Introduction to quantum mechanics

Maxwell's unification of electricity, magnetism, and light in the 1880s led to experiments on the interaction of light and matter. Some of these experiments had aspects which could not be explained. Quantum mechanics emerged in the early part of the 20th century from efforts to explain these results.[5]

Evidence of quanta from the photoelectric effect

Main article: Photoelectric effect

The seeds of the quantum revolution appear in the discovery by JJ Thomson in 1897 that cathode rays were not continuous but "corpuscles" now called electrons. Electrons had been named just six years earlier as part of the emerging theory of atoms. In 1900, Max Planck, a conservative physicist unconvinced by the atomic theory, discovered that he needed discrete entities like atoms or electrons to explain blackbody radiation.[6]

Blackbody radiation intensity vs color and temperature. The rainbow bar represents visible light; 5000K objects are "white hot" by mixing differing colors of visible light. To the right is the invisible infrared. Classical theory (black curve for 5000K) fails; the other curves are correct predicted by quantum theories.

Hot objects radiate heat; very hot objects – red hot, white hot objects – all look similar when heated to the same temperature. This temperature dependent "look" results from a common curve of light intensity at different frequencies (colors). The common curve is called blackbody radiation. The lowest frequencies are invisible heat rays – infrared light. White hot objects have intensity across many colors in the visible range. Continuous wave theories of light and matter cannot explain the blackbody radiation curve. Planck spread the heat energy among individual "oscillators" of an undefined character but with discrete energy capacity: the blackbody radiation behavior was then predicted by this model.

At the time, electrons, atoms, and discrete oscillators were all exotic ideas to explain exotic phenomena. But in 1905 Albert Einstein proposed that light was also corpuscular, consisting of "energy quanta", seemingly in contradiction to the established science of light as a continuous wave, stretching back a hundred years to Thomas Young's work on diffraction.

His revolutionary proposal started by reanalyzing Planck blackbody theory, arriving at the same conclusions by using the new "energy quanta". Einstein then showed how energy quanta connected to JJ Thomson's electron. In 1902, Philipp Lenard directed light from an arc lamp onto freshly cleaned metal plates housed in an evacuated glass tube. He measured the electric current coming off the metal plate, for higher and lower intensity of light and for different metals. This is the photoelectric effect. Lenard showed that amount of current – the number of electrons – depended on the intensity of the light, but that the velocity of these electrons did not depend on intensity. The continuous wave theories of the time would predict that more light intensity would accelerate the same amount of current to higher velocity contrary to experiment. Einstein's energy quanta explained the volume increase: one electron is ejected for each quantum: more quanta mean more electrons.[6]^: 23

Einstein then predicted that the electron velocity would increase in direct proportion to the light frequency above a fixed value that depended upon the metal. Here the idea is that energy in energy-quanta depends upon the light frequency; the energy transferred to the electron comes in proportion to the light frequency. The type of metal gives a barrier, the fixed value, that the electrons must climb over to exit their atoms, to be emitted from the metal surface and be measured.

Ten years elapsed before Millikan's definitive experiment[7] verified Einstein's prediction. During that time many scientists rejected the revolutionary idea of quanta.[8] But Planck's and Einstein's concept was in the air and soon affected other theories.

Throughout the first and the modern era of quantum mechanics the concept that classical mechanics must be valid macroscopically constrained possible quantum models. This concept was formalized by Bohr in 1923 as the correspondence principle. It requires quantum theory to converge to classical limits.[16]^: 29

One principal "paradox" is the apparent inconsistency between Newton's laws and quantum mechanics which can be explained using Ehrenfest's theorem, which shows that the average values obtained from quantum mechanics (e.g. position and momentum) obey classical laws.[17] However, Ehrenfest's theorem is far from capable of explaining all the counterintuitive phenomena (quantum weirdness) that have been observed, but rather is a mathematical expression of the correspondence principle.

The concept of wave–particle duality says that neither the classical concept of "particle" nor of "wave" can fully describe the behavior of quantum-scale objects, either photons or matter. Wave–particle duality is an example of the principle of complementarity in quantum physics.[18][19][20][21][22] An elegant example of wave-particle duality is the double-slit experiment.

The diffraction pattern produced when light is shone through one slit (top) and the interference pattern produced by two slits (bottom). Both patterns show oscillations due to the wave nature of light. The double slit pattern is more dramatic.

In the double-slit experiment, as originally performed by Thomas Young in 1803,[23] and then Augustin Fresnel a decade later,[23] a beam of light is directed through two narrow, closely spaced slits, producing an interference pattern of light and dark bands on a screen. If one of the slits is covered up, one might naïvely expect that the intensity of the fringes due to interference would be halved everywhere. In fact, a much simpler pattern is seen, a diffraction pattern diametrically opposite the open slit. The same behavior can be demonstrated in water waves, and so the double-slit experiment was seen as a demonstration of the wave nature of light.

Variations of the double-slit experiment have been performed using electrons, atoms, and even large molecules,[24][25] and the same type of interference pattern is seen. Thus it has been demonstrated that all matter possesses both particle and wave characteristics.

Even if the source intensity is turned down, so that only one particle (e.g. photon or electron) is passing through the apparatus at a time, the same interference pattern develops over time. The quantum particle acts as a wave when passing through the double slits, but as a particle when it is detected. This is a typical feature of quantum complementarity: a quantum particle acts as a wave in an experiment to measure its wave-like properties, and like a particle in an experiment to measure its particle-like properties. The point on the detector screen where any individual particle shows up is the result of a random process. However, the distribution pattern of many individual particles mimics the diffraction pattern produced by waves.

Werner Heisenberg at the age of 26. Heisenberg won the Nobel Prize in Physics in 1932 for the work he did at around this time.[26]

Suppose it is desired to measure the position and speed of an object—for example, a car going through a radar speed trap. It can be assumed that the car has a definite position and speed at a particular moment in time. How accurately these values can be measured depends on the quality of the measuring equipment. If the precision of the measuring equipment is improved, it provides a result closer to the true value. It might be assumed that the speed of the car and its position could be operationally defined and measured simultaneously, as precisely as might be desired.

In 1927, Heisenberg proved that this last assumption is not correct.[27] Quantum mechanics shows that certain pairs of physical properties, for example, position and speed, cannot be simultaneously measured, nor defined in operational terms, to arbitrary precision: the more precisely one property is measured, or defined in operational terms, the less precisely can the other. This statement is known as the uncertainty principle. The uncertainty principle is not only a statement about the accuracy of our measuring equipment but, more deeply, is about the conceptual nature of the measured quantities—the assumption that the car had simultaneously defined position and speed does not work in quantum mechanics. On a scale of cars and people, these uncertainties are negligible, but when dealing with atoms and electrons they become critical.[28]

Heisenberg gave, as an illustration, the measurement of the position and momentum of an electron using a photon of light. In measuring the electron's position, the higher the frequency of the photon, the more accurate is the measurement of the position of the impact of the photon with the electron, but the greater is the disturbance of the electron. This is because from the impact with the photon, the electron absorbs a random amount of energy, rendering the measurement obtained of its momentum increasingly uncertain, for one is necessarily measuring its post-impact disturbed momentum from the collision products and not its original momentum (momentum which should be simultaneously measured with position). With a photon of lower frequency, the disturbance (and hence uncertainty) in the momentum is less, but so is the accuracy of the measurement of the position of the impact.[29]

At the heart of the uncertainty principle is a fact that for any mathematical analysis in the position and velocity domains, achieving a sharper (more precise) curve in the position domain can only be done at the expense of a more gradual (less precise) curve in the speed domain, and vice versa. More sharpness in the position domain requires contributions from more frequencies in the speed domain to create the narrower curve, and vice versa. It is a fundamental tradeoff inherent in any such related or complementary measurements, but is only really noticeable at the smallest (Planck) scale, near the size of elementary particles.

The uncertainty principle shows mathematically that the product of the uncertainty in the position and momentum of a particle (momentum is velocity multiplied by mass) could never be less than a certain value, and that this value is related to Planck's constant.

Wave function collapse means that a measurement has forced or converted a quantum (probabilistic or potential) state into a definite measured value. This phenomenon is only seen in quantum mechanics rather than classical mechanics.

For example, before a photon actually "shows up" on a detection screen it can be described only with a set of probabilities for where it might show up. When it does appear, for instance in the CCD of an electronic camera, the time and space where it interacted with the device are known within very tight limits. However, the photon has disappeared in the process of being captured (measured), and its quantum wave function has disappeared with it. In its place, some macroscopic physical change in the detection screen has appeared, e.g., an exposed spot in a sheet of photographic film, or a change in electric potential in some cell of a CCD.

Because of the uncertainty principle, statements about both the position and momentum of particles can assign only a probability that the position or momentum has some numerical value. Therefore, it is necessary to formulate clearly the difference between the state of something indeterminate, such as an electron in a probability cloud, and the state of something having a definite value. When an object can definitely be "pinned-down" in some respect, it is said to possess an eigenstate.

In the Stern–Gerlach experiment discussed above, the spin of the atom about the vertical axis has two eigenstates: up and down. Before measuring it, we can only say that any individual atom has an equal probability of being found to have spin up or spin down. The measurement process causes the wave function to collapse into one of the two states.

The eigenstates of spin about the vertical axis are not simultaneously eigenstates of spin about the horizontal axis, so this atom has an equal probability of being found to have either value of spin about the horizontal axis. As described in the section above, measuring the spin about the horizontal axis can allow an atom that was spun up to spin down: measuring its spin about the horizontal axis collapses its wave function into one of the eigenstates of this measurement, which means it is no longer in an eigenstate of spin about the vertical axis, so can take either value.

Wolfgang Pauli

In 1924, Wolfgang Pauli proposed a new quantum degree of freedom (or quantum number), with two possible values, to resolve inconsistencies between observed molecular spectra and the predictions of quantum mechanics. In particular, the spectrum of atomic hydrogen had a doublet, or pair of lines differing by a small amount, where only one line was expected. Pauli formulated his exclusion principle, stating, "There cannot exist an atom in such a quantum state that two electrons within [it] have the same set of quantum numbers."[30]

A year later, Uhlenbeck and Goudsmit identified Pauli's new degree of freedom with the property called spin whose effects were observed in the Stern–Gerlach experiment.

Paul Dirac (1902–1984)

In 1928, Paul Dirac extended the Pauli equation, which described spinning electrons, to account for special relativity. The result was a theory that dealt properly with events, such as the speed at which an electron orbits the nucleus, occurring at a substantial fraction of the speed of light. By using the simplest electromagnetic interaction, Dirac was able to predict the value of the magnetic moment associated with the electron's spin and found the experimentally observed value, which was too large to be that of a spinning charged sphere governed by classical physics. He was able to solve for the spectral lines of the hydrogen atom and to reproduce from physical first principles Sommerfeld's successful formula for the fine structure of the hydrogen spectrum.

Dirac's equations sometimes yielded a negative value for energy, for which he proposed a novel solution: he posited the existence of an antielectron and a dynamical vacuum. This led to the many-particle quantum field theory.

Superposition of two quantum characteristics, and two resolution possibilities

The Pauli exclusion principle says that two electrons in one system cannot be in the same state. Nature leaves open the possibility, however, that two electrons can have both states "superimposed" over each of them. Recall that the wave functions that emerge simultaneously from the double slits arrive at the detection screen in a state of superposition. Nothing is certain until the superimposed waveforms "collapse". At that instant, an electron shows up somewhere in accordance with the probability that is the square of the absolute value of the sum of the complex-valued amplitudes of the two superimposed waveforms. The situation there is already very abstract. A concrete way of thinking about entangled photons, photons in which two contrary states are superimposed on each of them in the same event, is as follows:

Imagine that we have two color-coded states of photons: one state labeled blue and another state labeled red. Let the superposition of the red and the blue state appear (in imagination) as a purple state. We consider a case in which two photons are produced as the result of one single atomic event. Perhaps they are produced by the excitation of a crystal that characteristically absorbs a photon of a certain frequency and emits two photons of half the original frequency. In this case, the photons are interconnected via their shared origin in a single atomic event. This setup results in superimposed states of the photons. So the two photons come out purple. If the experimenter now performs some experiment that determines whether one of the photons is either blue or red, then that experiment changes the photon involved from one having a superposition of blue and red characteristics to a photon that has only one of those characteristics. The problem that Einstein had with such an imagined situation was that if one of these photons had been kept bouncing between mirrors in a laboratory on earth, and the other one had traveled halfway to the nearest star when its twin was made to reveal itself as either blue or red, that meant that the distant photon now had to lose its purple status too. So whenever it might be investigated after its twin had been measured, it would necessarily show up in the opposite state to whatever its twin had revealed.

In trying to show that quantum mechanics was not a complete theory, Einstein started with the theory's prediction that two or more particles that have interacted in the past can appear strongly correlated when their various properties are later measured. He sought to explain this seeming interaction classically, through their common past, and preferably not by some "spooky action at a distance". The argument is worked out in a famous paper, Einstein, Podolsky, and Rosen (1935; abbreviated EPR) setting out what is now called the EPR paradox. Assuming what is now usually called local realism, EPR attempted to show from quantum theory that a particle has both position and momentum simultaneously, while according to the Copenhagen interpretation, only one of those two properties actually exists and only at the moment that it is being measured. EPR concluded that quantum theory is incomplete in that it refuses to consider physical properties that objectively exist in nature. (Einstein, Podolsky, & Rosen 1935 is currently Einstein's most cited publication in physics journals.) In the same year, Erwin Schrödinger used the word "entanglement" and declared: "I would not call that one but rather the characteristic trait of quantum mechanics."[31] Ever since Irish physicist John Stewart Bell theoretically and experimentally disproved the "hidden variables" theory of Einstein, Podolsky, and Rosen, most physicists have accepted entanglement as a real phenomenon.[32] However, there is some minority dispute.[33] The Bell inequalities are the most powerful challenge to Einstein's claims.

The idea of quantum field theory began in the late 1920s with British physicist Paul Dirac, when he attempted to quantize the energy of the electromagnetic field; just like in quantum mechanics the energy of an electron in the hydrogen atom was quantized. Quantization is a procedure for constructing a quantum theory starting from a classical theory.

Merriam-Webster defines a field in physics as "a region or space in which a given effect (such as magnetism) exists".[34] Other effects that manifest themselves as fields are gravitation and static electricity.[35] In 2008, physicist Richard Hammond wrote:

Sometimes we distinguish between quantum mechanics (QM) and quantum field theory (QFT). QM refers to a system in which the number of particles is fixed, and the fields (such as the electromechanical field) are continuous classical entities. QFT ... goes a step further and allows for the creation and annihilation of particles ...

He added, however, that quantum mechanics is often used to refer to "the entire notion of quantum view".[36]^: 108

In 1931, Dirac proposed the existence of particles that later became known as antimatter.[37] Dirac shared the Nobel Prize in Physics for 1933 with Schrödinger "for the discovery of new productive forms of atomic theory".[38]

The physical measurements, equations, and predictions pertinent to quantum mechanics are all consistent and hold a very high level of confirmation. However, the question of what these abstract models say about the underlying nature of the real world has received competing answers. These interpretations are widely varying and sometimes somewhat abstract. For instance, the Copenhagen interpretation states that before a measurement, statements about a particle's properties are completely meaningless, while in the many-worlds interpretation describes the existence of a multiverse made up of every possible universe.[40]

Light behaves in some aspects like particles and 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 specific frequencies of light, a small set of distinct pure colors determined by neon's atomic structure. Quantum mechanics shows that light, along with all other forms of electromagnetic radiation, comes in discrete units, called photons, and predicts its spectral energies (corresponding to pure colors), and the intensities of its light beams. A single photon is a quantum, or smallest observable particle, of the electromagnetic field. A partial photon is never experimentally observed. More broadly, quantum mechanics shows that many properties of objects, such as position, speed, and angular momentum, that appeared continuous in the zoomed-out view of classical mechanics, turn out to be (in the very tiny, zoomed-in scale of quantum mechanics) quantized. Such properties of elementary particles are required to take on one of a set of small, discrete allowable values, and since the gap between these values is also small, the discontinuities are only apparent at very tiny (atomic) scales.

• Einstein's thought experiments
• Macroscopic quantum phenomena
• Philosophy of physics
• Quantum computing
• Virtual particle
• Teaching quantum mechanics
• List of textbooks on classical and quantum mechanics

• Bernstein, Jeremy (2005). "Max Born and the quantum theory". American Journal of Physics. 73 (11): 999–1008. Bibcode:2005AmJPh..73..999B. doi:10.1119/1.2060717.
• Beller, Mara (2001). Quantum Dialogue: The Making of a Revolution. University of Chicago Press.
• Bohr, Niels (1958). Atomic Physics and Human Knowledge. John Wiley & Sons]. ISBN 0486479285. OCLC 530611.
• de Broglie, Louis (1953). The Revolution in Physics. Noonday Press. LCCN 53010401.
• Bronner, Patrick; Strunz, Andreas; Silberhorn, Christine; Meyn, Jan-Peter (2009). "Demonstrating quantum random with single photons". European Journal of Physics. 30 (5): 1189–1200. Bibcode:2009EJPh...30.1189B. doi:10.1088/0143-0807/30/5/026. S2CID 7903179.
• Einstein, Albert (1934). Essays in Science. Philosophical Library. ISBN 0486470113. LCCN 55003947.
• Feigl, Herbert; Brodbeck, May (1953). Readings in the Philosophy of Science. Appleton-Century-Crofts. ISBN 0390304883. LCCN 53006438.
• Feynman, Richard P. (1949). "Space-Time Approach to Quantum Electrodynamics". Physical Review. 76 (6): 769–89. Bibcode:1949PhRv...76..769F. doi:10.1103/PhysRev.76.769.
• Feynman, Richard P. (1990). QED, The Strange Theory of Light and Matter. Penguin Books. ISBN 978-0140125054.
• Fowler, Michael (1999). The Bohr Atom. University of Virginia.
• Heisenberg, Werner (1958). Physics and Philosophy. Harper and Brothers. ISBN 0061305499. LCCN 99010404.
• Lakshmibala, S. (2004). "Heisenberg, Matrix Mechanics and the Uncertainty Principle". Resonance: Journal of Science Education. 9 (8): 46–56. doi:10.1007/bf02837577. S2CID 29893512.
• Liboff, Richard L. (1992). Introductory Quantum Mechanics (2nd ed.). Addison-Wesley Pub. Co. ISBN 9780201547153.
• Lindsay, Robert Bruce; Margenau, Henry (1957). Foundations of Physics. Dover. ISBN 0918024188. LCCN 57014416.
• McEvoy, J. P.; Zarate, Oscar (2004). Introducing Quantum Theory. Icon Books. ISBN 1874166374.
• Nave, Carl Rod (2005). "Quantum Physics". HyperPhysics. Georgia State University.
• Peat, F. David (2002). From Certainty to Uncertainty: The Story of Science and Ideas in the Twenty-First Century. Joseph Henry Press.
• Reichenbach, Hans (1944). Philosophic Foundations of Quantum Mechanics. University of California Press. ISBN 0486404595. LCCN a44004471.
• Schilpp, Paul Arthur (1949). Albert Einstein: Philosopher-Scientist. Tudor Publishing Company. LCCN 50005340.
• Scientific American Reader, 1953.
• Sears, Francis Weston (1949). Optics (3rd ed.). Addison-Wesley. ISBN 0195046013. LCCN 51001018.
• Shimony, A. (1983). "(title not given in citation)". Foundations of Quantum Mechanics in the Light of New Technology (S. Kamefuchi et al., eds.). Tokyo: Japan Physical Society. p. 225.; cited in: Popescu, Sandu; Daniel Rohrlich (1996). "Action and Passion at a Distance: An Essay in Honor of Professor Abner Shimony". arXiv:quant-ph/9605004.
• Tavel, Morton; Tavel, Judith (illustrations) (2002). Contemporary physics and the limits of knowledge. Rutgers University Press. ISBN 978-0813530772.
• Van Vleck, J. H.,1928, "The Correspondence Principle in the Statistical Interpretation of Quantum Mechanics", Proc. Natl. Acad. Sci. 14: 179.
• Westmoreland; Benjamin Schumacher (1998). "Quantum Entanglement and the Nonexistence of Superluminal Signals". arXiv:quant-ph/9801014.
• Wheeler, John Archibald; Feynman, Richard P. (1949). "Classical Electrodynamics in Terms of Direct Interparticle Action" (PDF). Reviews of Modern Physics. 21 (3): 425–33. Bibcode:1949RvMP...21..425W. doi:10.1103/RevModPhys.21.425.
• Wieman, Carl; Perkins, Katherine (2005). "Transforming Physics Education". Physics Today. 58 (11): 36. Bibcode:2005PhT....58k..36W. doi:10.1063/1.2155756.

The following titles, all by working physicists, attempt to communicate quantum theory to laypeople, using a minimum of technical apparatus.

• Jim Al-Khalili (2003). Quantum: A Guide for the Perplexed. Weidenfeld & Nicolson. ISBN 978-1780225340.
• Chester, Marvin (1987). Primer of Quantum Mechanics. John Wiley. ISBN 0486428788.
• Brian Cox and Jeff Forshaw (2011) The Quantum Universe. Allen Lane. ISBN 978-1846144325.
• Richard Feynman (1985). QED: The Strange Theory of Light and Matter. Princeton University Press. ISBN 0691083886.
• Ford, Kenneth (2005). The Quantum World. Harvard Univ. Press. Includes elementary particle physics.
• Ghirardi, GianCarlo (2004). Sneaking a Look at God's Cards, Gerald Malsbary, trans. Princeton Univ. Press. The most technical of the works cited here. Passages using algebra, trigonometry, and bra–ket notation can be passed over on a first reading.
• Tony Hey and Walters, Patrick (2003). The New Quantum Universe. Cambridge Univ. Press. Includes much about the technologies quantum theory has made possible. ISBN 978-0521564571.
• Vladimir G. Ivancevic, Tijana T. Ivancevic (2008). Quantum leap: from Dirac and Feynman, Across the universe, to human body and mind. World Scientific Publishing Company. Provides an intuitive introduction in non-mathematical terms and an introduction in comparatively basic mathematical terms. ISBN 978-9812819277.
• J. P. McEvoy and Oscar Zarate (2004). Introducing Quantum Theory. Totem Books. ISBN 1840465778'
• N. David Mermin (1990). "Spooky actions at a distance: mysteries of the QT" in his Boojums all the way through. Cambridge Univ. Press: 110–76. The author is a rare physicist who tries to communicate to philosophers and humanists. ISBN 978-0521388801.
• Roland Omnès (1999). Understanding Quantum Mechanics. Princeton Univ. Press. ISBN 978-0691004358.
• Victor Stenger (2000). Timeless Reality: Symmetry, Simplicity, and Multiple Universes. Buffalo NY: Prometheus Books. Chpts. 5–8. ISBN 978-1573928595.
• Martinus Veltman (2003). Facts and Mysteries in Elementary Particle Physics. World Scientific Publishing Company. ISBN 978-9812381491.

The Wikibook Quantum Mechanics has a page on the topic of: Introduction to Quantum Mechanics

• "Microscopic World – Introduction to Quantum Mechanics". by Takada, Kenjiro, Emeritus professor at Kyushu University
• The Quantum Exchange (tutorials and open-source learning software).
• Atoms and the Periodic Table
• Single and double slit interference
• Time-Evolution of a Wavepacket in a Square Well An animated demonstration of a wave packet dispersion over time.
• Carroll, Sean M. "Quantum Mechanics (an embarrassment)". Sixty Symbols. Brady Haran for the University of Nottingham.