6.3 The de Broglie Relation & the Heisenberg Uncertainty Principle
PRE-HEALTH PREP
de Broglie Equation
Heisenberg Uncertainty Principle
General Chemistry 6.3
The de Broglie equation (λ = h/mv) establishes that all matter possesses wave-like properties — a concept known as wave-particle duality — with wavelength inversely proportional to both mass and velocity. While macroscopic objects have immeasurably small de Broglie wavelengths, sub-atomic particles like electrons exhibit detectable matter waves, a phenomenon confirmed through electron diffraction experiments. The Heisenberg uncertainty principle (Δx · Δp ≥ h/4π) sets a fundamental limit on the simultaneous precision of position and momentum measurements, reflecting an inherent property of nature rather than a technological limitation.
What Is the de Broglie Relation?
The Story Behind the de Broglie Equation
Louis de Broglie holds a special place in the history of chemistry — and a special place in this course. His PhD thesis story is one of the most remarkable in all of science, so it is worth telling before diving into the de Broglie equation itself.
PhD theses are typically 50 to 100 pages long, and the defense is a rigorous oral examination. A committee of distinguished professors probes the candidate systematically: they begin with a question they are confident the candidate can answer, then escalate to one they suspect the candidate cannot. By bracketing the candidate's knowledge from below and above, the committee narrows in on exactly what the candidate truly understands. It is an intentionally grueling process designed to rattle even the most prepared student.
De Broglie's thesis, however, was a single page.
De Broglie's Argument (in one page): If light exhibits both wave-like and particle-like behavior, why should matter be any different? We have always treated matter as purely particle-like, but de Broglie proposed that matter also possesses wave-like properties — the basis of wave-particle duality — and that a characteristic wavelength can be calculated for any material object using his equation.
His thesis committee had no idea what to make of it. They sent de Broglie out of the room, debated among themselves, and ultimately decided to forward the thesis to Albert Einstein for an outside opinion. Einstein's response was characteristically direct: grant the man his PhD, stop wasting time, and send him to me.
De Broglie was, by any measure, smarter than his entire thesis panel.
The de Broglie Equation: λ = h / mv
The de Broglie relation states that any object with mass and velocity has an associated wavelength. This wave-particle duality is quantified by the de Broglie equation:
de Broglie Equation:
λ = h / mv
- λ = de Broglie wavelength of the particle (meters)
- h = Planck's constant (6.626 × 10⁻³⁴ J·s)
- m = mass of the particle (kilograms)
- v = velocity of the particle (m/s)
Two key relationships follow directly from the equation's structure. Because mass and velocity appear in the denominator, de Broglie wavelength is inversely proportional to both mass and velocity: as either increases, the wavelength decreases.
Why you can't see your own de Broglie wavelength: You have a de Broglie wavelength — every object does. But your mass is so large that the resulting wavelength is far too small to detect with any instrument on Earth. For sub-atomic particles like electrons, however, the mass is so extraordinarily small that the wavelength becomes measurable. This is why wave-particle duality matters most at the quantum scale.
Once physicists had the tools to test de Broglie's prediction, they confirmed it experimentally. A technique called electron diffraction exploits the wave nature of electrons to map the positions of atoms within crystalline materials — a direct, practical application of a one-page PhD thesis.
Worked Example: de Broglie Wavelength of a Baseball
To illustrate why matter waves are imperceptible at everyday scales, consider calculating the de Broglie wavelength of a baseball using the de Broglie equation.
Example: de Broglie Wavelength of a 1 kg Baseball at 50 m/s
Given:
- Mass: m = 1 kg
- Velocity: v = 50 m/s (approximately 112 mph — a solid fastball)
- Planck's constant: h = 6.626 × 10⁻³⁴ J·s
Calculation using λ = h / mv:
λ = (6.626 × 10⁻³⁴) / (1 × 50) = 1.33 × 10⁻35 m
Result: 1.33 × 10⁻35 m — far smaller than anything measurable. No instrument on Earth can detect a de Broglie wavelength this small.
Now contrast that with an electron. An electron's mass is approximately 9.11 × 10⁻³1 kg — about 30 orders of magnitude smaller than the baseball. Dividing Planck's constant by such a tiny mass produces a de Broglie wavelength large enough to measure experimentally, which is exactly what electron diffraction does.
On Planck's constant: 6.626 × 10⁻³⁴ J·s is not a constant most professors require students to memorize — it is almost always provided on exams. Check with your instructor, but do not lose sleep over it.
What Is the Heisenberg Uncertainty Principle?
Werner Heisenberg is another fascinating figure from the history of quantum mechanics. During World War II, Heisenberg was ordered by the Nazi government to lead Germany's atomic bomb program. According to the book The Catcher Was a Spy, a former major league catcher turned OSS operative was sent behind enemy lines to assess how close Heisenberg was to a working weapon — with orders to assassinate him if the answer was "close." Whether or not that account is entirely accurate, history records that Heisenberg never came close to completing the bomb. There are those who believe that was entirely intentional. Given what we know of Heisenberg's character and brilliance, that interpretation is easy to sympathize with.
The Heisenberg Uncertainty Equation: Δx · Δp ≥ h / 4π
The Heisenberg uncertainty principle establishes a fundamental limit on how precisely 2 related properties of a particle — position and momentum — can be known simultaneously. In a general chemistry course, the form you will encounter most often is:
Heisenberg Uncertainty Principle:
Δx · Δp ≥ h / 4π
- Δx = uncertainty in position
- Δp = uncertainty in momentum
- h = Planck's constant
- h / 4π = the minimum product of the 2 uncertainties (sometimes written as ℏ / 2, where ℏ is "h-bar")
Momentum is defined as mass times velocity (p = mv). For an electron, mass is constant, so changes in momentum reflect changes in velocity — both speed and direction.
The Heisenberg uncertainty principle inequality says that the product of the 2 uncertainties can never fall below h / 4π. This is not a limitation of our instruments; it is a fundamental property of nature. The more precisely you pin down where an electron is, the less you can know about where it is going — and vice versa.
The key insight into the Heisenberg uncertainty principle: Uncertainty in position and uncertainty in momentum are inversely linked. Decrease one and the other must increase to keep their product at or above the minimum value. You cannot simultaneously know both with unlimited precision — not because of technological limitations, but because the universe does not permit it.
Solving Heisenberg Uncertainty Problems Step-by-Step
Exam problems on the Heisenberg uncertainty principle typically follow a straightforward pattern: you are given one uncertainty and asked to find the minimum value of the other. This is a direct plug-and-chug application of the inequality.
4-Step Strategy for Heisenberg Uncertainty Problems
- Identify what you are given: either Δx (uncertainty in position) or Δp (uncertainty in momentum).
- If you are given a velocity uncertainty (Δv) instead of a momentum uncertainty, convert using Δp = m · Δv. The mass of the particle (usually an electron) will be provided.
- Set the product equal to the minimum: Δx · Δp = h / 4π
- Solve for the unknown quantity.
Velocity vs. momentum in Heisenberg uncertainty problems: When a problem gives you an uncertainty in velocity (Δv) rather than momentum, do not plug Δv directly into the formula. First multiply by the particle's mass to get Δp = mΔv, then proceed. The particle will almost always be an electron, and the mass will be provided in the problem.
The conceptual vocabulary is just as important as the arithmetic. Know these 2 terms cold: position and momentum. A question asking about the certainty of an electron's location is asking about position; a question about how fast it is moving and in what direction is asking about momentum. Recognizing which is which is often the hardest part of Heisenberg uncertainty problems.
6 Key Takeaways
- The de Broglie equation (λ = h / mv) states that all matter has an associated wavelength. De Broglie wavelength is inversely proportional to both mass and velocity.
- For macroscopic objects, the de Broglie wavelength is immeasurably small. For sub-atomic particles like electrons, it is large enough to detect — a phenomenon exploited in electron diffraction.
- Planck's constant h = 6.626 × 10⁻³⁴ J·s appears in both the de Broglie equation and the Heisenberg uncertainty principle; it is usually provided on exams rather than memorized.
- The Heisenberg uncertainty principle (Δx · Δp ≥ h / 4π) sets a fundamental lower limit on how precisely an electron's position and momentum can be simultaneously known.
- Position and momentum uncertainty are inversely linked: greater precision in one demands greater uncertainty in the other.
- In exam problems, if given a velocity uncertainty rather than a momentum uncertainty, convert first: Δp = mΔv, then apply the Heisenberg uncertainty inequality.