document provides a detailed walkthrough of how to non-dimensionalize an equation and discover its natural units.

Please read the following sections: First, carefully follow the "Engineer’s Work-a-Day Method" described in sections A. Finding Natural Units (Steps 1-6) on pages 7 through 9. This method involves rescaling the variables (e.g., x' = x/L), substituting them into the original equation, and then choosing the scales L, T, M such that the resulting "clusters" of constants become equal to one. For the falling body, this procedure leads to the natural scales L = E/mg, M=m, and T = \sqrt{E/(mg^2)}, and the beautifully simple dimensionless equation: \frac{1}{2} \left(\frac{dx'}{dt'}\right)^2 + x' = 1 Next, read the "Physicists Tricky 'Short-Cut'" in Section B on page 10. This is the method you will see and use most often in advanced physics. Instead of the formal rescaling, you simply assert that in your new unit system, the defining constants are unity. That is, you set m=1, g=1, and E=1 and solve for the units that make this true. The result is the same, but the path is more direct.

The physicist's shortcut is not cheating; it is a recognition that the choice of units is equivalent to setting the numerical value of the defining constants.

Another classic example is the simple pendulum. Its equation of motion is $θ ¨ = - (g / L) sin θ$ . The problem has an intrinsic time scale given by its linear period, $T_{0} \propto L / g$ . By defining a dimensionless time $τ = t / L / g$ , the equation of motion becomes $\frac{d ^{2} θ}{d τ ^{2}} = - sin θ$ . In these natural units, the equation is universal; it no longer depends on the specific values of $L$ or $g$ , and describes the motion of any simple pendulum.

Natural Units in Fundamental Physics: $c = ℏ = 1$

The most famous and powerful application of natural units is in quantum field theory and particle physics. The theories of relativity and quantum mechanics introduce two universal constants of nature:

The speed of light, $c$ , which connects space and time.
The reduced Planck constant, $ℏ$ , which connects energy and time (or momentum and length).

In this context, it is overwhelmingly convenient to adopt a system of units where $c = 1$ and $ℏ = 1$ . Let's be very precise about what this means.

A (Gentle) Introduction to Theoretical Physics & Natural Units

This video by Curt Jaimungal offers an exceptionally clear conceptual explanation of natural units, focusing on debunking the common misunderstandings around setting constants to one.

Please watch the following segments: First, watch the discussion on what units are, from "what are units?". The key idea is that standard units like the second or kilogram are based on arbitrary, human-made operational definitions. "Natural" units, by contrast, are based on quantities that are invariant across the universe. Next, focus on the crucial segment that addresses the central myth, from "myths about natural units". The presenter explains that saying "c=1" is a meaningless statement if taken literally. What it actually means is that we are choosing to measure velocity in units of the speed of light itself. The analogy to currency exchange or measuring height in units of "Lady Gagas" is very helpful for clarifying this point. You are not changing the physics; you are changing your ruler.

Once we adopt this system where $c = 1$ and $ℏ = 1$ , a profound simplification occurs. The fundamental equations that connect mass, energy, momentum, length, and time become simple conversion factors:

Special Relativity: $E = m c^{2}$ becomes $E = m$ .
- This implies that mass and energy are measured in the same units.
Quantum Mechanics: $E = ℏ ω$ becomes $E = ω = 1/ T$ .
- This implies energy has units of inverse time.
De Broglie Relation: $p = ℏ k = ℏ (2 π / λ)$ becomes $p = 2 π / λ \sim 1/ L$ .
- This implies momentum has units of inverse length.

Combining these, we find that we can express every physical quantity in terms of powers of a single, chosen unit. In high-energy physics, this unit is typically energy, measured in electron-volts (and its multiples, like GeV).

Mass: $[M] = [E]$
Length: $[L] = [E]^{- 1}$
Time: $[T] = [E]^{- 1}$

Why GeV? The mass of a proton is approximately 1 GeV/c². In a world where $c = 1$ , we can simply say the proton mass is about 1 GeV. This provides a cognitively graspable scale tied to a fundamental particle.

Let's see how this works in practice to find the "energy dimension" of any quantity.

A (Gentle) Introduction to Theoretical Physics & Natural Units

This part of the video demonstrates the conversion process and how to determine the units of any physical quantity in this system.

Watch these two segments: The Conversion Equations: From this section, see how the fundamental equations allow you to convert between mass, length, time, and energy, justifying the choice of GeV as the base unit. Analyzing Newton's Constant: From this practical example, the presenter walks through the dimensional analysis of Newton's constant, G. He starts with its SI units, decomposes them into fundamental dimensions (mass, length, time), and then replaces each with its energy equivalent (M \to E, L \to E^{-1}, T \to E^{-1}). This reveals that G has units of E^{-2}. You can do this for any physical quantity.

The table below summarizes the relationship between various unit systems.

This chart illustrates the conversion factors between Natural Units ($\hbar=c=1$), practical Hybrid Units, and cgs units for various physical quantities. The column "M" shows the power of mass/energy for each quantity in natural units.

Application: Energy Scales in a Molecule

This technique of using natural units to expose physical structure is not limited to high-energy physics. It is a powerful heuristic in condensed matter physics, as well. A prime example is understanding the hierarchy of energy scales in a molecule. A molecule has electronic, vibrational, and rotational degrees of freedom, and we can estimate their characteristic energies using these methods.

L17.4 Molecules and energy scales

In this MIT OpenCourseWare lecture, Prof. Hong Liu uses exactly these heuristic arguments to derive the hierarchy of energy scales in a molecule, a classic condensed matter system. The key small parameter is the ratio of the electron mass (m_e) to the nuclear mass (M).

Watch the following three estimation steps: Electronic Energy: From this segment, the lecturer estimates the characteristic electronic energy. Using the uncertainty principle, the momentum of an electron confined to a region of size a is p \sim \hbar/a. The electronic energy is then E_{el} \sim p^2/m_e = \hbar^2/(m_e a^2). This sets the fundamental energy scale for the entire system. Vibrational Energy: Next, watch the estimation of the nuclear vibrational energy. The nuclei oscillate in a potential created by the electrons. The effective spring constant k must depend on electronic properties, so by dimensional analysis, k \sim E_{el}/a^2. The vibrational frequency is \omega_{vib} \sim \sqrt{k/M}. The associated energy is E_{vib} = \hbar\omega_{vib}. Putting this together, you find the crucial result: E_{vib} \sim \sqrt{\frac{m_e}{M}} E_{el} Rotational Energy: Finally, watch the brief estimation of the rotational energy. The rotational energy is E_{rot} \sim L^2/I \sim \hbar^2/(M a^2). Comparing this to the electronic energy, we find: E_{rot} \sim \frac{m_e}{M} E_{el}

This is a beautiful demonstration of the power of this heuristic. Without solving any complex Schrödinger equation, but simply by using natural scales ( $ℏ$ , $a$ , $m_{e}$ , $M$ ), we have revealed the fundamental energy structure of any molecule:

E_{r o t} ≪ E_{v ib} ≪ E_{e l}

The energy scales are separated by powers of the small mass ratio $m_{e} / M$ . This hierarchy is the reason the Born-Oppenheimer approximation works, and it governs the entire field of molecular spectroscopy.

Conclusion

Today, we've seen that choosing natural units is a profound heuristic step that goes far beyond simple convenience. By aligning our unit system with the intrinsic scales of a physical problem, we simplify our equations and expose the core physics that was previously cluttered by arbitrary human conventions.

Key Takeaways:

Units are a choice: SI units are arbitrary conventions. Every physical problem suggests its own "natural" units based on its defining constants and parameters.
Simplification: In natural units, key physical constants (like $c$ , $ℏ$ , or problem-specific parameters) have a numerical value of 1. This drastically simplifies equations.
The $c = ℏ = 1$ System: In relativistic and quantum physics, setting $c = ℏ = 1$ is standard. This allows all physical quantities (mass, length, time) to be expressed in units of a single base quantity, typically energy (GeV).
Revealing Structure: The true power of natural units is in revealing the essential structure of a problem. We saw this in the clear hierarchy of energy scales—electronic, vibrational, and rotational—that emerged from a simple analysis of a molecule.

In our last few lessons, we have built a toolkit for analyzing the scales of a problem. We can identify characteristic scales, use them to constrain functions, form ratios to find dominant effects, and now, choose units to simplify our mathematics. In the next lesson, we will synthesize these ideas to estimate physical quantities in quantum and statistical systems using dimensional arguments, turning these scaling insights into concrete numerical predictions.

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