From Classical Physics to the Quantum World

How a few small wrinkles changed our view of nature

A world without quantum mechanics

Why does the wire in a toaster glow orange when it gets hot? Why do atoms not simply collapse? Why can molecules form stable structures, and why are solids solid?

These questions sound very different. Yet all of them have something in common: they cannot be answered within classical physics alone.

Quantum mechanics is sometimes introduced through its strangest consequences—particles that behave like waves, probabilities instead of definite outcomes, or states that can exist in superposition. But quantum mechanics is not merely an exotic theory for phenomena far removed from everyday experience. It explains the stability and structure of ordinary matter.

Modern technology also depends on it. The controlled motion of electrons in semiconductor devices makes computers and smartphones possible. Lasers and light-emitting diodes rely on transitions between quantum states. Much of modern medical imaging, photovoltaics, and countless other technologies ultimately depend on our ability to understand and manipulate matter at the quantum level.

How did physicists arrive at such a radically different description of nature?

To understand that, it is useful to go back to a time before quantum mechanics existed—to a period when the foundations of physics seemed remarkably secure.

1878: Physics is almost finished

Imagine Munich in 1878. A young student, about twenty years old, has begun studying physics at the university. His name is Max Planck.

Planck asks one of his teachers, Philipp von Jolly, about the prospects of pursuing physics. Jolly is an established experimental physicist, nearly seventy years old. Looking back many years later, Planck recalled the advice he received: physics appeared to be a highly developed, nearly mature science. Its basic structure seemed secure; what remained was largely a matter of filling in details.

Planck’s recollection

Writing in 1933, Planck recalled the advice Philipp von Jolly had given him when he began his university studies:

“As I began my university studies I asked my venerable teacher Philipp von Jolly for advice regarding the conditions and prospects of my chosen field of study. He described physics to me as a highly developed, nearly fully matured science, that through the crowning achievement of the discovery of the principle of conservation of energy it will arguably soon take its final stable form. It may yet keep going in one corner or another, scrutinizing or putting in order a jot here and a tittle there, but the system as a whole is secured, and theoretical physics is noticeably approaching its completion to the same degree as geometry did centuries ago. That was the view fifty years ago of a respected physicist at the time.”

Source: Max Planck, Wege zur physikalischen Erkenntnis: Reden und Vorträge
(S. Hirzel, Leipzig, 1933), p. 128.
Digitized German original at the TIB

From today’s perspective this sounds almost comically premature. Within only a few decades, relativity and quantum theory would transform the foundations of physics.

But dismissing Jolly’s assessment would be too easy. To understand why such confidence was possible, we should first appreciate just how successful nineteenth-century physics had become.

The triumph of classical physics

By the end of the nineteenth century, much of physics could be organized around three extraordinarily successful theoretical frameworks:

Mechanics — the motion of matter,
Electrodynamics — electricity, magnetism, and light,
Thermodynamics and statistical mechanics — heat, energy, and the collective behavior of matter.

Mechanics: predicting motion

The development of what we now call classical mechanics was closely tied to one of humanity’s oldest scientific problems: understanding the motion of objects in the sky.

In the early seventeenth century, Galileo Galilei used the newly developed telescope to observe, among other things, moons orbiting Jupiter. Johannes Kepler, using the exceptionally precise astronomical observations of Tycho Brahe, discovered that the planets travel around the Sun on elliptical orbits.

The decisive theoretical step came with Isaac Newton. In his Philosophiæ Naturalis Principia Mathematica, published in 1687, Newton provided a general framework connecting forces and motion. In modern notation, his second law is written

\[ \vec F = m\vec a, \]

while the gravitational attraction between two masses is

\[ F = G\frac{mM}{r^2}. \]

The same mechanics that describes a falling object on Earth also explains the motion of the Moon and the planets. Terrestrial and celestial physics had become part of a single framework.

Newton’s formulation was only the beginning. During the eighteenth and nineteenth centuries, physicists and mathematicians developed increasingly powerful and general formulations of mechanics.

Joseph-Louis Lagrange recast mechanics in terms of generalized coordinates and the principle of stationary action. For a system described by coordinates \(q_i\), the equations of motion can be written as

\[ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot q_i}\right) - \frac{\partial L}{\partial q_i} =0, \]

where \(L\) is the Lagrangian.

A further reformulation was developed by William Rowan Hamilton. Instead of describing a system in terms of coordinates and velocities, Hamiltonian mechanics uses coordinates \(q_i\) and their conjugate momenta \(p_i\). The central quantity is the Hamiltonian

\[ H(q_i,p_i,t), \]

which, for many familiar systems, corresponds to the total energy.

Hamilton’s equations,

\[ \dot q_i=\frac{\partial H}{\partial p_i}, \qquad \dot p_i=-\frac{\partial H}{\partial q_i}, \]

provide yet another compact and very general way of describing classical motion.

A name that will return

The Hamiltonian will become one of the central objects in quantum mechanics. There it is promoted from a classical function to an operator that governs the dynamics of a quantum system.

We do not need the details of either formulation here. The important point is the extraordinary generality of classical mechanics. Given the laws governing a system and its initial state, its subsequent motion is determined.

Classical mechanics could describe falling objects, projectiles, pendulums, gyroscopes, planetary systems, and much more. There were puzzles—most famously the small anomalous precession of Mercury’s orbit—but the successes were overwhelming.

Its practical reach was just as impressive. Classical mechanics underpinned engineering, machine design, ballistics, navigation, and increasingly precise astronomy. The same mathematical framework could be used to design a bridge, predict the trajectory of a projectile, or calculate the orbit of a planet.

Electrodynamics: one theory for many phenomena

The nineteenth century also transformed the understanding of electricity and magnetism.

At first, phenomena such as electrostatic forces, electric currents, magnetism, and electromagnetic induction looked like different subjects. Experiments by Coulomb, Gauss, Ampère, Faraday, and many others gradually revealed that they were deeply connected.

A stationary electric charge produces an electric field. Electric currents produce magnetic fields. Changing magnetic fields can induce electric currents. Electricity and magnetism were not independent phenomena.

Then, in 1865, the Scottish physicist James Clerk Maxwell published A Dynamical Theory of the Electromagnetic Field. Maxwell brought these discoveries together and completed the theoretical framework.

In modern notation, Maxwell’s equations are

\[ \nabla\cdot\vec E = \frac{\rho}{\epsilon_0}, \qquad \nabla\cdot\vec B = 0, \]

\[ \nabla\times\vec E = -\frac{\partial\vec B}{\partial t}, \qquad \nabla\times\vec B = \mu_0\left( \vec j+\epsilon_0\frac{\partial\vec E}{\partial t} \right). \]

The profound achievement was not simply that Maxwell found a compact set of equations. A single theory now described phenomena that had previously belonged to different areas of physics.

And the unification went even further.

In empty space, where there are no charges or currents, Maxwell’s equations imply a wave equation. For example,

\[ \nabla^2\vec E - \frac{1}{c^2} \frac{\partial^2\vec E}{\partial t^2} =0. \]

The propagation speed predicted by the theory is

\[ c=\frac{1}{\sqrt{\mu_0\epsilon_0}}. \]

The remarkable point was that this speed agreed with the measured speed of light.

Maxwell concluded that light itself is an electromagnetic wave.

Electricity, magnetism, and optics—once treated as different subjects—were now part of the same theory.

The story did not end with visible light. In the late 1880s, Heinrich Hertz produced and detected electromagnetic waves in the laboratory at frequencies far below visible light, confirming Maxwell’s prediction experimentally.

What we now call radio waves and visible light are not fundamentally different phenomena. They are different frequency ranges of the same electromagnetic field.

This was not merely an abstract theoretical triumph. Electromagnetism was also beginning to reshape everyday life and industry.

Generators converted mechanical motion into electrical energy. Electric motors did the reverse. Transformers and alternating-current systems made it possible to distribute electrical power over large distances. Electric lighting began to replace gas lighting in streets, factories, and homes. Telegraphy and telephony transformed long-distance communication, and the later development of radio extended communication into the wireless domain.

Inventors and engineers such as Nikola Tesla played an important role in developing practical alternating-current motors, polyphase power systems, transformers, and high-frequency electrical technology.

For someone entering physics around 1878, the lesson was difficult to miss:

Electromagnetism was not only explaining nature. It was unifying apparently unrelated phenomena and transforming the technological world at the same time.

Thermodynamics: from steam engines to atoms

The third pillar was thermodynamics, the study of heat, work, temperature, and energy.

Its development had an intensely practical motivation. During the Industrial Revolution, understanding heat was essential for improving steam engines—the machines that powered factories, mines, locomotives, and ships. But the effort to understand and improve these engines led to laws of remarkable universality.

In 1865, Rudolf Clausius summarized two central principles in memorable form:

The energy of the universe is constant.
The entropy of the universe tends to a maximum.

In modern terminology, the first statement expresses conservation of energy; the second captures the irreversibility of macroscopic processes—why many processes naturally proceed in one direction but not in reverse.

But an even deeper question remained:

What are temperature, pressure, and entropy actually telling us about matter?

The emerging kinetic theory of gases supplied an answer. A gas is made from enormous numbers of microscopic particles in continual motion. Pressure results from their collisions with the walls of a container. Temperature is related to their average kinetic energy.

For an ideal monatomic gas,

\[ \left\langle E_\mathrm{kin}\right\rangle = \frac{3}{2}k_\mathrm{B}T. \]

This was a remarkable change in perspective. Quantities such as temperature and pressure, which appear to describe matter as a whole, could emerge from the ordinary mechanical motion of microscopic particles.

What made the emerging statistical mechanics especially profound was that it did not require a new microscopic law of motion. The particles were still assumed to obey ordinary classical mechanics. No new force analogous to Newton’s gravitational law, Coulomb’s law, or Faraday’s law had to be introduced.

The new ingredient was instead statistical reasoning.

A macroscopic sample contains an enormous number of particles—typically on the order of \(10^{23}\). Following every microscopic trajectory is impossible in practice, but it is also unnecessary. Probability theory, combinatorics, and the mathematics of large numbers make it possible to extract reliable predictions for the collective behavior of the system.

In this sense, statistical mechanics occupies a somewhat unusual place among physical theories: rather than adding a new law governing the microscopic world, it shows how qualitatively new macroscopic behavior can emerge when the laws of mechanics are applied to very large numbers of particles.

Ludwig Boltzmann pushed this statistical interpretation much further. His famous relation

\[ S=k_\mathrm{B}\ln W \]

connects the entropy \(S\) of a macroscopic state to the number \(W\) of microscopic configurations compatible with it.

The significance is difficult to overstate. Entropy had entered thermodynamics as a macroscopic quantity associated with heat and the direction of thermodynamic processes. Boltzmann connected it to the number of microscopic ways in which the same macroscopic state can be realized.

Thermodynamics therefore began to look less like an independent set of phenomenological laws and more like something that could emerge from mechanics plus statistics.

This microscopic interpretation was controversial. The existence of atoms was not yet universally accepted. Boltzmann and other atomists treated matter as composed of microscopic particles, while opponents favored descriptions in terms of continuous matter and energy.

There is also an important subtlety in the word probability.

Statistical mechanics uses probabilities, but in the classical picture these probabilities did not necessarily mean that nature itself was fundamentally random. Each microscopic particle was still assumed to follow deterministic equations of motion. If one somehow knew the exact positions and momenta of every particle, classical mechanics would, in principle, determine their future motion.

Probability entered because no observer could realistically specify and follow the microscopic state of something like \(10^{23}\) particles.

In that sense, classical statistical mechanics was compatible with a fully deterministic universe. Its probabilities could be understood as describing our incomplete knowledge of an underlying mechanical reality—not as a fundamental indeterminacy in nature itself.

A clockwork universe

Put the pieces together.

Mechanics described matter and motion. Electrodynamics described electric and magnetic fields—and light. Thermodynamics and statistical mechanics connected the microscopic motion of matter to heat and macroscopic behavior.

The resulting picture was not merely successful. It suggested a particular philosophy of nature.

The fundamental equations of mechanics and electrodynamics are deterministic. Specify the state of a system at one instant and, in principle, its future is fixed by the equations of motion.

Chance need not be fundamental. Apparent randomness could simply reflect our ignorance of the exact initial conditions or our inability to follow an enormous number of particles.

More than half a century before Planck entered university, Pierre-Simon Laplace had expressed this ideal in especially striking language. He imagined an intellect that knew, at one instant, every force and every position in nature. With unlimited ability to analyze that information, such an intellect could know both past and future.

This hypothetical intellect later became known as Laplace’s demon.

The classical picture

At the end of the nineteenth century, it was reasonable to imagine nature as an enormous clockwork:

matter consists of particles following definite trajectories,
light is an electromagnetic wave,
physical systems evolve according to deterministic equations,
statistical uncertainty reflects incomplete knowledge rather than fundamental randomness.

It was a remarkably coherent picture—and an extraordinarily successful one.

Seen from 1878, the confidence of physicists such as Jolly becomes easier to understand.

Classical mechanics had unified terrestrial and celestial motion. Electrodynamics had brought electricity, magnetism, induction, light, and eventually radio waves into one framework while powering a rapidly expanding electrical technology. Thermodynamics and statistical mechanics were beginning to connect heat and macroscopic behavior to microscopic motion.

Physics was not only explaining more and more of nature—it was doing so through theories of striking unity and practical power. Mechanics supported engineering and navigation; thermodynamics grew alongside the steam-powered industrial world; and electromagnetism was producing electric power, lighting, motors, and new forms of communication.

Max Planck was therefore not discouraged by the suggestion that the foundations of physics were essentially complete. He wanted to study physics because he wanted to understand nature.

That decision turned out to be fortunate.

Because there were a few small wrinkles in the classical picture.

And one of them would eventually occupy Planck himself.

Small wrinkles

A successful physical theory should agree with observations, make testable predictions, and remain internally consistent. Classical physics passed these tests spectacularly across an enormous range of phenomena.

But not everywhere.

Toward the end of the nineteenth century and the beginning of the twentieth, several experimental results became increasingly difficult to reconcile with the classical picture.

At first, these problems did not necessarily look like signs that an entirely new mechanics of nature was needed. They were problems to be solved—perhaps missing details in an otherwise successful framework.

They turned out to be much more than that.

Wrinkle 1: How does a hot object glow?

Heat a piece of metal and it begins to glow. At first it appears dull red. As its temperature increases, the emitted light becomes brighter and shifts toward shorter wavelengths.

This is thermal radiation.

A particularly useful idealization is a black body: an object that absorbs all incident radiation and whose emitted spectrum depends only on its temperature.

By the late nineteenth century, the spectrum could be measured accurately. The challenge was to explain its shape.

By the 1890s, the shape of the thermal spectrum was known experimentally with increasing precision.

In 1896, Wilhelm Wien found a distribution law that described the high-frequency, short-wavelength side of the spectrum remarkably well. Wien’s work was an important theoretical and phenomenological success, but it did not amount to a complete derivation of the observed spectrum from the established foundations of classical physics.

Lord Rayleigh attacked the problem in a more direct and, in a sense, more fundamental way: he tried to reconcile Maxwell’s electrodynamics with classical thermodynamics and statistical mechanics.

Electromagnetic radiation inside a cavity can be decomposed into standing-wave modes. Maxwell’s theory tells us which electromagnetic modes are possible; classical statistical mechanics then tells us how thermal energy should be distributed among them. Through the equipartition theorem, each mode receives an average energy of order \(k_\mathrm{B}T\).

The number of electromagnetic modes available between frequencies \(\nu\) and \(\nu+d\nu\) increases as \(\nu^2\). Combining mode counting with equipartition leads to the Rayleigh–Jeans result

\[ u(\nu,T) = \frac{8\pi\nu^2}{c^3}k_\mathrm{B}T. \]

At low frequencies, this agrees well with experiment.

At high frequencies, however, it becomes absurd: the predicted energy density grows without limit as \(\nu\) increases. Integrating over all frequencies would imply an infinite amount of radiated energy.

This failure became known as the ultraviolet catastrophe.

This was a particularly disturbing kind of failure.

Rayleigh had not used an obviously crude or inappropriate model. He had simply brought together electrodynamics and statistical mechanics—two theories whose success had helped create the confidence that classical physics was nearing completion.

And together, they predicted something impossible.

And now Max Planck returns to our story.

In 1900, Planck was searching for a formula that could reproduce the full measured spectrum and connect the regimes in which the existing descriptions worked.

The result was

\[u(\nu,T) = \frac{8\pi h\nu^3}{c^3} \frac{1}{e^{h\nu/k_\mathrm{B}T}-1}. \]

To obtain it, Planck introduced an assumption that did not fit comfortably into classical physics: the oscillators exchanging energy with the electromagnetic field could possess energies only in discrete steps,

\[ E_n=nh\nu, \qquad n=0,1,2,\ldots \]

where \(h\) is a new fundamental constant—today called Planck’s constant.

Planck’s law reproduced the entire thermal spectrum, including both the low- and high-frequency limits.

The wrinkle had been repaired.

Or so it might have seemed.

A small mathematical fix?

The crucial step was the appearance of a discrete energy scale, \(E=h\nu\).

At first, this did not yet amount to the fully developed quantum mechanics we use today. But the idea that energy exchange might be fundamentally discrete was a sharp break with classical expectations.

Wrinkle 2: What is light?

Maxwell’s theory had given one of the great triumphs of nineteenth-century physics:

Light is an electromagnetic wave.

Interference, diffraction, and polarization all strongly support its wave character.

And yet another experiment was becoming difficult to understand.

When light strikes certain materials, electrons can be emitted from the surface. This is the photoelectric effect.

Classically, one might expect the energy transferred by a wave to depend primarily on its intensity. But the experiments instead revealed a decisive role for frequency.

In 1905, Albert Einstein took Planck’s energy relation seriously in a new way. He proposed that the electromagnetic field itself could exchange energy in localized quanta with

\[ E_\gamma=h\nu. \]

For the photoelectric effect,

\[ K_\mathrm{max}=h\nu-\Phi, \]

where \(\Phi\) is the work function of the material.

Increasing the light intensity increases the number of emitted electrons, but their maximum kinetic energy is controlled by the frequency.

A theory famous for establishing that light is a wave now had to coexist with experiments in which light behaved as though its energy arrived in discrete packets.

What, then, is light?

Planck had introduced discreteness in the exchange of energy between matter and radiation. Einstein had gone further and attached the quantum \(h\nu\) to light itself. The next clues came from the other side of the interaction: matter, and atoms in particular, also seemed to permit only very specific energies.

Wrinkle 3: What is an atom?

There was another difficulty—one even more directly connected to the subject of this course.

What is matter made of?

By the end of the nineteenth century, experiments had begun to reveal internal structure inside what had once been regarded as indivisible atoms. In 1897, J. J. Thomson identified the electron as a negatively charged constituent of matter.

If atoms are electrically neutral, positive charge must be present as well. But how are the positive and negative charges arranged?

Thomson proposed a model in which the positive charge was distributed throughout the atom, with electrons embedded within it.

Experiments by Hans Geiger and Ernest Marsden, under the direction of Ernest Rutherford, provided a dramatic test. Alpha particles were directed at a thin metal foil.

Most passed through with relatively small deflections.

But a tiny fraction scattered through surprisingly large angles.

In 1911, Rutherford interpreted the results as evidence that nearly all of the positive charge—and most of the mass—of an atom is concentrated in an extremely small nucleus.

The atom was mostly empty space.

That was a remarkable discovery.

It also created a serious problem.

If negatively charged electrons orbit a positive nucleus like planets orbiting the Sun, classical electrodynamics says that an accelerating charge should radiate electromagnetic energy. The electron should lose energy and spiral into the nucleus.

A classical planetary atom should therefore be unstable.

But atoms are stable.

Wrinkle 4: Why do atoms have colors?

Atoms presented another clue.

Excite a gas—for example in an electrical discharge—and analyze the emitted light with a prism or diffraction grating. Instead of producing a continuous rainbow, an atomic gas emits light only at particular wavelengths.

Each element has a characteristic line spectrum.

For hydrogen, the wavelengths follow a remarkably simple empirical pattern. The spectral lines can be described by the Rydberg formula,

\[ \frac{1}{\lambda} = R_\mathrm{H} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right), \qquad n_2>n_1. \]

Why should an atom emit only certain frequencies?

Classical mechanics and electrodynamics did not provide a convincing answer.

In 1913, Niels Bohr proposed a model of hydrogen that deliberately introduced nonclassical rules. Electrons were allowed to occupy only particular orbits, characterized by quantized angular momentum,

\[ L=n\hbar, \qquad n=1,2,3,\ldots \]

and light was emitted or absorbed when the atom changed between allowed states:

\[ h\nu = E_i-E_f. \]

This connected two puzzles that had initially looked separate. The quantum of light, \(h\nu\), now matched the difference between discrete atomic energies. Atomic spectra were no longer arbitrary collections of wavelengths: they were fingerprints of the allowed energies inside the atom.

The Bohr model reproduced the hydrogen spectrum with striking success.

But it was not a satisfactory fundamental theory. Why should only certain orbits be allowed? Why should an accelerating electron in one of these special orbits not radiate? Why did the model work so well for hydrogen but fail for more complicated atoms?

The rules worked—but the underlying physics was still missing.

Wrinkle 5: Is light a particle after all?

Einstein’s light quanta provided an explanation of the photoelectric effect, but the wave description of light remained indispensable.

Then, in 1922, Arthur Holly Compton studied the scattering of X-rays from electrons. The wavelength of the scattered radiation changed with scattering angle according to

\[ \lambda_f-\lambda_i = \frac{h}{m_ec}(1-\cos\theta). \]

The result can be understood remarkably naturally by treating the incoming light quantum as carrying not only energy,

\[ E_\gamma=h\nu, \]

but also momentum,

\[ p_\gamma=\frac{h}{\lambda}. \]

Light had been one of the strongest successes of classical wave physics.

Now experiments demanded that it also carry energy and momentum in discrete quanta.

Wrinkle 6: And are particles waves?

The classical categories were becoming increasingly difficult to maintain. Light, the paradigm of a wave, displayed particle-like energy and momentum.

In 1924, Louis de Broglie proposed a striking converse: perhaps particles of matter also possess a wavelength,

\[ \lambda=\frac{h}{p}. \]

For an electron, this wavelength is not merely a mathematical curiosity. A few years later, electron-diffraction experiments—most famously those of Clinton Davisson and Lester Germer—showed interference and diffraction effects characteristic of waves.

The symmetry was difficult to ignore:

Light, unquestionably wave-like, also behaves like particles. Matter, unquestionably particle-like, also behaves like waves.

The classical distinction between particles and waves was no longer sufficient.

Something has to give

By the early twentieth century, the small wrinkles had accumulated.

Classical physics remained enormously successful—but its basic concepts could no longer provide a consistent description of microscopic nature.

Thermal radiation suggested discrete energy exchange.
The photoelectric effect suggested light quanta.
Rutherford scattering revealed the atomic nucleus.
Classical electrodynamics predicted that a planetary atom should collapse.
Atomic spectra revealed sharply defined characteristic frequencies.
Bohr’s model introduced quantized states without explaining their deeper origin.
Compton scattering gave light quanta unmistakably particle-like momentum.
de Broglie’s hypothesis and electron diffraction showed that matter could display unmistakably wave-like behavior.

The solution would not be a small correction to Newton’s mechanics or Maxwell’s electrodynamics.

A new framework was needed.

During the 1920s, the clues accumulated into a new theoretical framework. The matter-wave idea of Louis de Broglie, together with the work of Werner Heisenberg, Erwin Schrödinger, Max Born, Paul Dirac, and others, grew into what we now call quantum mechanics.

And with that theory, questions that classical physics could not answer became accessible:

Why are atoms stable?

Why do atoms have discrete spectra?

How can electrons form stable many-electron atoms?

How do chemical bonds arise?

Why are solids solid?

And what does it actually mean for energy, angular momentum, or light to be quantized?

These are no longer peripheral wrinkles.

They lead directly to the physics of atoms, molecules, and light—and to the subject of this course.

Where we go from here

The historical path tells us why classical physics was not enough.

We now turn to the theory that replaced the classical description at microscopic scales. We will first establish the essential ideas and mathematical language of quantum mechanics and then use them to understand the structure and dynamics of atoms.