Quantum entanglement is a physical phenomenon that occurs when pairs or groups of particles are generated, interact, or share spatial proximity in ways such that the quantum state of each particle cannot be described independently of the state of the others when the particles are separated by a large distance. Measurements of physical properties such as position, momentum and polarization, performed on entangled particles are found to be correlated. For example, if a pair of particles is generated in such a way that their total spin is known to be zero, one particle is found to have clockwise spin on a certain axis, the spin of the other particle, measured on the same axis, will be found to be counterclockwise, as is to be expected due to their entanglement. However, this behavior gives rise to paradoxical effects: any measurement of a property of a particle performs an irreversible collapse on that particle and will change the original quantum state. In the case of entangled particles, such a measurement will be on the entangled system as a whole.
Such phenomena were the subject of a 1935 paper by Albert Einstein, Boris Podolsky, Nathan Rosen, several papers by Erwin Schrödinger shortly thereafter, describing what came to be known as the EPR paradox. Einstein and others considered such behavior to be impossible, as it violated the local realism view of causality and argued that the accepted formulation of quantum mechanics must therefore be incomplete. However, the counterintuitive predictions of quantum mechanics were verified experimentally in tests where the polarization or spin of entangled particles were measured at separate locations, statistically violating Bell's inequality. In earlier tests it couldn't be ruled out that the test result at one point could have been subtly transmitted to the remote point, affecting the outcome at the second location; however so-called "loophole-free" Bell tests have been performed in which the locations were separated such that communications at the speed of light would have taken longer—in one case 10,000 times longer—than the interval between the measurements.
According to some interpretations of quantum mechanics, the effect of one measurement occurs instantly. Other interpretations which don't recognize wavefunction collapse dispute that there is any "effect" at all. However, all interpretations agree that entanglement produces correlation between the measurements and that the mutual information between the entangled particles can be exploited, but that any transmission of information at faster-than-light speeds is impossible. Quantum entanglement has been demonstrated experimentally with photons, electrons, molecules as large as buckyballs, small diamonds; the utilization of entanglement in communication and computation is a active area of research. The counterintuitive predictions of quantum mechanics about correlated systems were first discussed by Albert Einstein in 1935, in a joint paper with Boris Podolsky and Nathan Rosen. In this study, the three formulated the EPR paradox, a thought experiment that attempted to show that quantum mechanical theory was incomplete.
They wrote: "We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete."However, the three scientists did not coin the word entanglement, nor did they generalize the special properties of the state they considered. Following the EPR paper, Erwin Schrödinger wrote a letter to Einstein in German in which he used the word Verschränkung "to describe the correlations between two particles that interact and separate, as in the EPR experiment."Schrödinger shortly thereafter published a seminal paper defining and discussing the notion of "entanglement." In the paper he recognized the importance of the concept, stated: "I would not call one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought." Like Einstein, Schrödinger was dissatisfied with the concept of entanglement, because it seemed to violate the speed limit on the transmission of information implicit in the theory of relativity.
Einstein famously derided entanglement as "spukhafte Fernwirkung" or "spooky action at a distance." The EPR paper generated significant interest among physicists which inspired much discussion about the foundations of quantum mechanics, but produced little other published work. So, despite the interest, the weak point in EPR's argument was not discovered until 1964, when John Stewart Bell proved that one of their key assumptions, the principle of locality, as applied to the kind of hidden variables interpretation hoped for by EPR, was mathematically inconsistent with the predictions of quantum theory. Bell demonstrated an upper limit, seen in Bell's inequality, regarding the strength of correlations that can be produced in any theory obeying local realism, he showed that quantum theory predicts violations of this limit for certain entangled systems, his inequality is experimentally testable, there have been numerous relevant experiments, starting with the pioneering work of Stuart Freedman and John Clauser in 1972 and Alain Aspect's experiments in 1982, all of which have shown agreement with quantum mechanics rather than the principle of local realism.
Until each had left open at least one loophole by which it was possible to question the validity of the results. However, in 2015 an experiment was performed that closed both the detection and locality loopholes, was heralded as "loophole-free".
Schrödinger's cat is a thought experiment, sometimes described as a paradox, devised by Austrian physicist Erwin Schrödinger in 1935. It illustrates what he saw as the problem of the Copenhagen interpretation of quantum mechanics applied to everyday objects; the scenario presents a hypothetical cat that may be both alive and dead, a state known as a quantum superposition, as a result of being linked to a random subatomic event that may or may not occur. The thought experiment is often featured in theoretical discussions of the interpretations of quantum mechanics. Schrödinger coined the term Verschränkung in the course of developing the thought experiment. Schrödinger intended his thought experiment as a discussion of the EPR article—named after its authors Einstein and Rosen—in 1935; the EPR article highlighted the counterintuitive nature of quantum superpositions, in which a quantum system such as an atom or photon can exist as a combination of multiple states corresponding to different possible outcomes.
The prevailing theory, called the Copenhagen interpretation, says that a quantum system remains in superposition until it interacts with, or is observed by the external world. When this happens, the superposition collapses into one or another of the possible definite states; the EPR experiment shows that a system with multiple particles separated by large distances can be in such a superposition. Schrödinger and Einstein exchanged letters about Einstein's EPR article, in the course of which Einstein pointed out that the state of an unstable keg of gunpowder will, after a while, contain a superposition of both exploded and unexploded states. To further illustrate, Schrödinger described how one could, in principle, create a superposition in a large-scale system by making it dependent on a quantum particle, in a superposition, he proposed a scenario with a cat in a locked steel chamber, wherein the cat's life or death depended on the state of a radioactive atom, whether it had decayed and emitted radiation or not.
According to Schrödinger, the Copenhagen interpretation implies that the cat remains both alive and dead until the state has been observed. Schrödinger did not wish to promote the idea of dead-and-alive cats as a serious possibility. However, since Schrödinger's time, other interpretations of the mathematics of quantum mechanics have been advanced by physicists, some of which regard the "alive and dead" cat superposition as quite real. Intended as a critique of the Copenhagen interpretation, the Schrödinger's cat thought experiment remains a defining touchstone for modern interpretations of quantum mechanics. Physicists use the way each interpretation deals with Schrödinger's cat as a way of illustrating and comparing the particular features and weaknesses of each interpretation. Schrödinger wrote: One can set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device: in a Geiger counter, there is a tiny bit of radioactive substance, so small, that in the course of the hour one of the atoms decays, but with equal probability none.
If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The first atomic decay would have poisoned it; the psi-function of the entire system would express this by having in it the living and dead cat mixed or smeared out in equal parts. It is typical of these cases that an indeterminacy restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can be resolved by direct observation; that prevents us from so naively accepting as valid a "blurred model" for representing reality. In itself, it would not embody anything unclear or contradictory. There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks. Schrödinger's famous thought experiment poses the question, "when does a quantum system stop existing as a superposition of states and become one or the other?" If the cat survives, it remembers only being alive. But explanations of the EPR experiments that are consistent with standard microscopic quantum mechanics require that macroscopic objects, such as cats and notebooks, do not always have unique classical descriptions.
The thought experiment illustrates this apparent paradox. Our intuition says that no observer can be in a mixture of states—yet the cat, it seems from the thought experiment, can be such a mixture. Is the cat required to be an observer, or does its existence in a single well-defined classical state require another external observer? Each alternative seemed absurd to Einstein, impressed by the ability of the thought experiment to highlight these issues. In a letter to Schrödinger dated 1950, he wrote: You are the only contemporary physicist, besides Laue, who sees that one cannot get around the assumption of reality, if only one is honest. Most of them do not see what sort of risky game they are playing with reality—reality as something independent of what is experimentally established, their interpretation is, refuted most elegantly by your system of radioactive atom + amplifier + charge of gun powder + cat in a box, in which the p
Quantum teleportation is a process by which quantum information can be transmitted from one location to another, with the help of classical communication and shared quantum entanglement between the sending and receiving location. Because it depends on classical communication, which can proceed no faster than the speed of light, it cannot be used for faster-than-light transport or communication of classical bits. While it has proven possible to teleport one or more qubits of information between two quanta, this has not yet been achieved between anything larger than molecules. Although the name is inspired by the teleportation used in fiction, quantum teleportation is limited to the transfer of information rather than matter itself. Quantum teleportation is not a form of transportation, but of communication: it provides a way of transporting a qubit from one location to another without having to move a physical particle along with it; the term was coined by physicist Charles Bennett. The seminal paper first expounding the idea of quantum teleportation was published by C. H. Bennett, G. Brassard, C.
Crépeau, R. Jozsa, A. Peres, W. K. Wootters in 1993. Quantum teleportation was first realized in single photons being demonstrated in various material systems such as atoms, ions and superconducting circuits; the latest reported record distance for quantum teleportation is 1,400 km by the group of Jian-Wei Pan using the Micius satellite for space-based quantum teleportation. In matters relating to quantum or classical information theory, it is convenient to work with the simplest possible unit of information, the two-state system. In classical information, this is a bit represented using one or zero; the quantum analog of a bit is qubit. Qubits encode a type of information, called quantum information, which differs from "classical" information. For example, quantum information can be neither destroyed. Quantum teleportation provides a mechanism of moving a qubit from one location to another, without having to physically transport the underlying particle to which that qubit is attached. Much like the invention of the telegraph allowed classical bits to be transported at high speed across continents, quantum teleportation holds the promise that one day, qubits could be moved likewise.
As of 2015, the quantum states of single photons, photon modes, single atoms, atomic ensembles, defect centers in solids, single electrons, superconducting circuits have been employed as information bearers. The movement of qubits does not require the movement of "things" any more than communication over the internet does: no quantum object needs to be transported, but it is necessary to communicate two classical bits per teleported qubit from the sender to the receiver; the actual teleportation protocol requires that an entangled quantum state or Bell state be created, its two parts shared between two locations. In essence, a certain kind of quantum channel between two sites must be established first, before a qubit can be moved. Teleportation requires a classical information channel to be established, as two classical bits must be transmitted to accompany each qubit; the reason for this is that the results of the measurements must be communicated, this must be done over ordinary classical communication channels.
The need for such classical channels may, at first, seem disappointing. What's more, Bell states are most shared using photons from lasers, so teleportation could be done, in principle, through open space, i.e. without the need to send the light through cables or optical fibers. The quantum states of single atoms have been teleported. An atom consists of several parts: the qubits in the electronic state or electron shells surrounding the atomic nucleus, the qubits in the nucleus itself, the electrons and neutrons making up the atom. Physicists have teleported the qubits encoded in the electronic state of atoms, it is therefore inaccurate to say "an atom has been teleported". The quantum state of an atom has. Thus, performing this kind of teleportation requires a stock of atoms at the receiving site, available for having qubits imprinted on them; the importance of teleporting nuclear state is unclear: nuclear state does affect the atom, e.g. in hyperfine splitting, but whether such state would need to be teleported in some futuristic "practical" application is debatable.
An important aspect of quantum information theory is entanglement, which imposes statistical correlations between otherwise distinct physical systems. These correlations hold when measurements are chosen and performed independently, out of causal contact from one another, as verified in Bell test experiments. Thus, an observation resulting from a measurement choice made at one point in spacetime seems to instantaneously affect outcomes in another region though light hasn't yet had time to travel the distance; however such correlations can never be used to transmit any information faster than the speed of light, a statement encapsulated in the no-communication theorem. Thus, teleportation, as a whole, can never be superluminal, as a qubit cannot be reconstructed until the accompanying classical information arrives. Understanding quantum teleportation requires a good grounding in finite-dimensional linear algebra, Hilbert spaces and projection matrixes. A qub
In physics, interference is a phenomenon in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive interference result from the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency. Interference effects can be observed with all types of waves, for example, radio, surface water waves, gravity waves, or matter waves; the resulting images or graphs are called interferograms. The principle of superposition of waves states that when two or more propagating waves of same type are incident on the same point, the resultant amplitude at that point is equal to the vector sum of the amplitudes of the individual waves. If a crest of a wave meets a crest of another wave of the same frequency at the same point the amplitude is the sum of the individual amplitudes—this is constructive interference. If a crest of one wave meets a trough of another wave the amplitude is equal to the difference in the individual amplitudes—this is known as destructive interference.
Constructive interference occurs when the phase difference between the waves is an multiple of π, whereas destructive interference occurs when the difference is an odd multiple of π. If the difference between the phases is intermediate between these two extremes the magnitude of the displacement of the summed waves lies between the minimum and maximum values. Consider, for example, what happens when two identical stones are dropped into a still pool of water at different locations; each stone generates a circular wave propagating outwards from the point where the stone was dropped. When the two waves overlap, the net displacement at a particular point is the sum of the displacements of the individual waves. At some points, these will be in phase, will produce a maximum displacement. In other places, the waves will be in anti-phase, there will be no net displacement at these points. Thus, parts of the surface will be stationary—these are seen in the figure above and to the right as stationary blue-green lines radiating from the centre.
Interference of light is a common phenomenon that can be explained classically by the superposition of waves, however a deeper understanding of light interference requires knowledge of wave-particle duality of light, due to quantum mechanics. Prime examples of light interference are the famous double-slit experiment, laser speckle, anti-reflective coatings and interferometers. Traditionally the classical wave model is taught as a basis for understanding optical interference, based on the Huygens–Fresnel principle; the above can be demonstrated in one dimension by deriving the formula for the sum of two waves. The equation for the amplitude of a sinusoidal wave traveling to the right along the x-axis is W 1 = A cos where A is the peak amplitude, k = 2 π / λ is the wavenumber and ω = 2 π f is the angular frequency of the wave. Suppose a second wave of the same frequency and amplitude but with a different phase is traveling to the right W 2 = A cos where φ is the phase difference between the waves in radians.
The two waves will superpose and add: the sum of the two waves is W 1 + W 2 = A. Using the trigonometric identity for the sum of two cosines: cos a + cos b = 2 cos cos , this can be written W 1 + W 2 = 2 A cos cos ; this represents a wave at the original frequency, traveling to the right like the components, whose amplitude is proportional to the cosine of φ / 2. Constructive interference: If the phase difference is an multiple of π: φ = …, − 4 π, − 2 π, 0, 2 π, 4 π, …
The Aharonov–Bohm effect, sometimes called the Ehrenberg–Siday–Aharonov–Bohm effect, is a quantum mechanical phenomenon in which an electrically charged particle is affected by an electromagnetic potential, despite being confined to a region in which both the magnetic field B and electric field E are zero. The underlying mechanism is the coupling of the electromagnetic potential with the complex phase of a charged particle's wave function, the Aharonov–Bohm effect is accordingly illustrated by interference experiments; the most described case, sometimes called the Aharonov–Bohm solenoid effect, takes place when the wave function of a charged particle passing around a long solenoid experiences a phase shift as a result of the enclosed magnetic field, despite the magnetic field being negligible in the region through which the particle passes and the particle's wavefunction being negligible inside the solenoid. This phase shift has been observed experimentally. There are magnetic Aharonov–Bohm effects on bound energies and scattering cross sections, but these cases have not been experimentally tested.
An electric Aharonov–Bohm phenomenon was predicted, in which a charged particle is affected by regions with different electrical potentials but zero electric field, but this has no experimental confirmation yet. A separate "molecular" Aharonov–Bohm effect was proposed for nuclear motion in multiply connected regions, but this has been argued to be a different kind of geometric phase as it is "neither nonlocal nor topological", depending only on local quantities along the nuclear path. Werner Ehrenberg and Raymond E. Siday first predicted the effect in 1949. Yakir Aharonov and David Bohm published their analysis in 1959. After publication of the 1959 paper, Bohm was informed of Ehrenberg and Siday's work, acknowledged and credited in Bohm and Aharonov's subsequent 1961 paper; the effect was confirmed experimentally, with a large error, while Bohm was still alive. By the time the error was down to a respectable value, Bohm had died. In the 18th and 19th centuries, physics was dominated by Newtonian dynamics, with its emphasis on forces.
Electromagnetic phenomena were elucidated by a series of experiments involving the measurement of forces between charges and magnets in various configurations. A description arose according to which charges and magnets acted as local sources of propagating force fields, which acted on other charges and currents locally through the Lorentz force law. In this framework, because one of the observed properties of the electric field was that it was irrotational, one of the observed properties of the magnetic field was that it was divergenceless, it was possible to express an electrostatic field as the gradient of a scalar potential and a stationary magnetic field as the curl of a vector potential; the language of potentials generalised seamlessly to the dynamic case but, since all physical effects were describable in terms of the fields which were the derivatives of the potentials, potentials were not uniquely determined by physical effects: potentials were only defined up to an arbitrary additive constant electrostatic potential and an irrotational stationary magnetic vector potential.
The Aharonov–Bohm effect is important conceptually because it bears on three issues apparent in the recasting of classical electromagnetic theory as a gauge theory, which before the advent of quantum mechanics could be argued to be a mathematical reformulation with no physical consequences. The Aharonov–Bohm thought experiments and their experimental realization imply that the issues were not just philosophical; the three issues are: whether potentials are "physical" or just a convenient tool for calculating force fields. Because of reasons like these, the Aharonov–Bohm effect was chosen by the New Scientist magazine as one of the "seven wonders of the quantum world", it is argued that Aharonov–Bohm effect illustrates the physicality of electromagnetic potentials, Φ and A, in quantum mechanics. Classically it was possible to argue that only the electromagnetic fields are physical, while the electromagnetic potentials are purely mathematical constructs, that due to gauge freedom aren't unique for a given electromagnetic field.
However, Vaidman has challenged this interpretation by showing that the AB effect can be explained without the use of potentials so long as one gives a full quantum mechanical treatment to the source charges that produce the electromagnetic field. According to this view, the potential in quantum mechanics is just as physical as it was classically. Aharonov and Rohrlich responded that the effect may be due to a local gauge potential or due to non-local gauge-invariant fields. Two papers published in the journal in 2017 Physical Review A have demonstrated a quantum mechanical solution for the system, their analysis shows that the phase shift can be viewed as generated by solenoid's vector potential acting on the electron or the electron's vector potential acting on the solenoid or the electron and solenoid currents acting on the quantized vector potential. The Aharonov–Bohm effect illustrates that the Lagrangian approach to dynamics, based on energies, is not just a computational aid to the Newtonian approach, based on forces.
Thus the Aharonov–Bohm effect validates the view that forces are an incomplete way to formulate physics, potential energies must be used instead. In fact R
Introduction to quantum mechanics
Quantum mechanics is the science of the small. It explains the behavior of matter and its interactions with energy on the scale of atoms and subatomic particles. By contrast, classical physics explains matter and energy only on a scale familiar to human experience, including the behavior of astronomical bodies such as the Moon. Classical physics is still used in much of modern technology. However, towards the end of the 19th century, scientists discovered phenomena in both the large and the small worlds that classical physics could not explain; the desire to resolve inconsistencies between observed phenomena and classical theory led to two major revolutions in physics that created a shift in the original scientific paradigm: the theory of relativity and the development of quantum mechanics. This article describes how physicists discovered the limitations of classical physics and developed the main concepts of the quantum theory that replaced it in the early decades of the 20th century, it describes these concepts in the order in which they were first discovered.
For a more complete history of the subject, see History of quantum mechanics. Light behaves in other aspects like waves. Matter—the "stuff" of the universe consisting of particles such as electrons and atoms—exhibits wavelike behavior too; some light sources, such as neon lights, give off only certain frequencies of light. Quantum mechanics shows that light, along with all other forms of electromagnetic radiation, comes in discrete units, called photons, predicts its energies and spectral intensities. A single photon is a quantum, or smallest observable amount, of the electromagnetic field because a partial photon has never been observed. More broadly, quantum mechanics shows that many quantities, such as angular momentum, that appeared continuous in the zoomed-out view of classical mechanics, turn out to be quantized. Angular momentum is required to take on one of a set of discrete allowable values, since the gap between these values is so minute, the discontinuity is only apparent at the atomic level.
Many aspects of quantum mechanics are counterintuitive and can seem paradoxical, because they describe behavior quite different from that seen at larger scales. In the words of quantum physicist Richard Feynman, quantum mechanics deals with "nature as She is – absurd". For example, the uncertainty principle of quantum mechanics means that the more one pins down one measurement, the less accurate another measurement pertaining to the same particle must become. Thermal radiation is electromagnetic radiation emitted from the surface of an object due to the object's internal energy. If an object is heated sufficiently, it starts to emit light at the red end of the spectrum, as it becomes red hot. Heating it further causes the color to change from red to yellow and blue, as it emits light at shorter wavelengths. A perfect emitter is a perfect absorber: when it is cold, such an object looks black, because it absorbs all the light that falls on it and emits none. An ideal thermal emitter is known as a black body, the radiation it emits is called black-body radiation.
In the late 19th century, thermal radiation had been well characterized experimentally. However, classical physics led to the Rayleigh–Jeans law, which, as shown in the figure, agrees with experimental results well at low frequencies, but disagrees at high frequencies. Physicists searched for a single theory; the first model, able to explain the full spectrum of thermal radiation was put forward by Max Planck in 1900. He proposed a mathematical model in which the thermal radiation was in equilibrium with a set of harmonic oscillators. To reproduce the experimental results, he had to assume that each oscillator emitted an integer number of units of energy at its single characteristic frequency, rather than being able to emit any arbitrary amount of energy. In other words, the energy emitted by an oscillator was quantized; the quantum of energy for each oscillator, according to Planck, was proportional to the frequency of the oscillator. The Planck constant written as h, has the value of 6.63×10−34 J s.
So, the energy E of an oscillator of frequency f is given by E = n h f, where n = 1, 2, 3, … To change the color of such a radiating body, it is necessary to change its temperature. Planck's law explains why: increasing the temperature of a body allows it to emit more energy overall, means that a larger proportion of the energy is towards the violet end of the spectrum. Planck's law was the first quantum theory in physics, Planck won the Nobel Prize in 1918 "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta". At the time, Planck's view was that quantization was purely a heuristic mathematical construct, rather than a fundamental change in our understanding of the world. In 1905, Albert Einstein took an extra step, he suggested that quantization was not just a mathematical construct, but that the energy in a beam of light occurs in individual packets, which are now called photons. The energy of a single photon is given by its frequency multiplied by Planck's constant: E = h f For centuri
The Franck–Hertz experiment was the first electrical measurement to show the quantum nature of atoms, thus "transformed our understanding of the world". It was presented on April 24, 1914, to the German Physical Society in a paper by James Franck and Gustav Hertz. Franck and Hertz had designed a vacuum tube for studying energetic electrons that flew through a thin vapor of mercury atoms, they discovered that, when an electron collided with a mercury atom, it could lose only a specific quantity of its kinetic energy before flying away. This energy loss corresponds to decelerating the electron from a speed of about 1.3 million meters per second to zero. A faster electron does not decelerate after a collision, but loses the same amount of its kinetic energy. Slower electrons bounce off mercury atoms without losing any significant speed or kinetic energy; these experimental results proved to be consistent with the Bohr model for atoms, proposed the previous year by Niels Bohr. The Bohr model was of the electron shell model of atoms.
Its key feature was that an electron inside an atom occupies one of the atom's "quantum energy levels". Before the collision, an electron inside the mercury atom occupies its lowest available energy level. After the collision, the electron inside occupies a higher energy level with 4.9 electron volts more energy. This means. There were no intermediate possibilities in Bohr's quantum model; this feature was "revolutionary" because it was inconsistent with the expectation that an electron could be bound to an atom's nucleus by any amount of energy. In a second paper presented in May 1914, Franck and Hertz reported on the light emission by the mercury atoms that had absorbed energy from collisions, they showed that the wavelength of this ultraviolet light corresponded to the 4.9 eV of energy that the flying electron had lost. The relationship of energy and wavelength had been predicted by Bohr. After a presentation of these results by Franck a few years Albert Einstein is said to have remarked, "It's so lovely it makes you cry."On December 10, 1926, Franck and Hertz were awarded the 1925 Nobel Prize in Physics "for their discovery of the laws governing the impact of an electron upon an atom".
Franck and Hertz's original experiment used a heated vacuum tube containing a drop of mercury. A contemporary Franck–Hertz tube is shown in the photograph, it is fitted with three electrodes: an hot cathode. The grid's voltage is positive relative to the cathode, so that electrons emitted from the hot cathode are drawn to it; the electric current measured in the experiment is due to electrons that pass through the grid and reach the anode. The anode's electric potential is negative relative to the grid, so that electrons that reach the anode have at least a corresponding amount of kinetic energy after passing the grid; the graphs published by Franck and Hertz show the dependence of the electric current flowing out of the anode upon the electric potential between the grid and the cathode. At low potential differences—up to 4.9 volts—the current through the tube increased with increasing potential difference. This behavior is typical of true vacuum tubes. At 4.9 volts the current drops almost back to zero.
The current increases once again as the voltage is increased further, until 9.8 volts is reached. At 9.8 volts a similar sharp drop is observed. While it isn't evident in the original measurements of the figure, this series of dips in current at 4.9 volt increments continues to potentials of at least 70 volts. Franck and Hertz noted in their first paper that the 4.9 eV characteristic energy of their experiment corresponded well to one of the wavelengths of light emitted by mercury atoms in gas discharges. They were using a quantum relationship between the energy of excitation and the corresponding wavelength of light, which they broadly attributed to Johannes Stark and to Arnold Sommerfeld; the same relationship was incorporated in Einstein's 1905 photon theory of the photoelectric effect. In a second paper and Hertz reported the optical emission from their tubes, which emitted light with a single prominent wavelength 254 nm; the figure at the right shows the spectrum of a Franck–Hertz tube. For reference, the figure shows the spectrum for a mercury gas discharge light, which emits light at several wavelengths besides 254 nm.
The figure is based on the original spectra published by Franck and Hertz in 1914. The fact that the Franck–Hertz tube emitted just the single wavelength, corresponding nearly to the voltage period they had measured, was important. Franck and Hertz explained their experiment in terms of elastic and inelastic collisions between the electrons and the mercury atoms. Moving electrons collide elastically with the mercury atoms; this means that the direction in which the electron is moving is altered by the collision, but its speed is unchanged. An elastic collision is illustrated in the figure, where the length of the arrow indicates the electron's speed; the mercury atom is unaffected by the collision because it is about four hundred thousand times more massive than an electron. When the speed of the electron exc