Heisenberg Made a Discovery in 1925. We Still Can't Explain It

0:00 Thank you to Novium, the team behind the hoverpen,

0:03 for supporting PBS 2025 is… was

0:06 the international year of quantum science and technology.

0:10 In celebration of the invention of our strangest true theory 100 years ago.

0:18 In 1925, quantum mechanics went from being a peculiar

0:21 set of ideas to describe some funny results from experiments,

0:26 to a full-blown theoretical framework that overturned

0:31 how we think reality really works.

0:35 So today, as the centenary year approaches its end I want to take you

0:40 on a little journey through We’ve got

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1:59 Now onto the episode Let’s start by understanding

2:02 where we stood at the beginning of 1925.

2:06 What was the scientific worldview that was about to be shattered?

2:11 So Albert Einstein has been the king of physics for a decade,

2:15 ever since he published his general theory of relativity

2:18 in 1915 and toppled the four-century reign of Newtonian mechanics.

2:23 GR completely supplanted our understanding of space, time, motion and gravity.

2:28 This was a revolution, no doubt, but in a way Einstein actually reinforced

2:34 the status quo rather than overturning it.

2:38 See in the Newtonian worldview, the motion of all particles could be perfectly

2:43 computed through a simple set of universal laws.

2:47 And all things are made of particles, so, surely,

2:50 the universe itself and all it contains is computable.

2:55 It’s all deterministic.

2:57 Know the present and you can calculate all past and all future states.

3:02 Newton’s achievement gave us a sense that the universe

3:07 is knowable to a degree of completeness never before imagined.

3:12 There were some few inconsistencies remaining—the fact

3:15 that Mercury’s orbit didn’t quite obey Newton,

3:18 some stuff about electromagnetism and the speed of light.

3:21 But the universe has to be axiomatically self-consistent.

3:26 And so Einstein was able to leverage

3:31 these seemingly minor glitches into a full-blown scientific revolution,

3:36 in which space and time were merged

3:38 and the resulting spacetime was no longer Newton’s static,

3:42 universal background, but rather a malleable, dynamic thing.

3:46 But the new picture that came out of Einstein’s

3:49 relativity still shared a fundamental quality with Newton’s worldview.

3:53 It was just as deterministic.

3:55 The equations of general relativity suffer no uncertainty.

3:59 The distribution of matter and energy

4:02 in spacetime perfectly defines the geometry of spacetime,

4:06 which in turn completely determines the motion of said matter and energy,

4:11 forever into the past and future.

4:14 Einstein also showed that simultaneity is relative,

4:18 destroying our delusion of an absolute sense of “now”.

4:22 So the already-deterministic past and future became as real as the present,

4:28 at least in some interpretations, and all of time was crystalized

4:33 into an eternal block for the entirety of existence.

4:38 On the one hand the universe under Einstein remained measurable and knowable,

4:44 but what is measured and what can be known came to depend on perspective;

4:50 where you are, how fast you’re moving,

4:54 the lightspeed-limited horizon of your embedded vision.

4:58 A hell of a perspective shift, but still firmly on the side of determinism.

5:03 In the early 1920s Einstein was a science rockstar.

5:08 There were still a few loose threads

5:11 in the fabric of our understanding of the world.

5:14 So you can imagine the hunger of a young physicist

5:17 of the era to pull at these threads and so,

5:22 perhaps, pull off another revolution.

5:25 Now a big outstanding mystery,

5:26 perhaps THE big one was the strange world of the atom,

5:31 and especially of the behavior of the humble electron.

5:35 From Rutherford we knew the rudimentary atomic nucleus.

5:38 You’ve all seen depictions of electrons whizzing around the nuclei like tiny

5:44 solar systems—a holdover from early assumptions

5:47 that the tiny universe resembles the gigantic.

5:50 But there were so many questions!

5:53 Why were only certain electron orbits allowed, and why those ones?

5:58 Why didn’t electrons spiral into their nuclei?

6:01 The uncanny orbit of the electron was surely the key to a new revolution,

6:07 just as the orbit of Mercury unlocked relativity.

6:10 First steps were tentative.

6:12 Neils Bohr, building on Planck and Einstein’s quantization of light,

6:16 gave us an empirical model of quantized electron energy levels.

6:21 Louis de Broglie, building on that same quantization,

6:24 guessed that all matter has a wave nature.

6:27 Bohr then realised that electron waves made sense of his atomic model.

6:32 The allowed orbits are those that perfectly fit an integer

6:37 number of electron wave cycles—these are self-reinforcing, standing waves.

6:42 While this gave a nice story for the quantization of electron orbit,

6:47 other problems remained,

6:48 perhaps the worst being that it only worked for the humble hydrogen atom.

6:53 So that’s where we stood going into 1925.

6:57 Bohr’s model was still very classical-feeling,

7:01 and so the Newtonian and Einsteinian worldviews still reigned.

7:05 We clung to the ambition to be aloof masters of a computable universe.

7:11 The spark that set this entire worldview

7:14 ablaze had already started—in Munich a few years

7:18 earlier when a 20 year old student showed up in the office of Arnold Sommerfeld.

7:25 Sommerfeld, famous both for his physics and his mentorship,

7:30 handed the student a trial problem—to solve one

7:33 of the many failings of the Bohr model,

7:36 in particular the distortions in hydrogen energy levels seen when

7:42 the stuff was in strong magnetic fields called the anomalous Zeeman effect.

7:47 So the thing about being a kid is that you’ll try anything.

7:51 You don’t yet know what you’re supposed to take as ground truth.

7:55 All common sense dictated that the math should be based

7:59 on integer numbers—the whole numbers of wavecycles in Bohr’s electron orbits.

8:05 But Sommerfeld’s naive student for whatever reason

8:11 tried a mathematical form involving physically irrelevant half-integers.

8:18 But that worked.

8:20 The student was Werner Heisenberg, and at 20 he’d discovered the first clue

8:25 to the strange phase symmetry of the electron,

8:29 a result of its yet undiscovered quantum spin.

8:33 This little project began Heisenberg’s obsession with the mysteries of the atom.

8:37 Three years later, now 1925,

8:39 Heisenberg was in Gottingen studying under Max Born,

8:43 but also in close connection with the now very famous Neils Bohr,

8:49 his future closest collaborator.

8:50 Surrounded by such giants of physics,

8:53 Heisenberg still turned to the wisdom of Einstein.

8:57 He asked himself, what would Einstein do?

9:00 In coming up with general relativity,

9:02 Einstein resolved to reject unfounded assumptions,

9:05 no matter how self-evident they seemed.

9:09 That meant putting aside ideas like the universality

9:14 of time and an unchanging and flat geometry of space.

9:19 So Heisenberg challenged himself to take this credo

9:22 seriously and push it as far as possible.

9:26 In his own words, “The aim of quantum theory should be

9:31 to describe only quantities which are

9:34 observable.” Make no assumptions about, for example,

9:37 what the electrons are really doing inside the atom; instead,

9:40 build a theory entirely around the results of what

9:43 we see when we do measurements on the atom.

9:46 In a cartoon view of an atom you can imagine knowing things like

9:51 where the electron is or how fast it’s moving at any point in time.

9:56 In the Bohr model that translates to a radius

9:59 and energy-slash-momentum for each orbit—a pair of concrete values,

10:04 each with an index to connect them to a list of possible electron states.

10:11 But Heisenberg realized that it’s not possible to actually

10:15 observe these orbits of the electrons within the atom.

10:18 The only thing we can measure is the light—the

10:21 photon—that shoots out when an electron changes orbit.

10:26 In fact, even the idea of orbits

10:29 is presumptuous—so we will call them electron states.

10:33 These invisible electron states couldn’t be the main players in the theory.

10:38 Instead, he committed to finding a law of motion that only describes

10:45 the observables—the frequency and intensity-slash-amplitude

10:47 of the photon produced when electrons change states.

10:52 The difference seems subtle but it really changes everything.

10:56 Instead of a formula to describe electron

10:59 properties for a single list of states,

11:02 Heisenberg sought a formula to describe the resulting photon properties

11:07 for every combination of initial and final state of the electron.

11:12 This sort of two-index list of properties is known as a matrix,

11:17 although Heisenberg didn’t know it at the time.

11:20 His job was to guess a formula relating these matrices,

11:25 and so he figured out the algebra of matrices on his own.

11:30 He was disconcerted to discover something that mathematicians

11:34 already knew—that matrix algebra defies standard multiplications rules.

11:40 For example, X*Y is not always the same as Y*X.

11:45 They are not commutable.

11:47 This turns out to be a central feature of quantum

11:52 mechanics and ultimately led Heisenberg not only to his solution,

11:56 but to the thing he’s most famous for—his uncertainty principle.

12:00 But for now, this unintuitive type of math was

12:04 just another challenge to Heisenberg’s resolution to reject prior assumptions,

12:10 and so he forged ahead.

12:13 Now his theory needed to do a couple of things:

12:16 to predict the photon frequencies and intensities—the “spectra” of atoms.

12:20 It also needed to be consistent with certain known fundamentals of the universe,

12:26 and the big one is the principle of energy conservation.

12:30 Heisenberg writes that he “knew only too well that my scheme

12:36 stood or fell by that principle.” So Heisenberg tested

12:40 his new theory as simply as he could—for a physical

12:43 system similar to a pendulum—the

12:45 anharmonic oscillator—chosen for some mathematical conveniences.

12:49 If energy conservation held in this case then his strange

12:55 theory must be connected to the true machinery of nature.

13:00 This was May 1925 and Gottingham was awash in springtime

13:07 pollen and Heisenberg’s own immune system tried to kill him.

13:11 He had a crippling attack of hay fever.

13:14 It swelled his face, weakened his body,

13:16 and left him barely able to multiply a matrix.

13:20 He begged his boss, Max Born,

13:22 for permission to travel to the one place he might find relief—a small,

13:27 barren island in the middle of the North Sea called Helgoland.

13:32 It’s an amazing place.

13:34 Helgoland—sometimes also Heligoland—from the High Frisian “holy land”—

13:40 with its stunning ocean cliffs and an elegant resort town.

13:45 Now the war changed everything—the town was obliterated by ally bombs and a Nazi

13:51 U-boat harbor and the entire tip

13:53 of the island destroyed in the largest non-nuke,

13:57 man-made explosion in all of history.

14:00 But Heisenberg came before all of that.

14:03 He spent 10 days walking the cliffs,

14:05 swimming in the ocean, and calculating his matrices.

14:08 Pages upon pages of them.

14:11 He talks about a mounting excitement as the scheme remained self-consistent,

14:16 of how that excitement led to many mistakes that had to be fixed.

14:21 And finally, on the night of June 9th 1925… well, let’s let Heisenberg say it:

14:29 It was 3 o’clock in the morning before

14:31 the final result of my computations lay before me.

14:35 The energy principle had held...

14:37 At first I was deeply alarmed.

14:40 I had the feeling that, through the surface of atomic phenomena,

14:45 I was looking at a strangely beautiful interior,

14:49 and felt almost giddy at the thought that I now had to probe

14:54 this wealth of mathematical structures nature

14:56 had so generously spread out before me.

14:59 I was far too excited to sleep, and so, as a new day dawned,

15:04 I made for the southern tip of the island… and waited for the sun to rise.

15:09 So cured of hayfever but now afflicted by revelation,

15:15 Heisenberg returned to Gottingen.

15:17 His friend Wolfgang Pauli encouraged him to show his work to Max Born,

15:22 who immediately recognized Heisenberg’s strange

15:25 mathematics as being matrix algebra.

15:28 From there, he worked with Jordan Pasqual and then with Pauli and Paul

15:33 Dirac to flesh out matrix mechanics—-the

15:36 first complete formulation of quantum mechanics.

15:40 But it was not the instant hit you might have imagined.

15:44 It was a strange theory—matrices were unfamiliar to most physicists of the era,

15:50 the non-commutivity felt alien,

15:53 but perhaps worst of all the theory seemed to tell

15:56 no story of what was really happening inside the electron.

16:00 The catalyzing philosophy of “only consider the observables” was uncomfortable.

16:07 But it did evolve into the Copenhagen interpretation of quantum mechanics,

16:12 developed primarily with Neils Bohr.

16:14 In this interpretation, the universe between measurements is unknowable—not just

16:20 in a practical sense because you didn’t measure,

16:24 but in the sense of being truly undefined.

16:27 This is the key fracture in the old worldview of Newton and Einstein.

16:33 Matrix mechanics implied that the world is not fundamentally knowable.

16:39 Perhaps not even “real” between observations

16:42 in the concrete manner that we were used to.

16:46 And any attempt at knowing was limited by Heisenberg’s uncertainty principle,

16:51 which he soon realized must follow from matrix non-commutivity.

16:56 There’s also the related randomness of quantum mechanics,

17:00 which is baked into the quantum laws of nature.

17:04 All of it was a violent takedown

17:08 of the deterministic and observer-independent Newtonian worldview.

17:12 So physicists were eager to claw back at least some

17:16 sense of realism—of a mechanism that’s independent of observer and measurement.

17:22 And a few months later, Erwin Schrodinger gave them that.

17:28 Where Heisenberg launched his chain of thought

17:30 from the quantization of energy levels,

17:33 Schrodinger started with de Broglie’s idea of particles as waves of matter.

17:38 Physicists had long pondered the connection between particles and waves,

17:43 and there was even a formulation

17:46 of classical mechanics—the Haminton-Jakobi equation—that

17:49 could represent particles as waves while

17:52 being formally equivalent to Newton’s laws.

17:54 In fact Schrodinger just substituted the so called “action”

17:59 in this Hamilton-Jakobi equation with what we now call the wavefunction,

18:05 to lead to the famous Schrodinger equation,

18:08 and in doing so he established wave mechanics

18:11 as a new way to formulate quantum mechanics.

18:14 Apparently seclusion in nature is key to quantum discoveries

18:19 because Schrodinger was on vacation high in the Alps,

18:23 in the Swiss town of Arosa, when he made his discovery.

18:27 He’d retreated there for the Christmas holidays with, as he reports,

18:32 a stack of de Broglie’s papers.

18:34 And with one of his many girlfriends

18:37 whose identity to this day remains a mystery.

18:40 So, not quite as secluded as Heisenberg.

18:42 Schrodinger may have come up with the final

18:45 version of the paper in December 1925 and so you may be watching this episode

18:52 at the centenary of the Schrodinger equation.

18:55 Certainly by January 1926 the work was complete because

18:59 by then he had returned to Zurich and submitted the paper.

19:03 So, by the start of 26,

19:06 we had not one by two complete formulations of quantum mechanics.

19:10 Schrodinger’s wave mechanics was an instant hit

19:14 in a way that matrix mechanics was not, and that's thanks to the familiarity

19:19 of the wave formulation versus the then-unfamiliar matrix mechanics.

19:23 Perhaps even more attractive was the fact that Schrodinger’s picture seemed

19:28 to tell the story of what was happening behind the math.

19:33 The theory described this continuous, deterministic object—the wavefunction.

19:38 It was something that you could imagined.

19:42 Something that “existed” between observations.

19:45 This comfort wouldn’t last.

19:47 Although the idea of waves moving through space was familiar,

19:51 Schrodinger’s formulation couldn’t tell us what these were waves of.

19:56 Max Born then showed that the wavefunction

19:59 could be thought of as representing probability amplitudes,

20:02 the square of which gives the actual probability of a given measurement.

20:08 But what does a wave of probability even mean?

20:11 Is it a thing that exists?

20:13 Or a statistical map of underlying activity?

20:16 Or some wishy-washy epistemic entity that’s uncomfortably tied to the observer?

20:22 Schrodinger and probably many others hoped that the wavefunction

20:25 would retain some sort of realist nature, a physical existence.

20:30 But no fully consistent approach to such

20:33 an interpretation emerged, nor has it yet.

20:37 Regarding this probabilistic interpretation, Schrodinger wrote “I don’t like it,

20:40 and I’m sorry I ever had anything to do with it.” In fact,

20:44 applying the Schrodinger equation directly to things like the double

20:48 slit experiment quickly told us that even the more

20:52 “realist” wave mechanics had the same underlying principle as matrix

20:57 mechanics—we can only know the input and output states,

21:01 and everything in between is pure potentiality.

21:04 And then, in 1927, Paul Dirac showed

21:07 that the Heisenberg and Schrodinger pictures are mathematically equivalent.

21:11 They really are two representations of the same system.

21:14 They predict the same amplitudes.

21:16 While wave mechanics did become the more famous due to its intuitive advantage,

21:22 the Heisenberg and Schrodinger pictures both have

21:25 their roles—both are useful in certain circumstances.

21:28 But it’s arguable that Heisenberg’s picture is more general,

21:33 closer to base truth.

21:36 That’s because the Schrodinger equation is inconsistent with special relativity.

21:41 Relativity treats time on the same footing as the dimensions of space,

21:46 while the Schrodinger equation assigns both space and time

21:50 a primary and universal status more like in Newtonian mechanics.

21:55 Of course Schrodinger knew his equation was an approximation,

22:00 valid only at low speeds.

22:02 But the principle of the equation was right,

22:05 and Paul Dirac would go on to publish

22:08 a relativistic version of the wave equation in 1928,

22:12 but that’s a story for another time.

22:14 In fact for a previous time, because we already covered it.

22:18 Now matrix mechanics on the other

22:20 hand is perfectly consistent with special relativity.

22:24 It doesn’t treat time and space separately

22:27 because it doesn’t treat space at all.

22:30 At least, space isn’t built into it.

22:33 Where wave mechanics describes evolution through space,

22:37 matrix mechanics describes evolution in something called Hilbert space.

22:41 That’s the space of states of the quantum system,

22:44 which can involve spacey things like position and momentum,

22:48 but can also be electron orbitals or spins or whatever.

22:52 The same abstractness that makes matrix mechanics so daunting

22:57 also frees it from assumptions about the nature of space,

23:00 and so it can be made consistent with Einstein.

23:04 Heisenberg’s “only the observables” led him to a more general,

23:09 if less straightforward, formulation than Schrodinger.

23:13 And it’s for this reason Heisenberg’s formulation also became

23:18 the foundation for the next evolution of quantum mechanics:

23:21 quantum field theory.

23:23 Following this crazy half year from June to December 1925,

23:27 a mad flurry of advances propelled the fringe

23:31 idea into a full-blown description of the subatomic world.

23:35 Now I mentioned the Dirac equation,

23:37 then came second quantization via the Heisenberg picture.

23:40 That gave us Quantum Field Theory, in which particles are excitations in fields,

23:46 shaped by the symmetries of nature,

23:48 and that lead to the Standard Model of particle physics.

23:51 The predictive power of this model has enabled so much incredible technology.

23:57 But all of this was all founded on core principles that were

24:02 set down in 1925 and have held true to this day.

24:06 The world forged by quantum mechanics—the world of the international year

24:10 of quantum science and technology—is one

24:14 of transistor-driven miracles—from smartphones to satellites.

24:18 It’s one of quantum chemistry and advanced materials and nuclear power.

24:22 100 years of building useful things from century-old epiphanies.

24:27 But it’s also a world whose deepest layers are,

24:32 if anything, less settled than in the previous era.

24:35 A century later, we still don’t know

24:38 what quantum mechanics means—what story to tell about

24:43 the mechanisms at play beneath the observables—if

24:46 such a story can be told at all.

24:49 Questions like “what exists” or “what can

24:52 be known” are no longer clearly answerable.

24:55 One more Heisenberg quote:

24:57 "What we observe is not nature itself but nature exposed to our method

25:03 of questioning.” There’s a boundary between

25:07 external reality and our observations of it.

25:10 Quantum mechanics makes us question whether even

25:14 our most powerful theories can ever cross that boundary.

25:18 It’s been a giddying era of equally profound insight and confusion.

25:25 So here’s to a century of quantum mechanics,

25:28 and while we’re at it, to another year of space time.

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