The Physics of Energy, page 25
Box 7.1 The World Needs Quantum Mechanics
Quantum physics may seem rather obscure and largely irrelevant to everyday life. But without quantum mechanics, atomic structure, chemical interactions between molecules, materials that conduct electricity, energy production in stars, and even the existence of structure in the universe would not be possible in anything like their current form.
Without quantum mechanics, matter as we know it would not be stable. An electron circling a proton in a hydrogen atom would gradually radiate away energy until the atomic structure collapsed. Quantum mechanics allows the electron to remain in a lowest energy ground state from which no further energy can be removed.
The rich variety of chemical structures and interactions depends critically on the atomic orbital structure of the atoms within each molecule (§9). Water, DNA, and neurotransmitters would neither exist nor function in a purely classical world.
The production of energy in stars occurs through nuclear fusion, which is mediated by the strong nuclear force at short distances (§18). Without quantum physics, protons in the Sun would not overcome their electrostatic repulsion and come close enough to be affected by strong nuclear interactions until the temperature was nearly 1000 times greater (Box 7.2).
The existence of structure in our universe – galaxies, nebulae, stars, and planets – is believed to have arisen from microscopic quantum fluctuations in the early universe that expanded over time and seeded the inhomogeneous distribution of matter that we see currently in the universe (§21).
Quantum mechanics also plays an important role in separating energy scales in a way that helps scientists to systematically understand physical systems. Quantum mechanics allows us to ignore levels of structure deep within everyday objects (§9.2). The fact that nuclei, for example, are complex systems with internal degrees of freedom is irrelevant when we warm a cup of coffee in the microwave, because microwaves do not have enough energy to make a quantum excitation of the nucleus. In classical systems, any energy, no matter how small, can excite a system at any scale. This quantum separation of energy scales not only makes life easier for scientists, but also avoids structural problems such as the UV catastrophe (§22.2.2) that would arise if electromagnetism were governed by classical physics at short distance scales.
(Galaxy image credit: NASA and STSCI)
Box 7.2 Quantum Tunneling and Solar Energy
Quantum mechanics allows processes to occur that are forbidden by the rules of classical mechanics. Such processesare critical ingredients in some energy systems. While classically a particle can be kept in or out of a particular region of space by a sufficiently high potential barrier, the quantum wavefunction of a particle cannot be stopped in this way. The quantum wavefunction for a particle with fixed energy will in general be nonzero, though small, even in regions where the particle is classically forbidden to go. This can allow the quantum particle to tunnel through a classical barrier.
The core of the Sun contains protons (ionized hydrogen atoms) at high temperature and pressure. Coulomb repulsion between the protons acts to keep them apart. If the protons can get sufficiently close, however, nuclear forces take over and they can combine through the process of nuclear fusion into a more massive particle (a deuteron), giving off a great deal of energy. The potential barrier keeping the protons apart is so high that protons at the temperature of the solar core could not fuse without the benefit of quantum tunneling. The cartoon at right shows the Coulomb barrier, outside which the potential is repulsive, and, at shorter distances, a region of nuclear attraction. The kinetic energy E of the protons is not high enough to surmount the barrier classically, but the quantum wavefunction has a small amplitude inside the barrier. Thus quantum mechanics literally enables the Sun to shine as it does. The mechanism of tunneling is described in more detail in §15, and the application to solar fusion is described in §22.
7.1Motivation: The Double Slit Experiment
7.1.1 Classical and Quantum Physics
The laws of classical physics have a structure that fits naturally with human intuition for how the world operates. In classical mechanics, objects move through space, residing at a specific point in space at every moment of time. Classical mechanics and classical electromagnetism are deterministic: two identical systems prepared in identical initial configurations will evolve identically in the absence of outside interference under the time evolution given by Newton’s laws of mechanics or Maxwell’s equations.
Underlying the apparent classical world, however, the natural world operates according to a much more alien set of principles. At short distances the quantum laws of nature violate our classical intuition. Quantum particles do not have well-defined positions or momenta. And at the quantum level, the laws of physics do not uniquely predict the outcome of an experiment, even when the initial conditions are precisely fixed – the laws of quantum physics are non-deterministic (see Box 7.3).
Box 7.3 Radioactive Decay and Non-determinism in Quantum Physics
According to quantum mechanics, even if two systems are in exactly the same quantum state, the result of performing the same measurement on each of the two systems need not be the same. Suppose that a quantum system is prepared in a superposition of two states , and suppose that the measurement of an observable A – such as energy – on would give a value and on would give . Then according to Axiom 3, measuring A on a set of similarly prepared systems would give or randomly with probabilities and respectively. No additional probing of the system would allow an observer to obtain more certainty about this measurement. This is a radical departure from classical physics, where any system, in principle, can be described in terms of deterministic equations.
Radioactive decay provides a simple example of a random process in quantum mechanics – one that can be observed with the aid of a simple radiation detector such as a Geiger counter, which measures the passage of energetic charged particles. To be specific, consider a particular nucleus known as plutonium-238 (denoted ), which is used as an energy source on space exploration missions that extend far from the Sun (see Box 10.1). decays by emitting α particles. The α particles are stopped by other atoms in the plutonium, generating heat at a rate of ~500 W/kg. The average lifetime of a nucleus is about 127 years. The inset figure shows a histogram, in one-second bins, of Geiger counter hits made by α particles from the decay of roughly nuclei of . It is a random sequence of integers with a mean of approximately two per second (see Box 15.1 for more about this distribution).
In quantum mechanics, the nucleus has a small, but non-vanishing probability of being in a state consisting of a nucleus plus an α particle. The probability controls the rate at which decays are recorded by the Geiger counter. We study nuclear decays in more detail in §17. Similar radioactive processes are central to the operation of nuclear reactors (§19), are the source of a significant fraction of geothermal energy (§32), and underlie mechanisms for the measurement of Earth’s climate in past eras (§35).
One might imagine additional degrees of freedom within the nucleus whose detailed dynamics could be responsible for the occurrence and timing of nuclear decay in an entirely classical framework. These are examples of the hidden variable theories mentioned in §7.5.3. There is no experimental evidence for any such degrees of freedom, and they are not needed or present in the quantum description of the decay process. At this point our best explanation for the easily observed, but apparently random, phenomenon of radioactive decay lies in the non-deterministic world of quantum physics.
Another way in which classical and quantum physics differ is that many quantities, including energy, that can take arbitrary values in classical physics are restricted to discrete quantized values in quantum physics. One of the first places in which this became apparent historically is in the context of blackbody radiation (§6.3.3). From a classical point of view, one might expect that fluctuations of definite frequency ν in the electromagnetic field could hold an arbitrary, continuously variable amount of energy. Problems with the classical description of blackbody radiation (discussed further in §22) led the German physicist Max Planck to propose in 1900 that the amount of energy contained in light of a given frequency is quantized in units proportional to the frequency,
(7.1)
where h is Planck’s constant, which is often expressed as
(7.2)
This relation between energy and frequency of oscillation plays a central role in quantum physics. In fact, almost all of the physics of quantum mechanics can be captured by the statement that every quantum system is a linear system in which states of fixed energy oscillate with a frequency given by eq. (7.1); much of the rest of this chapter involves unpacking this statement.
There is no inconsistency between classical and quantum physics. At a fundamental level, physical systems are governed by quantum mechanics; classical physics emerges as an approximation to the quantum world that is highly accurate under everyday conditions.
In the remainder of this section we consider a set of parallel scenarios in which the differences between classical and quantum processes are highlighted.
Quantum Physics
Classical mechanics and electromagnetism are only approximate descriptions of a more fundamental physical reality described by quantum physics.
Quantum systems violate our classical intuition in many ways. Quantum particles do not have definite position or momentum, but rather are described by distributed wavefunctions.
Unlike the deterministic laws of classical physics, quantum processes are non-deterministic, so that even with perfect knowledge of initial conditions, the outcome of experiments involving quantum systems cannot be predicted with certainty.
Energy and other quantities that classically take arbitrary values in a continuous range are quantized and take only certain discrete values in quantum physics. For example, energy in an electromagnetic wave of angular frequency ω is quantized in units
This relation between energy and frequency is fundamental to quantum physics.
7.1.2 The Double Slit Experiment
Feynman described a pedagogical “thought experiment” in [40] to illustrate the weirdness of quantum mechanics and suggest ways to think about the quantum world. Feynman’s “experiment” actually consists of the same type of experiment repeated three times: first with “bullets” (i.e. classical particles), second with water waves, and third with electrons. The experimental apparatus is a screen with two holes (or slits) through which the bullets, waves, or electrons can pass. On one side is an appropriate source: a gun, an ocean, or a cathode ray tube. On the other side is a screen that can register the arrival of the bullets, wave energy, or electrons.
The scale of the apparatus is different for the three cases. For water waves, we imagine a calm harbor and a breakwater that is impenetrable to water except at two gaps separated by several times the gap length. Waves impinge on the breakwater with their crests parallel to the barrier; the wavelength between successive peaks is comparable to the size of the gaps. For bullets, we imagine an impenetrable shield with holes several times the size of a bullet. The bullets are sprayed out randomly by a poor marksman, covering the shield with a more or less uniform distribution of hits. For electrons, the screen is also impenetrable except for two slits; like the bullets, the electrons are sprayed over the screen more or less evenly. The size of the electron apparatus is discussed below.
First, consider the bullets. They move in straight lines and either go through a hole or are blocked. If they get through, they arrive on the observing screen in a location determined by simple geometry. Perhaps a few deflect slightly off the edges of the holes, but most end up behind the hole they passed through. The bullets arrive one by one and eventually build up patterns as shown in Figure 7.1. If only one hole is open, the pattern of bullets that hit the screen is centered behind the open hole (a or b), while if both holes are open the distribution of bullets corresponds to the sum of the patterns from the two individual holes (c).
Figure 7.1 Double slit experiment with bullets. When either hole is open, the bullets pile up behind the gap. A few ricochet off the edges (some of those trajectories are shown) and they end up at the edges of the pile. When both gaps are open (c) the pattern is the sum of the single hole patterns (a) and (b). Some bullets in flight are shown, but the many bullets stopped by the shield are not.
Next, consider water waves. First block one gap. As described in §4.5, the water waves diffract through the other gap and fan out in a circular pattern. The wave energy arrives continuously on the observing screen, forming a diffraction pattern similar to Figure 4.10(a). When the wavelength of the water waves is large compared to the size of the gap, the pattern is dominated by a single broad peak as in Figure 7.2(a). If we block the other gap instead, a similar pattern appears (Figure 7.2(b)). If, however, both gaps are open, quite a different pattern is observed on the screen: the circular waves coming from the two gaps interfere. When they meet crest-to-crest or trough-to-trough the intensity is four times as great as in the individual waves; when they meet crest-to-trough there is zero energy delivered to the screen. An interference pattern, shown in Figure 7.2(c), is formed on the observing screen. The pattern is most striking when the wavelength of the waves is comparable to the distance between the slits. This is what classical wave mechanics predicts.
Figure 7.2 Double slit experiment with water waves. When either gap is open ((a) or (b)) a broad diffraction pattern is forms on the observing screen. When both gaps are open (c) the waves coming from the two gaps give rise to an interference pattern (see Figure 4.8). The incident water waves are shown, but the waves reflected from the barrier are not.
To be more explicit, the amplitude of the wave pattern at the observing screen when the upper gap is open can be described by a time-dependent waveform , where x is the position on the observing screen. The energy density in a wave is proportional to the (time-averaged) square of the amplitude, , which (unlike the amplitude) is always positive. When the bottom gap is open, the wave pattern on the screen is described by a different waveform . And when both are open, the amplitudes of the waves add or superpose, . The energy density in the interference pattern, proportional to , is given by the sum of energies in the two separate patterns plus an interference term proportional to . The interference term can be either positive or negative and gives rise to the spatial oscillations observed in Figure 7.2(c).
Finally, consider electrons (see Figure 7.3). Remarkably, if the slits are small enough, the electrons diffract like waves, but arrive on the observing screen like particles. First, suppose one slit is blocked. The electrons arrive on the screen one by one. Eventually they build up a pattern peaked roughly behind the open slit. This could be explained by electrons deflecting off the edges of the slit, or it could be a diffraction pattern. If the other slit is blocked and the first opened, a similar pattern builds up behind the open gap. The surprise occurs when both slits are open: Although the electrons arrive one by one like bullets, they build up an interference pattern on the arrival screen like the one formed by water waves. Even if the electrons are fired so slowly that only one is in the apparatus at a time, still the interference pattern is formed. A single electron in the double slit experiment thus behaves like a wave that has passed through both slits and interfered with itself. Here, perhaps, is where quantum mechanics seems most alien to the classically trained mind. Our classical intuition suggests that a given electron either passes through one slit or through the other, but not both. Any attempt, however, to follow the electron trajectories in detail – by shining light on them, for example – disturbs the electrons’ motion so much that the interference pattern is destroyed.
Figure 7.3 Double slit experiment with electrons. When either slit is open ((a) or (b)) the electrons pile up behind the slit in a broad peak. Though they arrive one by one, when both slits are open (c) the distribution of electrons displays the same kind of interference pattern as water waves. Neither the incident electrons nor those reflected from the screen are shown.
The appearance of an interference pattern for a single electron suggests that the motion of the electron should be described in some way in terms of the propagation of a wave. Electron diffraction, however, is only observable on very small distance scales. When American physicists Clinton Davisson and Lester Germer first demonstrated electron diffraction in 1927, they found – as proposed earlier by the French physicist Louis de Broglie – that the spacing between slits needed to see the interference effect is inversely proportional to the electron momentum p. Since interference in the diffraction pattern is most easily seen when the wavelength and slit separation are comparable, as for water waves, this suggests that electrons are characterized by waveforms with wavelength inversely proportional to momentum. Experiments show that the wavelength of a matter wave, known as the de Broglie wavelength, is given by
(7.3)
The small size of Planck’s constant in SI units indicates that quantum effects are directly observable only at very small distances. For an electron traveling at 100 m/s, for example, the wavelength is m, so the size of slit needed to see interference for such an electron is quite small – but easily within experimental range. An image of an actual electron diffraction pattern is shown in Figure 7.4.
Figure 7.4 A diffraction pattern formed by electrons with energy 50 keV passing through two slits separated by 2.0 μm [41].
The quantum wave function description of particles is the only explanation that naturally fits the observed phenomenon of electron diffraction and other experimental data. Unlike classical physics, which in principle allows us to compute the exact trajectory of a particle given its initial position and velocity, quantum physics provides only a probabilistic prediction for where a given electron will impact the screen.
