By Dr. John Millam1
What do most people think of when they hear the words quantum mechanics? Most likely they would say that it is mysterious and extremely difficult to understand. That it is beyond the grasp of the common person. While quantum mechanics is indeed challenging, much of it can be understood by anyone and without using lots of mathematical formulas. I hope to prove just that here and in four subsequent installments. Buckle your seatbelts to begin this mind-blowing tour of the quantum world.
What is quantum mechanics? In simplest terms, it is a fundamental theory of physics that describes the behavior of matter at the atomic scale. Quantum mechanics evolved out of a series of discoveries in the early twentieth century that could not be explained by the laws of physics as they were understood at that time. The prevailing understanding of the physical universe was set forth by Isaac Newton in the seventeenth century, now known as classical physics. What makes quantum mechanics and the behavior of the sub-atomic world so fascinating is how startlingly different it is from what we experience in our daily lives. At the quantum level, objects behave in unfamiliar, counter-intuitive, and bizarre ways with a whimsical quality reminiscent of Alice in Wonderland. So strange are some of these ideas that Niels Bohr, one of the founders of quantum mechanics, neatly summarized, “Those who are not shocked when they first come across quantum theory cannot possibly have understood it.”2
To help grasp how bizarre quantum mechanics is, let us picture what life would be like if quantum effects were magnified to such an extent that they were readily visible in our daily affairs. Three simple examples:
- Quantization. Imagine riding a bicycle and only being able to travel at certain specific speeds. For example, going from rest to 5 mph, then 10 mph, jumping from one speed to the next without intermediate speeds.
- Quantum superposition. Imagine being able to be in more than one place at a time.
- Quantum tunneling. Imagine just passing through a closed door, rather than having to open it first.
These examples highlight just a few unexpected and seemingly magical aspects of quantum mechanics.
Quantum mechanics is not an obscure feature of the world reserved only for high-level physicists; it surrounds us and undergirds much of our modern technology. (For example, read “10 Examples Of Quantum Physics In Everyday Life.”)
1 Ph.D. in Theoretical Chemistry from Rice University. Full permission is given to reproduce or distribute this document, or to rearrange/reformat it for other media, as long as credit is given, and no words are added or deleted from the text.
2 Original quote from a 1952 conversation with Werner Heisenberg and Wolfgang Pauli in Copenhagen; quoted in Werner Heisenberg, Physics and Beyond, (New York: Harper & Row, 1971), p. 206.
2
Overview of Modern Physics
Let us begin by considering how quantum mechanics fits into the broader tapestry of physics. This understanding will help elucidate how quantum mechanics compares historically and scientifically to other major theories. For this study, we will only concern ourselves with significant developments beginning with the work of Isaac Newton in the seventeenth century.
Newtonian (or “Classical”) Mechanics
In 1687, Isaac Newton published the three laws of motion and universal law of gravitation in his masterwork, Philosophiæ Naturalis Principia Mathematica. These equations represent the first unified mathematical formulation of the laws of physics. It was a revolutionary breakthrough that would dominate scientific inquiry until the twentieth century. Today, however, Newtonian mechanics is generally referred to as classical mechanics to distinguish it from the two later theories—quantum mechanics and general relativity—which would eventually supersede it.
Isaac Newton helped demystify the world by showing how various phenomena were governed by natural forces and predictable by mathematical calculations. His equations gave scientists a new set of tools for understanding all manner of natural phenomena. However, his conceptions gave rise to the notion of a clockwork universe, one where everything ran with the smooth precision of a clock. Newton’s work was not merely a revolution in physics; it created a paradigm shift across science and even into philosophy.
One significant philosophical implication of the classical (clockwork) paradigm was the notion of determinism. Given Newton’s equations of motion, if one were to know the exact position and momentum of all the particles in a system and all the forces involved, then in principle, one could exactly predict everything about the system at any time, both past and future. For example, modern computer programs based on Newton’s equations of motion can accurately predict the locations of the planets in our solar systems for billions of years in the past and likewise into the future. This concept implies the future is already determined by the laws of physics, then we can have no freewill.
The next critical breakthrough came in 1861-1862 when James Clerk Maxwell published a set of four equations unifying electricity and magnetism into a single framework called electromagnetism. Maxwell’s equations were a significant triumph for classical mechanics. His achievement completed Newton’s “revolution of science” by fully describing the one remaining force known at that time. (The strong nuclear force and weak nuclear force would not be discovered until much later in history.)
One crucial aspect of Maxwell’s work is that it described light as an electromagnetic wave traveling at a precisely defined, universal speed. The speed of light in a vacuum is approximately 300,000 km/sec or 186,000 miles/sec. That speed sets the speed limit for physical phenomena in the universe.
At the beginning of the twentieth century, classical mechanics seemed like an invincible juggernaut. For more than 200 years, it had been used to successfully model an ever-increasing range of phenomena. Because of this, there was a growing sense in the scientific community that nothing significant was left to discover. But this complacency would soon be shattered as classical mechanics came under attack from two entirely different directions. Two new laws would eventually need to be formulated to plug the holes in Newton’s model.
Theory of Relativity
One hint that classical mechanics was in trouble was a recognition by scientists that Newton’s laws and Maxwell’s equations were in direct conflict with each other. To understand the problem, imagine standing by the side of the road as I drive by in a car at 70 mph. Just as I pass you, I turn on my headlights. From my perspective in the car, the light from my headlights would travel away from me at the speed of light as set forth by Maxwell’s equations. From your
3
perspective alongside the road, Newtonian mechanics predicts that the light would be observed to be traveling just a bit faster—specifically 70 mph faster.
In fact, for this type of scenario each observer would determine a different speed for the light depending on how fast they were moving relative to the light source. In contrast, Maxwell’s equations insist the speed of light in a vacuum is a universal constant so that all observers should measure the same universal speed. Because the two theories make different predictions, one had to be wrong. But which one?
Though it was designed to address a different issue, the Michelson-Morley experiment in 1887 provided an answer to that question. The pair of scientists had been attempting to measure Earth’s motion relative to previously hypothesized luminiferous aether. Scientists had proposed aether to explain why we could see the light from distant stars even though there is no matter in outer space for it to propagate through.3 Michelson and Morley used a technique called interferometry to make very sensitive measurements comparing Earth’s motion along perpendicular directions. Despite all their efforts, they found no difference in velocity regardless of how they oriented the apparatus. This result meant that the speed of light was constant regardless of the Earth’s motion vindicating Maxwell while refuting Newton.
As a result, Newton’s equations of motion had to be modified so that the speed of light in a vacuum would be invariant, that is, that all observers would measure the same speed regardless of their reference frame as required by Maxwell’s equation. In 1905, Albert Einstein published the solution: the special theory of relativity, in 1905. He would go on to publish his general theory of relativity in 1915.
Quantum Mechanics
Quantum mechanics also emerged in the twentieth century as a result of attempts to patch up classical mechanics for issues dealing with atomic-scale phenomena. But unlike relativity, quantum mechanics was developed piecemeal over several decades by multiple scientists. That history will be covered in more detail a little bit later.
3 Traditionally, waves were understood as energy traveling through some material medium. For example, sound waves represent pressure waves in air. Because of this need for a medium, sound waves cannot travel through the vacuum of space. To explain why light can travel through space to reach us, scientists hypothesized the existence of luminiferous aether as a special material pervading space and capable of transmitting light waves. The failure of the Michelson-Morley experiment to detect any motion of Earth with respect to the aether ultimately led scientists to reject its existence.
Revolutionary Implications of the Theory of Relativity - United the concepts of space and time into a single framework known as space-time.
- While Newton had treated space and time as absolute entities, Einstein showed that they were malleable. For example, time dilation is a phenomenon where time passes at different rates depending on how fast one is moving relative to another.
- The universe is expanding. Working backward, this means the universe must have had a beginning in what we call the big bang.
- Energy and matter are interchangeable as shown by Einstein’s famous equation, E = mc2. This led to the development of nuclear energy and nuclear bombs.
4
Toward a Theory of Everything
Here at the dawn of the twenty-first century, we have two central pillars of modern physics—relativity and quantum mechanics. Each discipline emerged from attempts to resolve failures in Newtonian mechanics, but they are remarkably different theories because they were developed to solve different problems. Each approach works beautifully within the domain for which it was designed. The results for both quantum mechanics and the general theory of relativity have been demonstrated to an amazing level of accuracy. The problem arises when one theory is applied to the other’s domain. We discover that these two theories are fundamentally incompatible. We can characterize the two domains as follows: - The General Theory of Relativity. Needed for objects moving at speeds close to the speed of light or located in strong gravitational fields.
- Quantum Mechanics. Needed for atomic-scale phenomena.
These two domains are non-overlapping with two main exceptions: black holes and the origin of the universe. Both cases involve massive gravitational forces applied over a tiny region of space. To correctly model these, we need a super-theory reconciling relativity and quantum mechanics into a single coherent framework. Such a theory could be called a “theory of everything” because it should be able to describe every aspect of how the universe works.
Much work has been done to combine the two approaches, yet a theory of everything (TOE) remains elusive. String theory and loop quantum gravity are currently the leading contenders for a solution. Unfortunately, because we have no way to generate theoretical predictions that can be tested with our current technology, validating either of these theories has remained elusive.
Historical Origin of Quantum Mechanics
Planck’s Formula (1900)
The story of quantum mechanics begins at the start of the twentieth century. At this time, one of the big unresolved questions in physics was developing a precise model for black-body
Revolutionary Implications of the Quantum Mechanics - Microscopic reality is quantized, rather than continuous. Only certain outcomes are permitted.
- Quantum entities can behave like waves under some circumstances and like particles in others. This is known as wave-particle duality.
- Quantum outcomes are fundamentally probabilistic. That means that we cannot precisely predict the future but only what would probably be true. This refutes classical determinism and its argument against free will.
- Particles do not always possess well defined properties. Only when particle’s property is measured is it forced to take a definite value.
- Quantum mechanics (when combined with special relativity) predicted the existence of antimatter. Every fundamental particle has an antimatter twin with the opposite charge.
- Measurement has an inherent uncertainty so that it is impossible to perfectly measure some pairs of quantities.
5
radiation.4 In simplest terms, black-body radiation is the spectrum of light emitted by an object due to its temperature. For example, if you heat a metal bar hot enough, it will start to glow red. Incandescent light bulbs and the sun are two common examples of this phenomenon. One useful benefit is that measuring an object’s black-body spectrum is that it can be used to infer its temperature. For example, this allows astronomers to determine the temperature of stars and is the principle behind touchless thermometers.
Figure 1 shows the black-body radiation curves for objects at several different temperatures. The vertical axis shows the spectral radiance, the total amount of light emitted at a given wavelength. The horizontal axis describes the wavelength of the emitted light. Ordinary visible light (wavelengths 0.4-0.8 μm) is shown with a vertical rainbow stripe representing the range of visible colors. To the right of visible (wavelengths > 0.8 μm) is infrared light, and to the left (wavelengths < 0.4 μm) is ultraviolet light. The main body of the graph shows the black-body radiation curve for materials at three different temperatures: 3000 K (red curve), 4000 K (green curve), and 5000 K (blue curve).5 From this, we can identify two significant trends in how the black-body radiation curve changes as a function of increasing temperature. First, more light is emitted at all wavelengths. Second, the peak shifts to the left (to shorter wavelengths). Objects at room temperature (300 K) will only emit tiny amounts of infrared light that is invisible to the human eye. At temperatures around 773 K, some of the radiation spills into the red portion of the visible spectrum resulting in a dull red glow. At higher temperatures, the color will change as more of the visible spectrum is included in the output. For example, at 3000 K (red curve) an object glows a reddish-orange, at 4000 K (green curve) it is light orange, and at 5000 K (blue curve) is nearly white. Our sun’s surface temperature is approximately 5800 K, so it shines with an almost pure white light. To begin, we need to understand something about what black-body radiation is and what it represents. The temperature of an object is a measure of its thermal energy. Thermal energy is a measure of vibrational motion of the individual atoms that make up the object. The atoms in hot objects vibrate more vigorously than cold objects. Hot objects therefore possess more thermal energy which is expressed as the emission of electromagnetic radiation. By understanding how this thermal energy is distributed among the different vibrational modes and how these vibrations emit and absorb light, scientists can determine the precise mathematical formula describing the spectrum of light emitted at a given temperature. The earliest attempts to model black-body radiation were based on classical mechanics. One early example was the Wien approximation (sometimes called Wien’s Law) in 1896. The equation performed correctly for short wavelengths but was slightly off at longer wavelengths. 4 The term “black body” refers to an ideal material that absorbs all incoming light. This property ensures that the material’s particular properties does not influence its thermal spectrum thus making the resulting spectrum general and material independent. The best way to generate a black-body spectrum is to heat a hollow metal sphere with a small hole in it and observe light coming out of it. No light reaches the interior of the sphere so that it behaves as if it is truly black. 5 Temperatures here are reported in Kelvin (K), degrees above absolute zero. To convert from Kelvin to Celsius (°C), just subtract 273. Figure 1: Black-body radiation curves. Credit: Wikipedia commons. 6 The next attempt at modeling black-body radiation was the Rayleigh-Jeans law in 1900 (see “Classical theory” curve in Figure 1). It worked at long wavelengths but failed spectacularly at short wavelengths. It wrongly predicted that objects should emit unlimited amounts of radiation as short wavelengths. If that prediction was accurate, then even cool objects should emit so much ultraviolet and x-ray radiation that it would kill anyone nearby. This prediction was so glaringly wrong that scientists called it the ultraviolet catastrophe. Max Planck was the scientist who ultimately resolved this issue in 1900. His first step was recognizing the Rayleigh-Jeans law worked for long wavelengths while the Wien approximation was successful at short wavelengths. Taking the best parts of both formulas, he was able to piece together the correct formula that worked at all wavelengths. The problem was that he needed to justify this result by deriving it from fundamental principles. But how to do that? Planck’s critical insight was to look for a way to prevent the growth of high-frequency (short wavelength) light that gave rise to the ultraviolet catastrophe. He found that he could do this by making the frequency emitted proportional to energy. This change penalized higher frequency vibrations because those are the ones that require more energy to be produced. After much work, he found that he could derive the desired equation if he introduced two rather unorthodox assumptions: First assumption: Vibrations are restricted to very specific energy levels (E): E = nhν, Equation 1 where n = 1,2,3, …, h is Planck’s constant, and ν is the oscillation frequency. Second assumption: These vibrations do not radiate energy continuously, but only in specific “jumps” or quanta. Light is emitted when the vibration jumps from one energy level to another. For the smallest jump (n = 1), the energy emitted (E) would therefore be: E = hν, Equation 2 where h is Planck’s constant, and ν is the frequency of the emitted light. Although this trick worked, Planck was convinced that it was nothing more than a mathematical hack. He expected later scientists to develop a better and more physically meaningful solution. With the benefit of hindsight, we know that Planck’s work was the first step in the development what would become quantum mechanics. No one in his day would have foreseen this because no one was expecting to overthrow classical mechanics. Planck’s constant (symbolized with an “h”) is now recognized as the universal constant governing quantum behavior. Because its value is extremely small, quantum effects are generally too small to observe in our daily lives. Cosmic Microwave Background As an aside, the most famous black-body radiation curve in all history is something that most people have heard of, but typically not in this context. It is the black-body radiation curve for the entire universe, better known as the cosmic microwave background (CMB) It was discovered by accident in 1964 by Arno Penzias and Robert Wilson and is a relic from an earlier epic in the universe’s history. The universe started insanely hot but has since expanded and cooled. As can be seen in Figure 2a, this background radiation precisely corresponds to the theoretical black-body radiation curve of an object with a temperature of 2.725 K or just a few degrees above absolute zero. Because of this low temperature, its black-body spectrum is entirely in the microwave range rather than visible or infrared. 7 This background radiation is uniform in all directions except for minute differences on the order of 1 part in 100,000. These subtle variations have been carefully mapped by the Cosmic Background Explorer (COBE) (1989-1993), the Wilkinson Microwave Anisotropy Probe (WMAP) (2001-2010), and the Planck (2009-2013) satellites. Analyzing this data has given cosmologists insights into the early universe. Figure 2b shows the small temperature differences in the microwave background in different directions as measured by WMAP with red marking higher temperatures and blue showing lower temperatures. Photoelectric Effect (1905) The next significant development in our understanding of quantum mechanics came from an unexpected direction: the photoelectric effect. It was none other than Albert Einstein who published his explanation in 1905. Amazingly, in addition to developing the theory of relativity, he is a co-founder of quantum mechanics. Curiously, Einstein earned his Nobel Prize for his work on the photoelectric effect rather than for relativity. The photoelectric effect occurs when metal surfaces are exposed to ultraviolet light causing electrons to be ejected. (Electrons were previously discovered by J. J. Thomson in 1897.) The main difficulty in understanding the photoelectric effect was determining how to correctly model how light (electromagnetic waves) transferred its energy to electrons causing them to be ejected from the metal. At that time, light was understood to be a classical wave, so it was expected to behave like common waves, such as water waves. To illustrate, imagine being out on a lake in a boat on a calm day. Another boat passes by at high speed, generating a wave. The resulting wave is an undulating series of peaks and troughs. When the first part of the wave arrives, it causes your vessel to rock slightly back-and-forth. The remaining portions of the wave simply add to the rocking motion. As this picture illustrates, classical waves impart their energy incrementally over time. With this in mind, let us turn to a different picture to illustrate the photoelectric effect from a classical (wave) perspective. Imagine a bathtub that is empty except for a rubber ducky at the bottom. The rubber ducky represents an electron, and the bathtub represents the attraction of the electron to the metal atom that keeps it bound to it. Now imagine closing the drain and turning on the water spigot. As the tub fills with water (representing light), the rubber ducky floats atop the rising water. If this continues, the water will overflow the tub liberating the ducky. This event correlates with the ejection of the electron in the photoelectric effect. a) b) Figure 2: (a) Cosmic microwave background intensity curve and (b) variations in the cosmic microwave background as measured by the WMAP satellite. Credit: Wikipedia commons. 8 Based on the classical wave model, there should have been a slight interval between the time when the light strikes the metal and when the electrons are ejected. However, no delay was found. Einstein resolved this by arguing that light must impart its energy all at once, like a particle. Consider the following illustration. Imagine a pool table with a shallow depression, and that depression contains a single pool ball. In our thought experiment, the ball is an electron, and the depression is the electron’s attraction to the metal atom holding it in place. Now imagine propelling the cue ball (representing the light) so that it collides with the pool ball. If the cue ball strikes it hard enough, the pool ball will be ejected from the depression and allowed to fly free (akin to the photoelectric effect). But if the cue ball does not impart enough energy, the pool ball will only make it part way out of the depression before rolling back to the center again. This understanding resolved a second issue: light with a frequency below a certain threshold (depending on the metal) did not produce the photoelectric effect regardless of its intensity. Einstein adopted Planck’s notion of quantization but took it one step further. While Planck had quantized vibrational motion, he still treated light as a wave. Einstein insisted that light was quantized—that it was emitted or absorbed in discrete packets, as if it was a particle. He called light packets “quanta,” although today we call them photons. Following Planck’s formula, the energies of these light quanta were proportional to their frequency: E = hν, Equation 3 where E is the energy, h is Planck’s constant, and ν is the frequency of light. Low-frequency light did not give rise to the photoelectric effect because it lacked the energy to liberate electrons. Einstein’s analysis yielded yet another important insight: the kinetic energy of the electron was the excess energy from the photon beyond what was needed to liberate it. Consequently, the liberated electron’s kinetic energy should increase with increasing light frequency, as was later experimentally verified by Robert Millikan in 1916. The Spectrum of Hydrogen (1913) In the nineteenth and early twentieth centuries, an important tool for analyzing the nature of atoms was the emerging field of spectroscopy. Spectroscopy involves the analysis of light emitted or absorbed by materials to learn their nature and composition. To study a particular chemical element, scientists would place a small sample in a vacuum-sealed glass discharge tube and run a current through it. This action causes the atoms to heat up Photoelectric Effect: Expectation versus Experiment Classical expectation 1: There should be a time delay between the light hitting the metal and the electrons being ejected. Experimental result: The photoelectric effect takes place immediately. Classical expectation 2: Effect should occur even with low frequency light. Experimental result: Low frequency light does not generate the effect regardless of its intensity. Classical expectation 3: The kinetic energy of the ejected electrons should depend on the intensity of the light. Experimental result: The kinetic energy of the ejected electrons is determined by the light’s frequency. 9 and glow with a characteristic color. To gain more information about the light, scientists used a diffraction grating to spread it out into individual wavelengths (analogous to a prism separating sunlight into a rainbow of colors). The resulting plot of the intensity of the emitted light as a function of wavelength or frequency is called a spectrum. When scientists studied the spectra of different elements, two details stood out. First, the spectra were discrete, that is, only certain specific wavelengths were present. (In contrast, blackbody radiation is a continuous spectrum where all frequencies are allowed.) Second, the set of wavelengths for each element is unique. Because of this, scientists can examine the spectra of various materials to determine their chemical composition. For example, the element helium was discovered in the sun’s spectrum before it was isolated on Earth. (The name helium comes from the Greek helios for “sun.”) One of the first spectra to be extensively studied was that of hydrogen because it is the simplest element. In 1885, Johann Balmer identified six distinct emission lines for hydrogen in the visible range (see Figure 3). Together, these emission lines are known as the Balmer series. Balmer was subsequently able to show that these wavelengths could be computed using a simple mathematical formula known as the Balmer formula, with each wavelength specified by a simple integer. In 1888, Johannes Rydberg generalized Balmer’s formula by including a second integer and therefore predicting additional wavelengths: H 2 2 1 1 1 R m n = − , Equation 4 where λ is the wavelength, RH is the Rydberg constant, and m and n are integers. The Balmer series corresponded to m = 2 and n > m. Another series of lines known as the Lyman
series (corresponding to m = 1) was discovered between 1906 and 1914 in the ultraviolet range.
Additional sets of emission lines were later found in the infrared, including the Paschen (m = 3),
Brackett (m = 4), and Pfund (m = 5) series.
Einstein’s 1905 formula for the energy of a photon (E = h) led to a new understanding for
Rydberg’s formula. Using it along with another basic formula (c = λ), scientists could re-express
it in terms of energy. Thus, the energy of the emitted photon could be understood as the difference
in energy between the electron’s starting orbit (specified by n) and final orbit (specified by m):
H 2 H 2
1 1
E(photon) E(m) E(n) hcR hcR
m n
= − = −
, Equation 5
where E is energy, h is Planck’s constant, c is the speed of light, RH is the Rydberg constant,
and m and n are integers.
This discovery revealed that the electrons in hydrogen could only occupy orbits with energies
given by E(n) = -hcRH/n2 and the spectral lines are just transitions between these levels (see Figure
4). That greatly simplified the picture of the hydrogen atom, but questions remained. Why are
Figure 3: The “visible” hydrogen emission spectrum lines in the Balmer series.
Credit: Wikipedia commons.
10
only these energies allowed? How can one derive these energy levels based on theoretical principles?
Danish physicist, Niels Bohr (1913) provided a theoretical justification for the experimental results. His solution was to quantize these orbits in the same spirit as Planck’s quantization of molecular vibrations. He explained the features of hydrogen atoms using three postulates:
- Electrons are only allowed in fixed orbits and jump directly from one allowed orbit to another.
- The orbital angular momentum of an electron is quantized, i.e., angular momentum = nh/2, where n is a positive integer.
- The frequency of light emitted or absorbed is given by the difference in energy levels, i.e., E = h.
Bohr’s atomic model was based on ad hoc rules, but this was sufficient to explain the features of the hydrogen spectrum. The electrons in the atoms were viewed as following perfectly circular orbits with a radius defined by the quantized angular momentum. Bohr’s work introduced the notion of a quantum number, n, that specifies each of the allowed orbits.
Bohr’s model successfully explained the spectra of hydrogen, but it did not work for any other elements. Nevertheless, his general framework did apply to all elements. Electrons in atoms are restricted to a small set of quantized orbits, and they can only move between these orbits. The problem was that predicting the energy levels for multi-electron elements was more complicated than for hydrogen because of the interaction between the electrons. Non-hydrogen elements would not be properly described until the development of Schrödinger’s equation in 1926.
Quantum Mechanics and the Stability of Atoms
Bohr’s model of the hydrogen atom was revolutionary, but it also resolved a significant conceptual difficulty with the previous model for atoms. Just two years earlier, Ernest Rutherford had proposed that atoms consisted of a diffuse cloud of negatively charged electrons surrounding a small, dense, positively charged nucleus. In his model, electrons orbit the nucleus like planets around the sun.
The problem with the Rutherford’s model was that Maxwell’s equations predicted that negatively charged electrons moving in a curved path should emit electromagnetic radiation. This radiation would remove energy from the electrons, causing them to spiral inward until they reached the nucleus. Given this, atoms should immediately collapse in a flash of radiation. That means that life would not be possible anywhere in the universe! No atoms, no physical life.
Bohr’s quantum model prevented this collapse by restricting the electrons to fixed orbits. They could only jump from one orbit to another and nowhere else. This restriction effectively
Figure 4: Energy levels and transitions for hydrogen.
Credit: Wikipedia commons.
11
prevented electrons from spiraling inward because that would not correspond to one of the fixed orbits. Bohr also predicted the existence of a minimum energy orbit preventing the electrons from falling into the nucleus. Saving the atom was a major triumph for quantum mechanics.
Wave-Particle Duality (1924)
In classical antiquity, entities could be organized into two mutually exclusive categories: waves and particles. This classification scheme for objects was analogous to biological classification, where at the kingdom level, living creatures are traditionally divided into plants and animals.
For most of history, light was treated as a classical wave. The whole field of optics relies heavily on the wave nature of light. So, it came as a shock when Einstein proposed that light consisted of individual particle-like entities called quanta (or photons). While this created quite a stir, his idea did not immediately gain widespread acceptance. It was too radical of a proposal to be embraced based on just one isolated example. This changed in 1923 when Arthur Compton surprised the scientific community by providing a second example of light behaving like a particle. He showed that when light scattered off electrons, it behaved like a classical particle rather than a wave. More importantly, Compton scattering established that light had momentum like a particle. (Momentum is the tendency of an object to remain in motion.) With this second example, the dual nature of light could no longer be so easily dismissed.
It was Louis de Broglie who put all the pieces of the puzzle together. In 1924, he introduced the notion of wave-particle duality, a radical proposal where wave and particle categories are not incompatible but are two faces of the same coin. Light could therefore behave like a wave when passing through a glass lens and act like a particle in the photoelectric effect and in Compton scattering.
Building a theoretical bridge between these two very distinct concepts was central to de Broglie’s thesis. The principal challenge was that particles and waves were traditionally described in very different ways. Particles in Newton’s equations were primarily characterized by two quantities: momentum (p) and energy (E). In contrast, electromagnetic waves governed by Maxwell’s equations were principally defined by two other quantities: wavelength (λ) and frequency (). How can we relate these two sets of quantities?
De Broglie recognized that Einstein had already described the relationship between energy and frequency (Equation 3):
E = hv, Equation 6
where E is energy, h is Planck’s constant, and ν is the frequency.
De Broglie completed this picture by introducing a relationship between momentum and wavelength:
λ = h/p, Equation 7
where λ is the wavelength, h is Planck’s constant, and p is momentum.
De Broglie showed that light could simultaneously be described as both a wave and a particle, and scientists could now interconnect these two perspectives.
But de Broglie did not stop there. He recognized that the bridge he had created was a two-way street. If light waves could be treated as particles, then fundamental particles could also be treated as waves. The logical consequence of his equations is stunning. Previously viewed as a point-like object, an electron could now be thought of as a matter wave. Most scientists found this hard to accept.
12
Because everyday objects are made of fundamental particles, they too must be treated as
waves. A thrown baseball is governed by the same rules as the quantum world and therefore
oscillates with a definite wavelength. Even human bodies have an associated wavelength! Of
course, the wavelengths of ordinary objects are incredibly small—at least a trillion times smaller
than the diameter of an atom. This means that their wave properties would be too small to be
detectable. Therefore, we do not have to contend with these wave properties in everyday life, only
at the atomic scale.
Matter particles behaving like waves was a radical idea, but how could it be tested?
Fortunately, electrons have wavelengths comparable to the distances between atoms in solid
materials. In 1927, G. P. Thomson, along with parallel work by Clinton Davisson and Lester
Germer, took advantage of this property and bombarded thin metal foils with electrons and
observed diffraction—a wave property. The diffraction pattern produced by the electrons could
be directly compared with those made by x-rays, thus demonstrating wave-particle duality.6
Scientists have since provided additional examples of diffraction using neutrons, neutral atoms,
and even small molecules, such as buckminsterfullerene (C60).
Schrödinger’s Equation (1926)
De Broglie’s work provided the critical missing piece needed by Erwin Schrödinger in
1926 to develop an equation describing the behavior and evolution of quantum systems. Known
as Schrödinger’s equation, it was a monumental achievement because it provided a general model
that could be applied to any system. It superseded the previous work of Planck, Einstein, and
Bohr. Those ad hoc rules were invented to match specific observations and are now referred to as
the “old quantum” theory.
For this review, we will only consider the time-independent version of the Schrödinger
equation:
ˆH
(r) = E (r)
n n n
, Equation 8
where Ĥ is the Hamiltonian operator, n(r) is the set of wavefunctions, En are the associated
energies, and n is an integer.
The Ĥ is the Hamiltonian operator that describes the energy of the system. It is constructed using
classical (non-relativistic) expressions but with each term modified in a specific way. n(r) is the
set of possible wavefunctions derived by solving the equation. These wavefunctions are a
mathematical description of the various possible quantum states. Each of these solutions is unique
and described by one (or more) integer quantum numbers, n. En gives the energy of the system
corresponding to wavefunction, n(r).
One of the greatest difficulties was understanding the meaning of the resulting
wavefunction, because it had no correspondence in classical mechanics to act as a guide. Initially,
Schrödinger understood the wavefunction as a matter wave describing the particle itself. More
study, however, showed that this interpretation simply did not work. Max Born in 1926 laid out
the now-accepted interpretation—the square of the wavefunction gives the probability density. So,
instead of describing the particle’s location, the wavefunction simply provides a probability of
where we might find the particle.
6 This work eventually led to the development of the electron microscope. Electrons can be used to examine objects
in far greater detail than optical microscopes because electrons have a much smaller wavelength.
13
Dirac’s Equation (1928)
Just two short years after Schrödinger published his seminal equation defining quantum mechanics, Paul Dirac modified it to satisfy Einstein’s special theory of relativity. (Note, it is not consistent with the general theory of relativity and therefore is not a theory of everything.) The precise details of the Dirac equation are beyond the scope of this review. Schrödinger’s equation is still preferred for most applications because it is mathematically much easier to solve and is sufficiently accurate for most cases (i.e., those in which relativistic effects are not significant). Schrödinger’s approach can be referred to as standard quantum mechanics and Dirac’s as relativistic quantum mechanics. For this discussion, we will only focus on two critical discoveries derived from the Dirac equation.
Because relativity operates in four dimensions (three space dimensions plus time), Dirac’s equation yields four distinct solutions (rather than just one for standard quantum mechanics). But what to make of this abundance of results? Physicists immediately recognized that two of the solutions represented a property called spin. Spin is a form of intrinsic angular momentum possessed by all particles. It was previously postulated to explain certain experimental results, but now that proposition rested on solid theoretical grounds.
At first, physicists were puzzled by the remaining solutions. For a particle like an electron, these extra solutions were nearly identical to a regular electron, but with a negative energy (instead of a positive one) and a positive charge. These solutions were eventually identified as antimatter. Thus, every fundamental particle has an antimatter twin that is identical except the twin has the opposite charge. For example, a negatively charged electron has a positively charged antimatter partner called a positron. The positron was discovered experimentally in 1931. We know today that every fundamental particle has an antimatter twin. This prediction of the previously unknown antimatter was a major triumph for theoretical physics.
Quantum Superposition
Schrödinger’s equation fundamentally transformed quantum mechanics into a powerful tool for understanding the sub-atomic world. Scientists could now solve the equation for various cases, but often struggled to properly understand the meaning of the resulting wavefunction. One particularly difficult and puzzling aspect of quantum mechanics is superposition, a complex state constructed from more than one simple state. Solving Schrödinger’s equation (Equation 8) yields a set of “pure” solutions, n(r), each with a clearly defined meaning. Mathematically, any linear combination (superposition) of these solutions is also a valid solution of Schrödinger’s equation and so represents a physically realizable possibility. Yet these solutions are classified as indeterminate because they do not correspond to a single measurable outcome. How are we to understand this?
To illustrate this situation, consider the following example. Imagine taking photos of a cat using a film camera. You take one picture of the cat lying down and one of it standing up, not realizing that the film is not advancing. When you develop the film, you see the cat in two different positions in the same picture (see Figure 5). Each shot of the cat represents a pure state with a clear and familiar meaning. The double exposure is a superposition and does not correspond to a situation we can observe in real life because a cat cannot be simultaneously laying down and standing up.
Now let us consider a quantum coin representing the simplest possible quantum system. Like an ordinary coin, it has two pure states: heads and tails. In addition to these pure states, a
Figure 5: Cat in a superposition state.
Credit: Wikipedia commons (cropped).
14
quantum coin can exist in a continuum of superposition states. Examples of superposition states include 50% heads and 50% tails, or 70% heads and 30% tails, and so forth.
The challenge is that we cannot passively observe a quantum system in the same way we can for ordinary household objects. The only way to gain information about a quantum system is to measure it. But when we measure it, we only detect pure states, never a superposition. For the quantum coin, that means we only measure it to be heads or tails. But that begs the question, how do we even know that the quantum coin was in a superposition prior to measurement? One way to learn is by examining the distribution of outcomes from a group of coins, rather than a single outcome. For example, if we prepare a set of quantum coins in the superposition 50% heads and 50% tails and then measure them, half will come up heads and the other half tails. Superposition is real, not just theoretical.
Schrödinger’s Cat Paradox (1935)
The possible existence of superposition states bothered many theoretical physicists, including Schrödinger, one of the founders of quantum mechanics. To highlight this seeming absurdity lying at the heart of quantum mechanics, Schrödinger proposed his famous cat thought experiment in 1935. The setup is fiendishly simple (see Figure 6). Start by placing a cat (A) in a sealed box with a radioactive atom (B) having a 50% chance of decaying during the period of study. If the atom decays, it will trigger a Geiger counter (C) that would cause a hammer (D) to smash a vial of poison gas (E) and kill the cat. But if the atom does not decay, the cat remains alive.
According to quantum mechanics, the radioactive atom should be in a superposition state of 50% undecayed and 50% decayed. This superposition would then extend to the Geiger counter, hammer, poison vial, and finally to the cat. That is, the cat should be in a superposition of 50% alive and 50% dead! This theoretical prediction is obviously nonsensical and conflicts with human experience. This is the paradoxical situation that Schrödinger was trying to highlight.
Of course, we can never directly observe an object in a superposition state. Instead, its wavefunction somehow “collapses” to yield a single state that is ultimately observed. But how and when does that happen? This conundrum is known as the measurement problem and remains unresolved. Many different schools of thought have been developed to explain this paradox and other challenging quantum issues with no clear favorite. It is to this subject that we turn next.
Interpretations of Quantum Mechanics
Quantum mechanics undoubtedly works! It has shown itself to be correct whenever it has been experimentally tested. And that includes the more extreme claims, such as quantum tunneling and quantum entanglement (to be described more later). Yet some critical theoretical difficulties remain. Even after a century of study, we still do not fully understand how quantum mechanics works at the deepest level. It is like a black box where we can correctly predict the output for a given input, yet we do not fully understand what is going on inside the box.
Figure 6: Diagram of Schrödinger’s cat experiment.
Credit: Wikipedia commons.
15
Many interpretations of quantum mechanics have been proposed over the years, but only a few of the most influential ones will be discussed here. And the differences between these models are not minor. For example, some contested ideas regarding quantum mechanics:
Part 5: Spin and the Quantum Atom.
Appendix: Resources on Quantum Mechanics
Robert Gilmore, Alice in Quantumland: An Allegory of Quantum Physics.
Erica W. Carlson, The Great Courses: Understanding the Quantum World.
James Kakalios, The Amazing Story of Quantum Mechanics.
Alex Montwill and Ann Breslin, The Quantum Adventure: Does God Play Dice?
Brian Clegg, The Quantum Physics Bible: The Definitive Guide to 200 Years of Subatomic Science.
Banesh Hoffmann, The Strange Story of the Quantum.
Amir D. Aczel, Entanglement: The Greatest Mystery in Physics.
George Musser, Spooky Action at a Distance.
Are quantum outcomes probabilistic or deterministic?
Is the wavefunction something real and physical, or is it just a mathematical construct?
Does the wavefunction collapse?
What is the role of the observer?
Currently, there is no way to determine which interpretation of quantum mechanics is correct because they all make the same basic predictions for values that can be measured.
Copenhagen Interpretation
The Copenhagen interpretation is the earliest and most influential of the various interpretations. This view was primarily developed by Niels Bohr and Werner Heisenberg at the University of Copenhagen in the 1920s-1930s. It was quickly adopted as the orthodox interpretation of quantum mechanics and is still commonly taught in university classes. This approach is the simplest and allowed early quantum theorists to make progress by not tackling the more subtle aspects. As such, this view has been humorously summarized as “shut up and calculate!”7 Its position as the leading view is weakening as other views are gaining more adherents.
One of the interpretation’s central features is that when the object is not being observed, its wavefunction may exist in a superposition state. That means that the object lacks definite properties. It is the act of measuring the system that causes the wavefunction to collapse into a definite state. The observed state is determined according to probability rules set out by Max Born.
Nature of the wavefunction. The wavefunction is just a mathematical construct that represents the limits of our knowledge about the system.
Schrödinger’s cat paradox. The cat will be in a superposition of 50% alive and 50% dead until someone opens the box to observe it. After that, it has an equal chance of being either alive or dead.
Bohm’s Interpretation/Pilot Wave Theory
Louis de Broglie developed pilot wave theory in the 1920s, but later abandoned it. David Bohm resurrected and extended it in 1952. Bohm’s interpretation treats particles as actual point-like objects governed by a real wavefunction as defined by Schrödinger’s equation. The wavefunction guides the particle, thereby giving it quantum behaviors. Unlike the Copenhagen interpretation, it does not treat observation/measurement as a distinct activity. A major strength of this view is that it is strictly deterministic, rather than probabilistic. The primary objection is that it is explicitly non-local; that is, particles can be influenced by certain interactions regardless of distance. This non-locality results from the wavefunction that guides the particle being spread out in space. Changes to the wavefunction at one location will alter the wavefunction everywhere and therefore change the particle’s behavior regardless of where along the wavefunction it is located.
The best analogy for this view would be to think of a surfer riding on a large ocean wave. The surfer represents the particle, and the wave is the wavefunction as defined by Schrödinger’s equation. As the wave moves, it drives the surfer forward. The wavefunction can represent multiple states (a superposition) while the particle is always in a definite state.
7 Original quote by David Mermin in 1989 but often misattributed to Richard Feynman.
16
For Bohm’s interpretation of Schrödinger’s cat thought experiment, imagine the surfer on the ocean wave encountering an obstacle, such as a large rock formation in the water. To the right of the rock represents the cat alive, and to the left, the cat is dead. When the wave encounters the rock, half of the wave will flow to the right and the other half to the left. According to Bohm, the surfer is randomly located somewhere along the wave and has equal probabilities of being swept to the right (i.e., the cat is alive) or to the left (i.e., the cat is dead). The wave follows both paths, but the surfer only takes one.
Nature of the wavefunction. The wavefunction is a real entity that guides a real particle imparting it with quantum behavior.
Schrödinger’s cat paradox. The wavefunction represents a superposition of alive and dead, but the cat itself will either be dead or alive. The cat is always in a definite state, although we don’t know which until we open the box to observe it.
Many-Worlds Interpretation
One of the most radical interpretations of quantum mechanics comes from Hugh Everett in 1957. Bryce DeWitt popularized the formulation in the 1970s and named it the many-worlds interpretation. In this view, instead of the wavefunction collapsing, each possible outcome of the superposition occurs but in a separate “world” or “universe.” Thus, every quantum event causes the universe to fork, and each possible outcome is realized in one of them (see Figure 7). This interpretation avoids the collapse of the wavefunction because all the states are manifested, albeit in different universes. The observer is also present in each of these universes and will see one of the outcomes according to which universe they are in. This interpretation makes the outcome deterministic rather than probabilistic. The main philosophical challenge for this model is accepting the existence of an infinite number of simultaneously existing alternative histories that we cannot observe.
To help grasp this, let us consider a simple word picture. Think of the trunk of a large tree that represents a system in superposition. The trunk splitting into multiple branches represents all possible outcomes (pure states). Someone sitting on branch A would only see outcome A but would never see the other branches/outcomes. Each branch (quantum outcome) is entirely independent, without knowledge of any other branch. Only if we could see all the branches together would we not observe a collapse of the wavefunction.
Nature of the wavefunction. The universal wavefunction is objectively real and never collapses, so all possible outcomes of quantum measurements are physically realized in some “world” or universe.
Schrödinger’s cat paradox. When the atom decays, the universe forks—the cat is alive in one universe but dead in another. Each universe will be occupied by a version of the observer who only sees one particular outcome: a living cat with a happy scientist in one universe and a dead cat with a sad scientist in another.
Some Other Interpretations
Decoherence/Objective Collapse Theories. There are several versions of this idea, but they all hold that the superposition will eventually collapse to a single state independent of measurement.
Figure 7: Schrödinger’s cat according to the Many-Worlds Interpretation.
Credit: Wikipedia commons.
17
Simple particles can remain in a superposition indefinitely but the larger and more complex the system is the faster it will collapse. This would explain why we never see superposition in everyday objects because they would always collapse before they could be observed.
Consciousness Causes Collapse (von Neumann–Wigner Interpretation). This view holds that a human observer is critical for the collapse of the wavefunction. It was proposed in the 1960s but was later abandoned by most physicists. It is sometimes used (abused) by New Age or Eastern mystical groups (quantum mysticism).
Time-symmetric Theories. This theory is based on retrocausality. Retrocausality posits that events in the future can affect past events, precisely as past events affect ones in the future. The collapse of the wavefunction is, therefore, not a physical change to the system. It is a change in our knowledge of the system due to a later measurement.
And, of course, there are more than a dozen other possible interpretations!
Quantum Weirdness
Most people have at least heard of the more exotic aspects of quantum mechanics. A brief description of some of these will be presented here.
Quantization
Quantization is a fundamental aspect of quantum mechanics where only certain states or outcomes are allowed. Planck’s quantization of vibrations and Bohr’s quantization of atomic orbitals are two specific examples. This fundamental idea runs utterly contrary to classical mechanics and our everyday experience. To help illustrate this, we can compare classical mechanics to a volume control knob where all values are allowed. Quantum mechanics, in contrast, is like an AM/FM tuner—only specific frequencies correspond to a radio program.
Heisenberg Uncertainty Principle
Werner Heisenberg developed his uncertainty principle in 1927. It holds that certain pairs of properties cannot be simultaneously measured to perfect precision due to the wave nature of particles. For these cases, there will always be some inherent uncertainty in the measured value. The most common example links momentum (p) and position (x):
xp ≥ h/4π, Equation 9
where x is position, p is momentum, and h is Planck’s constant.
The uncertainty derives in part from the fact that the very act of measuring a system inherently changes the system. (This contrasts with classical mechanics, where it is always, in principle, possible to measure the system without disturbing it.) Because of this change, you get a different answer if you measure the particle’s position first and momentum second, then if you did the reverse.
For another way to think about this, consider that in classical mechanics, position and momentum were separate variables, each of which would have a clearly defined value. Within quantum mechanics, both position and momentum are derived from the wavefunction of the system. Because of this, the wavefunction cannot simultaneously be in a pure momentum state and a pure position state at the same time.
A second uncertainty relationship involves energy (E) and time (t):
18
Et ≥ h/4π, Equation 10
where E is energy, t is time, and h is Planck’s constant.
According to this, for very short periods of time (t), the uncertainty in the energy (E) can be significant enough for a particle and antiparticle pair to appear and disappear spontaneously.
Probabilistic Outcomes
Both classical and relativistic mechanics are strictly deterministic—knowing the state of the system at one point in time and all the forces involved allows one to predict both the past and future of the system. In contrast, quantum mechanics is fundamentally probabilistic, where it is impossible to predict the outcome of quantum events, only the probability. According to the Copenhagen interpretation of quantum mechanics, when a system is observed or measured it will collapse into a definite state. The resultant state is randomly determined according to the mathematical probabilities as laid out by Max Born. This randomness is intrinsic to quantum mechanics.
Despite being a co-founder of quantum mechanics, Albert Einstein disliked this aspect of the theory. It disturbed him so greatly that he famously quipped, “God does not play dice.”
Quantum Tunneling
Quantum tunneling is a phenomenon where a particle can cross a barrier even though classically it lacks enough energy to do so. It plays a role in a wide variety of phenomena, such as nuclear fission, nuclear fusion, and certain chemical reactions. In these cases, tunneling allows the process to proceed where classically it would not be permitted. This unexpected behavior was predicted theoretically in 1927 and used to explain the radioactive decay rates of certain elements in 1928.
Related technology: quantum computing and scanning tunneling microscopy.
Quantum Entanglement
Quantum entanglement is by far the most mysterious yet most intriguing aspect of quantum mechanics. When particles are made to interact in certain ways, their properties are no longer fully independent. When one of the particles is measured, its entangled partners are simultaneously affected regardless of the distance between them. This phenomenon deeply troubled Einstein dismissing it as “spooky action at a distance.” Entanglement was hotly debated in the early days of quantum mechanics but was experimentally demonstrated (1970s-80s) and is now widely accepted.
Related technology: quantum computing, quantum radar, and quantum teleportation.
Bose-Einstein Condensate
All elementary particles have an intrinsic property called spin, a form of internal angular momentum. Compound particles and atomic nuclei also carry this property. All these particles can be classified into one of two categories based on their spin:
Fermions (particles with half-integer spin). Described by Enrico Fermi and Paul Dirac and consist of matter particles, such as protons, neutrons, and electrons.
Bosons (particles with integer spin). Described by Satyendra Nath Bose and Albert Einstein and consist of force-carrying particles, such as photons.
The main difference between these two types of particles is that fermions obey the Pauli exclusion principle, while bosons do not. This exclusion principle means that two fermions of the same type cannot occupy the same orbit unless they have opposite spins (i.e., they cannot share the same
19
quantum numbers). For electrons in atoms, this means that once an orbital is filled, subsequent electrons must move to higher energy orbitals.
The nuclei of atoms containing an even number of protons and neutrons behave as bosons. If a gas of such atoms is cooled to very close to absolute zero temperature, particles will behave as a Bose–Einstein condensate (BEC), also known as the fifth state of matter. In this state, a significant fraction of the atoms will share the lowest energy state and therefore lose their individuality and behave like one big collective super-atom. This behavior was predicted theoretically in 1924, but not experimentally demonstrated until 1995.
Related technology: Superconductivity, superfluidity, and lasers.
Conclusion
In this overview, we have provided a very brief outline of the origin of quantum mechanics and many of its principal features. In the remaining parts of this series, we will cover four key aspects of quantum mechanics in detail:
Part 2: Wave-particle Duality.
Part 3: Quantum Tunneling.
Part 4: Quantum Entanglement.
Part 5: Spin and the Quantum Atom.