Wave-Particle Duality

We’ve encountered a puzzling situation. Light behaves as electromagnetic waves, exhibiting diffraction and interference. Yet the photoelectric effect demonstrates that light also behaves as discrete particles (photons), each carrying energy E = hν. These aren’t two different types of light. The same beam of light exhibits both behaviors depending on how we observe it.

This phenomenon, called wave-particle duality, challenged physicists in the early 20th century. How can something be both a wave and a particle? The answer forced a complete rethinking of the nature of reality at small scales.

The Double-Slit Experiment

The most striking demonstration of wave-particle duality comes from the double-slit experiment, one of the most famous experiments in physics. In 1801, English physicist Thomas Young first performed this experiment to demonstrate that light behaves as a wave, settling a long-standing debate about the nature of light. But in the early 20th century, after Einstein proposed photons, physicists realized this simple experiment reveals something far stranger than Young imagined.

Watch: Dr. Quantum’s Double-Slit Experiment video.

Watch: Space Time (PBS) The Quantum Experiment that Broke Reality video.

The experiment is straightforward. Light passes through two narrow slits and hits a screen behind them. What pattern appears on the screen?

If light were purely particles (like bullets), we’d expect two bright bands on the screen, one behind each slit. Particles go through one slit or the other.

If light behaved purely as particles, we would expect two bands on the screen.

If light were purely waves, we’d expect an interference pattern: alternating bright and dark bands across the screen. Waves from the two slits overlap, interfering constructively in some places (bright bands) and destructively in others (dark bands).

$Double-slit experiment showing wave behavior: waves diffract through both slits and interfere, producing an interference pattern with alternating bright and dark bands on the screen$ $Double-slit experiment showing wave behavior: waves diffract through both slits and interfere, producing an interference pattern with alternating bright and dark bands on the screen$

If light behaves as waves, we expect an interference pattern with alternating bright and dark bands.

What actually happens? An interference pattern appears. Light behaves as a wave.

But we can dim the light source until photons arrive one at a time at the screen. With only one photon traveling at a time, each photon must go through one slit or the other, right? We’d expect two bands.

But the interference pattern still builds up, one photon at a time. Each individual photon behaves like a particle (it creates a single dot on the screen), but the probability distribution of where photons land follows a wave pattern. Somehow, each photon “knows” about both slits. The photon doesn’t travel through both slits in any classical sense. Its wave function, which determines where the photon might land, passes through both. The interference pattern reflects this probability, not a classical trajectory.

What if we try to determine which slit each photon passes through? We could place detectors at the slits to track each photon’s path. When we make this measurement, the interference pattern vanishes. Instead of alternating bright and dark bands, we see two bright bands, exactly what we’d expect for particles.

The photon cannot simultaneously exhibit wave-like interference and particle-like which-path information. If we know which slit it went through (particle information), we lose the interference pattern (wave behavior). If we don’t measure which slit, we observe interference but cannot say which path was taken. The two behaviors are complementary: mutually exclusive aspects of the same phenomenon that cannot be observed simultaneously. We can measure one or the other, but never both at once.

This isn’t about human consciousness affecting reality. Detecting which slit a photon passes through requires physically interacting with it, typically by scattering another photon off it or using some other probe. This interaction imparts momentum to the photon, disrupting the phase relationships between the two possible paths. (Phase describes where a wave is in its cycle: peak, trough, or somewhere between. Interference requires the waves from both paths to maintain a consistent phase relationship.) It’s not that experimenters are clumsy; any measurement capable of determining “which slit” necessarily disturbs the system enough to destroy the interference pattern.

The double-slit experiment reveals the core mystery of quantum mechanics. Light (and matter, as we’ll see) isn’t simply waves or simply particles. These are quantum objects that exhibit properties of both depending on the measurement we make.

Matter Waves

If light has particle-like properties (photons with energy E = hν), a natural question arises: Does matter have wave-like properties?

In 1924, French physicist Louis de Broglie proposed exactly this. He suggested that all matter exhibits wave-like behavior, with a wavelength given by

\[\lambda = \dfrac{h}{mv}\]

where

λ is the de Broglie wavelength
h is Planck’s constant (6.626 × 10⁻³⁴ J s)
m is the mass of the particle
v is the velocity of the particle

The product mv is the particle’s momentum. Notice that wavelength is inversely proportional to both mass and velocity. Heavier, faster-moving objects have shorter wavelengths.

Louis Victor Pierre Raymond, 7th Duc de Broglie: French theoretical physicist (1892–1987) and recipient of the Nobel Prize in Physics in 1927 “for his discovery of the wave nature of electrons”. (Source: Wikipedia)

Example

Calculate the de Broglie wavelength for

an electron moving at 10⁶ m s⁻¹ (in nm in standard notation)
a baseball (m = 0.145 kg) moving at 40 m s⁻¹ (in m in normalized scientific notation)

Solution

Electron

The mass of an electron is m_e = 9.109 × 10⁻³¹ kg.

\[ \begin{align*} \lambda &= \dfrac{h}{mv} \\[1.5ex] &= \dfrac{6.626\times 10^{-34}~\mathrm{J~s}} {(9.109\times 10^{-31}~\mathrm{kg})(10^{6}~\mathrm{m~s^{-1}})} \\[1.5ex] &= 7.27\bar{3}6 \times 10^{-10}~\mathrm{m} \\[1.5ex] &= 0.727~\mathrm{nm} \end{align*} \]

This wavelength is comparable to atomic dimensions (atoms are roughly 0.1–0.5 nm in diameter). Wave effects for electrons are significant and measurable.

Baseball

\[ \begin{align*} \lambda &= \dfrac{h}{mv} \\[1.5ex] &= \dfrac{6.626\times 10^{-34}~\mathrm{J~s}} {(0.145~\mathrm{kg})(40~\mathrm{m~s^{-1}})} \\[1.5ex] &= 1.14\bar{2}4 \times 10^{-34}~\mathrm{m} \end{align*} \]

This wavelength is unimaginably small, far smaller than any measurable length scale. For macroscopic objects, wave effects are completely negligible.

The de Broglie wavelength has been experimentally confirmed. In 1927, Clinton Davisson and Lester Germer demonstrated electron diffraction by scattering electrons off a nickel crystal. The electrons produced a diffraction pattern, just as X-rays do. Electrons behave as waves.

Matter, like light, exhibits wave-particle duality.

Connection to the Bohr Model

The de Broglie wavelength provides the physical picture missing from Bohr’s model. Recall that Bohr postulated that angular momentum is quantized (L = nh/2π), but he offered no physical explanation for why. It worked mathematically, but why should angular momentum come in discrete units? Why are only certain orbits allowed?

De Broglie showed that stable electron orbits correspond to standing waves. A standing wave forms when a wave interferes with itself in a confined space, creating fixed points (nodes) and oscillating regions (antinodes). Think of a vibrating guitar string: only certain wavelengths fit on the string, producing distinct musical notes. Similarly, only certain electron wavelengths fit around an orbit.

For an electron wave to fit around a circular orbit without destructively interfering with itself, the circumference of the orbit must equal an integer number of wavelengths:

\[2\pi r = n\lambda\]

where n = 1, 2, 3, … and r is the orbital radius.

Substituting the de Broglie wavelength λ = h/(mv), which connects wavelength (a wave property) to momentum (a particle property):

\[2\pi r = n\left(\dfrac{h}{mv}\right)\]

Rearranging:

\[mvr = \dfrac{nh}{2\pi}\]

The left side (mvr) is the angular momentum of the electron. By requiring the wavelength to fit the orbit (a wave condition), we’ve constrained the momentum (a particle property). This is exactly the quantization condition Bohr postulated, but now it emerges from treating electrons as waves rather than being assumed without explanation.

Why does fitting an integer number of wavelengths matter? When the circumference equals exactly nλ, the electron wave reinforces itself constructively as it goes around the orbit. If the wavelength doesn’t fit, the wave interferes destructively with itself and cancels out. The electron cannot exist in such an orbit.

This derivation has limitations. It assumes electrons travel in circular paths, which turns out to be wrong. Quantum mechanics will replace Bohr’s circular orbits with three-dimensional probability distributions called orbitals. But de Broglie’s insight remains valid: electrons exhibit wave behavior, and this wave nature explains why only certain energy states are allowed. The integer n survives in quantum mechanics as the principal quantum number.

Why Don’t We See Quantum Effects in Daily Life?

Consider a dust particle with mass 10⁻⁹ kg moving at 1 mm s⁻¹ (barely drifting):

\[\lambda = \dfrac{6.626\times 10^{-34}}{(10^{-9})(10^{-3})} = 6.6 \times 10^{-22}~\mathrm{m}\]

Even for this tiny, slow-moving particle, the de Broglie wavelength is billions of times smaller than a proton.

For everyday objects, the wavelengths are even more absurdly small:

To put these in perspective: a proton is about 10⁻¹⁵ m across. The de Broglie wavelength of a walking person is 10²¹ times smaller than a proton. There is no conceivable experiment that could detect wave behavior at such scales.

The pattern is clear. The de Broglie wavelength depends on mass:

\[\lambda = \dfrac{h}{mv}\]

For macroscopic objects, large mass means very small wavelength. When the wavelength is much smaller than the size of the object, wave effects become unmeasurable and classical (particle-like) behavior dominates.

For microscopic objects like electrons, atoms, and molecules, small mass means wavelength comparable to object size. Wave effects are significant, observable, and a quantum mechanical description becomes necessary.

Planck’s constant h sets the scale. At h ≈ 10⁻³⁴ J s, quantum effects are imperceptible for everyday objects but dominant at atomic scales.

The Uncertainty Principle

In 1925, Werner Heisenberg made a discovery that overturned a basic assumption of classical physics. In classical mechanics, the order of multiplication doesn’t matter: 3 × 5 equals 5 × 3. Position times momentum equals momentum times position. But Heisenberg found that at the quantum level, this basic rule fails.

Werner Karl Heisenberg: German theoretical physicist (1901–1976) and recipient of the Nobel Prize in Physics in 1932 “for the creation of quantum mechanics”. (Source: Wikipedia)

When Heisenberg developed his matrix mechanics formulation of quantum theory, he discovered that the mathematical objects representing position and momentum do not commute. If Q represents position and P represents momentum, then QP ≠ PQ. The order of measurement matters. Measuring an electron’s position and then its momentum gives a different result than measuring momentum first and then position.

This wasn’t a mathematical curiosity. Heisenberg showed that this non-commutativity has a direct physical consequence: you cannot simultaneously know both the position and momentum of a particle with arbitrary precision. The more precisely you measure one, the less precisely you can know the other.

Going Deeper: The Commutator Relation

In quantum mechanics, physical quantities are represented by mathematical operators. The commutator of position (Q) and momentum (P) is:

\[[Q, P] = QP - PQ = i\hbar\]

where i is the imaginary unit and ℏ = h/(2π) is the reduced Planck constant.

In classical physics, this commutator would be zero (order doesn’t matter). The fact that it equals iℏ rather than zero is what forces the uncertainty principle. This mathematical structure, discovered by Heisenberg in 1925, meant that the rules of ordinary arithmetic break down at the quantum scale.

In 1927, Heisenberg published his famous paper formalizing this insight as the uncertainty principle. The mathematical statement is:

\[\Delta x \cdot \Delta p \geq \frac{\hbar}{2}\]

where

Δx is the uncertainty in position
Δp is the uncertainty in momentum
ℏ = h/(2π) = 1.055 × 10⁻³⁴ J s (the reduced Planck constant)

The product of the uncertainties has a minimum value. If you reduce Δx (measure position more precisely), Δp must increase (momentum becomes less certain), and vice versa. You cannot make both uncertainties arbitrarily small at the same time.

Heisenberg, W. “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik.” Z. Phys. 1927, 43, 172–198.

Going Deeper: Why the Uncertainty Exists

The uncertainty principle isn’t a statement about measurement limitations. It’s a consequence of what waves fundamentally are.

Start with a single pure wave, one with a perfectly defined wavelength. This wave extends forever in both directions. It has no beginning and no end. If you ask “where is this wave?”, the answer is everywhere. A pure wave with definite wavelength has completely undefined position.

Since momentum is related to wavelength through the de Broglie relation (λ = h/p), a wave with one wavelength has one momentum. So: definite momentum means indefinite position.

Now suppose you want to create a wave that exists here but not there, a localized “bump.” How do you build such a thing from waves?

You add waves of different wavelengths together. When waves overlap, they interfere. In some places they reinforce each other (constructive interference), and in other places they cancel (destructive interference). By combining waves carefully, you can create a pattern where the waves add up in one region and cancel everywhere else.

The more wavelengths you combine, the more complete the cancellation outside the central region. The bump gets narrower. Position becomes better defined.

But here is the tradeoff: if you’ve combined many wavelengths to build that localized bump, the result no longer has a single wavelength. It has many. And since wavelength determines momentum, the momentum is now spread across many values. Better-defined position means less-defined momentum.

To create a perfectly localized particle, a single point in space, you would need to combine waves of all possible wavelengths. Only then would the destructive interference cancel the wave everywhere except at one exact location. But a combination of all wavelengths has no defined wavelength at all. The momentum would be completely uncertain.

This tradeoff is built into the mathematics of waves. It has nothing to do with our instruments or our cleverness as experimenters. You cannot construct a wave that is both localized in space and has a single wavelength. The uncertainty principle quantifies this limitation.

Audio Analogy: A pure musical tone (single frequency) sustains indefinitely, like a tuning fork. It has definite pitch but extends forever in time. A short percussive click is localized in time but contains many frequencies. The shorter the click, the more frequencies it contains. An infinitely short click contains all frequencies equally. Localization in time requires spread in frequency. The same mathematics governs position and momentum in quantum mechanics.

The uncertainty tradeoff. Left column: a wave packet localized in position has uncertain momentum. Right column: a wave packet spread in position has well-defined momentum. The product Δx · Δp is always at least ℏ/2.

Going Deeper: Understanding Wave Packets

The visualization above shows wave packets, localized wave-like disturbances that represent quantum particles. Understanding these plots requires some background on wave functions and probability.

Wave Functions and Probability Amplitudes

The wave function Ψ(x) is a complex-valued function that contains all information about a particle’s quantum state. However, Ψ(x) itself is not directly measurable. What we can measure is where the particle is found, and the probability of finding it at position x is given by

\[|\Psi(x)|^2 = \Psi^*(x) \cdot \Psi(x)\]

where Ψ(x) is the complex conjugate of Ψ(x). The absolute value squared ensures the result is real and non-negative, as probabilities must be. For a complex number z* = a + bi, we have |z|² = a² + b², not z² = a² - b² + 2abi (which is still complex).

This is the Born interpretation: |Ψ(x)|² is the probability density for finding the particle at position x.

Position and Momentum Representations

A quantum state can be described equivalently in position space or momentum space:

Position representation Ψ(x): The probability density |Ψ(x)|² tells you where the particle might be found.
Momentum representation φ(p): The probability density |φ(p)|² tells you what momentum the particle might have.

These are two views of the same quantum state, like describing a musical chord as either “notes over time” or “frequencies present.”

Fourier Transforms Connect Them

The mathematical connection between Ψ(x) and φ(p) is the Fourier transform:

\[\phi(p) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \Psi(x) \, e^{-ipx/\hbar} \, dx\]

\[\Psi(x) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \phi(p) \, e^{ipx/\hbar} \, dp\]

This is pure mathematics, not quantum mechanics. Fourier transforms are used in audio processing, image compression, and signal analysis. The key property is: a function that is narrow in one domain must be wide in the other domain. There is no way around this. Narrowing one always spreads the other.

This is why the uncertainty principle is unavoidable. It is not a limitation of measurement technology or experimental skill. It is a mathematical consequence of waves.

Why Gaussians Are Special

The wave packets shown are Gaussian wave packets, where the envelope has the form e^{-x²/(2σx²)}. Gaussians have a special property: the Fourier transform of a Gaussian is another Gaussian. If the position-space width is σx, the momentum-space width is

\[\sigma_p = \frac{\hbar}{2\sigma_x}\]

This gives σx · σp = ℏ/2, which is the minimum possible uncertainty product. Gaussian wave packets are called minimum uncertainty states because they achieve equality in the uncertainty relation. Any other wave shape gives Δx · Δp > ℏ/2.

The Oscillations Encode Momentum

In the top row of the visualization, you see oscillations inside the wave packet envelope. These oscillations come from the complex exponential factor e^ik₀x in the wave function:

\[\Psi(x) = A \cdot e^{-x^2/(2\sigma_x^2)} \cdot e^{ik_0 x}\]

The oscillation frequency k₀ determines the average momentum via de Broglie’s relation p₀ = ℏk₀. When you take the Fourier transform, these oscillations shift the center of the momentum distribution to p = p₀. The width of the momentum distribution depends only on how localized the envelope is, not on the oscillation frequency.

This is why the momentum distribution φ(p) appears as a smooth Gaussian without oscillations: the oscillations in position space become a shift in momentum space, not oscillations in momentum space.

Reading the Visualization

Row 1 [Ψ(x)]: The wave function with oscillations. The oscillation wavelength encodes average momentum. The envelope width shows position localization.
Row 2 [|Ψ(x)|²]: Position probability density. Squaring removes the oscillations, leaving just the envelope squared. This shows where the particle is likely to be found.
Row 3 [|φ(p)|²]: Momentum probability density. Shows what momenta are likely. The width is inversely related to the position width.

Left column (localized packet): Small Δx, large Δp. The particle’s position is well-defined, but its momentum is uncertain.

Right column (spread packet): Large Δx, small Δp. The particle’s momentum is well-defined, but its position is uncertain.

Read the Original Paper

Heisenberg’s 1927 paper, On the Intuitive Content of Quantum Theoretical Kinematics and Mechanics, is where the uncertainty principle was first presented. In it, Heisenberg explains why the “1s orbit of the electron in the hydrogen atom makes no sense” and why we must replace orbits with probability distributions.

Read the annotated translation →

What the Uncertainty Principle Is Not

The uncertainty principle is often misunderstood. It is not about:

Measurement errors. This isn’t a statement about clumsy experimenters or imperfect instruments. Even with perfect instruments, the uncertainty exists.
Disturbing the system by measurement. While measurement does disturb quantum systems, the uncertainty principle is more fundamental. It’s not that we could know both position and momentum if only we were more careful. The particle does not have a precisely defined position and momentum simultaneously. (To measure position precisely, you bounce a photon off the particle. But that photon carries momentum. Shorter wavelength photons give finer position resolution but carry more momentum, kicking the particle harder. You can’t win.)
Consciousness affecting reality. Human observation plays no special role. The uncertainty exists whether or not anyone is watching.

Why It Matters for Chemistry

The uncertainty principle explains why electrons cannot have definite orbits around atoms. In the Bohr model, we imagined electrons traveling in circular paths with specific radii. But if an electron had a definite circular orbit, we would know both its position (on the circle) and its momentum (tangent to the circle) precisely. The uncertainty principle forbids this.

Instead, we describe electrons using probability distributions. We cannot say exactly where an electron is, only where it is likely to be found. These probability distributions are called orbitals, and they replace the sharp orbits of the Bohr model. An orbital isn’t a path the electron follows; it’s a map of where the electron might be.

The uncertainty principle also explains zero-point energy. Even at absolute zero temperature, particles cannot be completely at rest. If a particle were stationary (zero momentum, perfectly known), its position would be completely uncertain. The ground state of any quantum system represents a balance between localizing the particle (which increases kinetic energy through momentum uncertainty) and spreading it out (which may increase potential energy). This minimum-energy state still has motion, called zero-point motion.

Implications

Wave-particle duality forced physicists to abandon classical concepts. An electron isn’t a tiny ball orbiting a nucleus, nor is it a spread-out wave. It’s a quantum object that exhibits particle-like properties when we measure position and wave-like properties when we measure interference patterns.

This led to the development of quantum mechanics, a new framework for describing nature at atomic scales. Key features:

Wave functions (Ψ) describe the state of quantum systems
The Schrödinger equation governs how wave functions evolve
Probability distributions replace definite trajectories: we can only predict the probability of finding a particle at a given location
The uncertainty principle sets fundamental limits on how precisely we can know certain pairs of properties simultaneously

These concepts go beyond General Chemistry, but the foundation has been laid. Energy is quantized (E = hν), matter has wave-like properties (λ = h/(mv)), and Planck’s constant h sets the scale at which quantum effects become important.

The de Broglie picture, with electron waves wrapping around circular orbits, is a useful starting point. But it’s incomplete. What equation governs these matter waves? What do electron waves actually look like in three dimensions? The next chapter develops the mathematical framework, the Schrödinger equation, that answers these questions and reveals where quantum numbers come from.

Practice 1

Calculate the de Broglie wavelength of:

a proton (m = 1.673 × 10⁻²⁷ kg) moving at 1.0 × 10⁶ m s⁻¹
an alpha particle (m = 6.645 × 10⁻²⁷ kg) moving at the same speed

Which particle has the longer wavelength?

Solution

Proton

\[ \begin{align*} \lambda &= \dfrac{h}{mv} \\[1.5ex] &= \dfrac{6.626\times 10^{-34}~\mathrm{J~s}} {(1.673\times 10^{-27}~\mathrm{kg})(1.0\times 10^{6}~\mathrm{m~s^{-1}})} \\[1.5ex] &= 3.96 \times 10^{-13}~\mathrm{m} \\[1.5ex] &= 0.396~\mathrm{pm} \end{align*} \]

Alpha particle

\[ \begin{align*} \lambda &= \dfrac{h}{mv} \\[1.5ex] &= \dfrac{6.626\times 10^{-34}~\mathrm{J~s}} {(6.645\times 10^{-27}~\mathrm{kg})(1.0\times 10^{6}~\mathrm{m~s^{-1}})} \\[1.5ex] &= 9.97 \times 10^{-14}~\mathrm{m} \\[1.5ex] &= 0.0997~\mathrm{pm} \end{align*} \]

The proton has the longer wavelength. Since λ = h/(mv), wavelength is inversely proportional to mass. At the same speed, the lighter particle (proton) has the longer de Broglie wavelength.

Practice 2

An electron is confined to a region the size of an atom, approximately Δx = 1.0 × 10⁻¹⁰ m (100 pm).

What is the minimum uncertainty in the electron’s momentum?
What is the minimum uncertainty in its speed?

Solution

Part a: Minimum uncertainty in momentum

The uncertainty principle states:

\[\Delta x \cdot \Delta p \geq \frac{\hbar}{2}\]

Solving for Δp:

\[ \begin{align*} \Delta p &\geq \dfrac{\hbar}{2\Delta x} \\[1.5ex] &= \dfrac{1.055\times 10^{-34}~\mathrm{J~s}}{2(1.0\times 10^{-10}~\mathrm{m})} \\[1.5ex] &= 5.28 \times 10^{-25}~\mathrm{kg~m~s^{-1}} \end{align*} \]

Part b: Minimum uncertainty in speed

Since p = mv, we have Δp = mΔv (for constant mass). Therefore:

\[ \begin{align*} \Delta v &= \dfrac{\Delta p}{m_e} \\[1.5ex] &= \dfrac{5.28\times 10^{-25}~\mathrm{kg~m~s^{-1}}}{9.109\times 10^{-31}~\mathrm{kg}} \\[1.5ex] &= 5.8 \times 10^{5}~\mathrm{m~s^{-1}} \end{align*} \]

This uncertainty in speed (580 km s⁻¹) is enormous, about 0.2% of the speed of light. Confining an electron to atomic dimensions means we cannot know its speed precisely. This is why the Bohr model, which assigns electrons definite velocities in definite orbits, cannot be correct.

Summary

Wave-Particle Duality

Light and matter exhibit both wave and particle properties depending on the measurement. The double-slit experiment demonstrates this: photons create an interference pattern (wave behavior), yet each photon strikes the screen at a single point (particle behavior). Measuring which slit a photon passes through destroys the interference pattern.

De Broglie Wavelength

All matter has an associated wavelength given by the de Broglie relation:

\[\lambda = \dfrac{h}{mv}\]

For electrons, λ is comparable to atomic dimensions and wave effects are significant. For macroscopic objects, λ is negligibly small and classical behavior dominates.

Connection to the Bohr Model

Stable electron orbits correspond to standing waves where an integer number of wavelengths fits around the circumference:

\[2\pi r = n\lambda\]

This condition naturally produces Bohr’s quantization of angular momentum (mvr = nh/2π) without postulating it.

The Uncertainty Principle

Position and momentum cannot both be known precisely at the same time:

\[\Delta x \cdot \Delta p \geq \frac{\hbar}{2}\]

This is not a measurement limitation but a fundamental property of waves. Electrons cannot have definite orbits; instead, we describe them using probability distributions called orbitals.