1. What Did 23-Year-Old Heisenberg Begin While Recovering on Helgoland Island?
On June 7, 1925, 23-year-old Heisenberg traveled to the island of Helgoland in the North Sea to recover from severe hay fever [1]. It was a trip to heal in a place with less pollen. On the island, he worked on stopping the idea of "invisible orbits" describing how electrons circle inside an atom. Instead, he tried to explain atoms using only quantities that can actually be measured, such as the frequency and intensity of the light an atom emits.
Heisenberg recalled in later years that the final result of his calculation came out near 3:00 AM [2]. Since this is a memory told later, treat the time as an estimate. Once he decided to build a theory using only measurable quantities, a question arose: What does it actually mean to "measure the position of an electron"?
2. What Does "Seeing" Mean in Physics? Einstein's Remark to Heisenberg
Heisenberg recalled in later years that Einstein told him, "It is the theory that decides what can be observed" [4]. Since this is a story from later years, it is best to treat it as an anecdote. This comment raised a doubt: If you decide to build a theory using only what can be measured, you must really think about what "measuring" means. What is the specific procedure for seeing the position of an electron?
So, he devised an experiment to be done only in his head.
3. What Happens When You Look at an Electron with a Gamma-Ray Microscope?
In March 1927, Heisenberg published the uncertainty relation at Bohr’s institute in Copenhagen, stating there is a limit to measuring paired quantities like position and momentum at the same time [1]. The paper is dated March 27 and uses the thought experiment of a gamma-ray microscope [4].
To see the position of an electron clearly, you need light with a short wavelength, such as gamma rays. However, when that light hits the electron, the electron is kicked, and its momentum is disturbed [3]. The more you try to clarify the position, the blurrier the momentum becomes. A familiar analogy is looking for a rolling ball in a dark room by touch. The moment you touch it, the ball moves. However, the "kick" for an electron is much more essential than for a ball. What happens with an electron is different from the analogy of touching a ball with your hand.
Heisenberg estimated the size of this blur as Δp · Δx ~ h [3]. Note that the "~" symbol indicates a rough estimate. This raises a question: Is this "blur" caused only by kicking the electron with light in an attempt to see it? Note that whether this "disturbance by measurement" is the only cause is a problem that was later sorted out differently (see Section 5).
4. Why Electron Waves, Not Just a "Kick," Limit Measurement
Actually, electrons behave like particles and also like waves. If represented in wave form, the spread of the wave shape and the distribution of which wavelengths are mixed in cannot both be sharp at the same time. This is explained as a fact derived from a mathematical relationship called the Fourier transform [5].
Position and momentum are a "pair" in this relationship. Therefore, in a prepared state of an electron, you cannot make the spread of position and the spread of momentum small at the same time. This explanation does not involve a gamma-ray microscope. Even without kicking with light, the wave nature alone explains why position and momentum cannot be sharply determined at the same time. This is a separate issue from measurement error or disturbance.
So, are these two explanations just different ways of saying the same formula, or are they different things?
5. Two Ways to Read the Uncertainty Relation: Spread in Preparation vs. Disturbance in Measurement
In the same year, 1927, Kennard proved a strict inequality for the spread (standard deviation) of position and momentum. It states that the product of the spread in position and the spread in momentum is at least half of ħ (h-bar), which is Planck’s constant h divided by 2π [5]. This is the current standard formula.
The Stanford Encyclopedia of Philosophy organizes uncertainty into two types [4]. One is preparation uncertainty. If you measure position and momentum separately for many electrons made in the same state, the product of their spreads is ħ/2 or more. Kennard’s formula corresponds to this. The other is the relationship between measurement error and disturbance, which is close to what Heisenberg wanted to say with the microscope story.
These two are different things. Therefore, it is possible for one of the formulas to have a flaw.
6. Was Ozawa's 2003 Inequality on Measurement Error and Disturbance Confirmed by Experiment?
In 2003, Masanao Ozawa from Nagoya University proposed the "Ozawa inequality," which represents the relationship between measurement error and disturbance. He also showed that Heisenberg’s original inequality could be broken depending on the state before measurement [6].
In 2012, Yuji Hasegawa (Vienna University of Technology) and others reported that in an experiment measuring neutron spin, results following Ozawa’s formula were observed, deviating from Heisenberg’s error and disturbance formula [7]. The results were published in Nature Physics (January 15, 2012) [6].
Here, it is important to note that what was broken was the formula for "measurement error and disturbance." The uncertainty spread expressed by Kennard’s formula was not broken. It is good to read this as an experiment confirming how well the 1927 idea holds up nearly 100 years later.
By the way, Heisenberg received the 1932 Nobel Prize in Physics not for the uncertainty relation itself, but for the creation of quantum mechanics [8].
7. How Can You Look Up the Uncertainty Relation Yourself?
What you can do today: Write in your own words what it means to "see." We see things in a dark room because light hits the object and reaches our eyes. So, what happens to this mechanism when we look at small things like electrons?
Materials in Japanese: There are articles introducing the Ozawa inequality and the 2012 experiment in Japanese materials, such as explanations from domestic universities and research institutions, and articles on the JST Science and Technology Portal. You can access the original sources via the URLs in the source list.