Introduction to quantum mechanics

Quantum mechanics

Uncertainty principle

Introduction to... Mathematical formulation of... Background Classical mechanics Old quantum theory Interference · Bra-ket notation Hamiltonian Fundamental concepts Quantum state · Wave function Superposition · Entanglement Measurement · Uncertainty Exclusion · Duality Decoherence · Ehrenfest theorem · Tunneling Experiments Double-slit experiment Davisson–Germer experiment Stern–Gerlach experiment Bell's inequality experiment Popper's experiment Schrödinger's cat Elitzur-Vaidman bomb-tester Quantum eraser Formulations Schrödinger picture Heisenberg picture Interaction picture Matrix mechanics Sum over histories Equations Schrödinger equation Pauli equation Klein–Gordon equation Dirac equation Bohr Theory and Balmer-Rydberg Equation Interpretations Copenhagen · Ensemble Hidden variable theory · Transactional Many-worlds · Consistent histories Relational · Quantum logic · Pondicherry Advanced topics Quantum field theory Quantum gravity Theory of everything Scientists Planck · Einstein · Bohr · Sommerfeld · Bose · Kramers · Heisenberg· Born · Jordan · Pauli · Dirac · de Broglie ·Schrödinger · von Neumann · Wigner · Feynman · Candlin · Bohm · Everett · Bell · Wien

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Life creates preconceptions that fail drastically when experience is extended to the very massive and the very fast, or when extended to the very small and the very cold. The large scale requires relativity theory, and the small scale requires quantum mechanics. Quantum physics deals with "Nature as She is—absurd."^[1]

Quantum physics deals with unexpected realities of "neither-nor," where the usual picture of reality breaks down. Photons (discrete units of light) and other very small things are neither waves nor particles. They have spectra, but the spectra are chopped up instead of being continuums. The energies carried by particles are discontinuous and color coded. The energies, the colors, and the spectral intensities of electromagnetic radiation produced by something like a neon light bulb, are all interconnected by laws. But the same laws ordain that the more closely one pins down one measure the more wildly another measure relating to the same thing must fluctuate. Even more disconcerting, particles can be created as twins and therefore as entangled entities -- which means that doing something that pins down one characteristic of one particle will determine something about its entangled twin even if it is millions and millions of miles away.

Contents 1 General trends of development 2 An elegant example 3 How the unexpected came to light 3.1 Planck and the constant h 3.2 Spectroscopy and onward 3.3 Consequences of contemporaneous research 3.4 Double-slit experiment and complementarity 3.5 Planck's constant and Einstein 3.6 Bohr atom 3.7 Wave-particle duality 4 Development of modern quantum mechanics 4.1 Full quantum mechanical theory 4.2 Schrödinger wave equation 4.3 Uncertainty principle 4.4 Wavefunction collapse 4.5 Eigenstates and eigenvalues 4.6 The Pauli exclusion principle 4.7 Dirac wave equation 4.8 Quantum entanglement 5 Quantum electrodynamics 6 Interpretations 7 Summary 8 See also 9 Further reading 10 References 11 Notes 12 External links

General trends of development

At least up to the physics of Schrödinger, progress came by way of difference equations that mirrored the way quantum events in atoms depend on the difference between the energy state that the electron left to the energy state where the electron arrived. (In the simplest model, the Bohr atom, an electron is said to jump from one orbit to another.) The original development appeared to be just the discovery of a kind of arbitrary sequence of permitted wavelengths that seemed to imply an orbital structure that might account for vibrations of the appropriate frequency to create those wavelengths. Soon that model was generalized to account for the spectrum of hydrogen that goes beyond visible light, next the general idea of difference equations was applied to dispersion phenomena, and with Heisenberg's paper of 1925 the model gained a new feature in what has been called "Heisenberg's law for multiplying transition amplitudes together" to account for the varying intensities of the components of the hydrogen bright-line spectrum. The mathematical operations involved constituted de facto rules for laying out matrices that could encode all that was then known about quantum phenomena in one coherent system. The picture laid out for inspection did not look anything like the world of everyday experience. By the time Schrödinger made his equation, the number of variables (quantum numbers) involved in models of sub-atomic events was making the mathematics very complicated.

In 1927 Niels Bohr wrote: "Anyone who is not shocked by quantum theory does not understand it."

An elegant example

Light does not go in a straight line from light to detection screen. Note 3 fringes at right.

The most elegant character on the quantum stage is the double-slit experiment. It highlights several features of quantum mechanics. A single photon emitted by a laser or an electron emitted by a cathode will behave differently depending on whether one or two slits lie in its path. With two slits present, what arrives at a remote detection screen will be a superposition of two wave functions. As the illustration shows, a wave from the top slit and another from the bottom slot will fall on top of each other on the detection screen, and so they are superimposed. Where a photon or electron actually shows up on the detection screen will indicate the resolution of those wave functions, sometimes called their "collapse", and the location where this individual "collapse" occurs will be determined in that it must appear in one of the bright "fringes" that will show up when many photons are run through the apparatus, but it will be entirely unpredictable as to which of the fringes it contributes to.

Photons function as though they are waves as they go through the slits. When two slits are present, the "wave function" pertaining to each photon goes through each slit. The wave functions are superimposed all across the detection screen, yet at the detection screen only one particle, a photon, shows up and its position is in accord with strict probability laws. All of these points will be revisited below.

How the unexpected came to light

Isaac Newton believed that light consisted of infinitesimally small particles which he called "corpuscles." In 1827, Thomas Young and Augustin Fresnel conducted experiments on light interference that found results inconsistent with a corpuscular theory of light. All theoretical and empirical results through the late 19th century seemed inconsistent with Newton's corpuscular theory of light.

Later experiments identified phenomena, such as the photoelectric effect, that were consistent only with a packet or "quantum" model of light.

How could nature reconcile these seemingly incompatible characteristics? Humans found that they needed to alter their conceptual scheme to more closely model nature.

In 1900, Lord Kelvin gave a lecture entitled “Nineteenth-Century Clouds over the Dynamical Theory of Heat and Light Light,” in which he said that the "beauty and clearness of theory" was overshadowed by "two clouds, the null result of the Michelson-Morley experiment and the problems of black body radiation.”^[2]

The second of these two was also known as the ultraviolet catastrophe. Classical physics implied that a heated object such as a cannon ball in a bonfire would produce electromagnetic radiation at all frequencies. According to classical theory, it should produce an infinite flux of ultraviolet and even higher frequencies of light.

Planck and the constant h

Energies are multiples of h.

Beset by the problem of the threatened Ultraviolet Catastrophe, Max Planck very unwillingly handled the problem by deciding that energy must come in integral multiples of a small unit, h, that is now known as Planck's constant. This process yielded the equation

E = hf

(energy is the product of h and the frequency. So if a photon has a frequency of x, the energy it carries will be equal to hx.

In this way, Planck discovered something fundamental about nature that would change physics forever.

Philipp Lenard discovered that light shining on a metal plate could cause electrons to fly off one electrode and onto another -- but the threshold for this effect to occur was related to the frequency of the light rather than to its intensity. That phenomenon is known as the photoelectric effect.

According to the quantum explanation, when a photon strikes a metal surface (such as that in a light meter), it delivers a definite amount of energy related to its frequency and wavelength. A photon of ultraviolet light, having a short wavelength, will deliver a high amount of energy -- enough to contribute to a sunburn for instance. A photon of infrared light, having a long wavelength, will deliver a low amount of energy -- only enough to warm one's skin. So a very large infrared light can warm a large surface, perhaps even large enough to keep people comfortable in a cold room or even make someone too hot, but it cannot give anyone a sunburn. What is needed for an electron to be freed from its original atom and jump a spark gap or do something else requiring a certain voltage to be present is for the frequency to be high enough.

Spectroscopy and onward

(m = 3,4,5...)
The law behind the wavelengths in the bright line spectrum of hydrogen was one of the few seemingly minor mysteries that dotted the landscape of late 19th century physics. There seemed to be no simple rule behind the wavelengths that glowing hydrogen gas would produce, but all hydrogen produces the same set of wavelengths so it could not be an accident. Balmer worked with the numbers until he came up with a rule that could produce a series of numbers. These numbers had to be multiplied by some constant to produce the characteristic wavelengths in the visual spectrum of hydrogen. The constant was derived by dividing the empirical value of the wavelengths of any line in the spectrum by the number produced by the core part of the rule. Then given that constant and the core part of the rule, one could derive the wavelengths of any of the other lines in the visual spectrum.

In 1888, Rydberg generalized and greatly increased the explanatory utility of Balmer's formula. He wrote:

where λ stands for the wavelength, R is Rydberg's constant, and n can be any integer greater than 2.

The basic rule produced by Balmer was soon improved by Rydberg and Ritz by solving a related formula for the inverse of the the wavelength, and calculating the theoretical results provided good predictions. So it seemed that the universe could once again be shown to behave in stable and predictable ways. See how the Rydberg-Ritz formula works. Visit Hyperphysics and scroll down past the middle of the page.

Ritz also discovered what has come to be known as the Ritz combination principle that demonstrates how new intervals among frequencies in a bright line spectrum can be discovered because there are several differences of frequencies between the energy states (or orbits) of electrons that keep repeating themselves. This principle is implicit in Heisenberg's breakthrough formulation of the new quantum mechanics in 1925.

Consequences of contemporaneous research

Red: (high amperage) low voltage. Violet: high voltage.

Other experiments showed that when light shone upon a metal surface it would drive electrons away. The result can be seen in the light meters designed for photographic use. A beam of light creates an electrical potential (a voltage), and that voltage will cause a current with a certain amperage to flow through an external part of the circuit. In a light meter, the voltage induced by a beam of light induces an electric current of a certain voltage and amperage that powers a little electromagnet that moves the needle in the light meter. The strange thing, the thing that did not jibe at all well with the idea of light waves, was that the voltage produced by a beam of light did not change when the intensity of the beam of light was changed. Only the current in the circuit changed. But if one substituted beams of a single frequency, red beams and violet beams of equal intensity would produce different voltages. These experiments showed that longer wavelength light produces lower voltage, i.e., it puts less of a "kick" on individual electrons, and shorter wavelength light produces a greater force on individual electrons. The next mystery was how light, which had clearly defined wavelengths and frequencies, could behave as though it were not a wave phenomenon. The only way that seemed able to explain its action was to say that light delivers individual "packets" of energy to specific points, and that the amount of energy delivered is an exact integer multiple of a quantity called Plank's constant. Albert Einstein gave these bundles of energy the name "photon."

Red light warms; violet excites strongly; ultraviolet penetrates.

Double-slit experiment and complementarity

If pumpkins were photons, nobody could hide behind a post.

Scientists were forced to draw a seemingly very self-contradictory conclusion. Light behaves like a wave in some situations, and yet it performs like particles in other situations. The quantum physicists enunciated the principle of complementarity, i.e., the idea that light cannot be adequately characterized by the wave interpretation, but it also cannot be adequately characterized by the particle interpretation. One cannot stand without the other, at least not when talking about things on an atomic scale. In the quantum world, a photon may be emitted as the result of an interaction within a single atom, and end up by being absorbed by a single atom in the detection screen. But where on that detection screen it appears depends very strongly on whether there is a single path between the point of origin and the screen, or there are two or more paths. If there are two or more paths then something wavelike passes through both slits and then interferes with itself. The fact of self interference determines the probabilities of the photon's potential points of appearance.

The brightest spot occurs directly opposite the center post.

The double-slit experiment is a very compelling example of both quantum weirdness and complementarity. A beam of light from a laser pointer can be shown through two narrow slits. If photons were like tiny bullets and the laser were like a rather poor rifle that puts a fairly broad splatter of bullets into a target, then some of the photon bullets ought to go through the left slit and end up more-or-less at one spot to the left of center of the target screen, and some of the photon bullets ought to go through the right slit and end up to the right of center of the target. But that is not at all what happens. The most photons end up directly behind the solid part between the slits, and then a pattern of dark and bright fringes is manifest across the a very broad swath. The results cannot be explained by a particle interpretation of light, but only as some kind of wave phenomenon in which light interferes with itself both positively and negatively. So in the region of the two slits, light has to be "behaving like" a wave. The weird thing is that if light is permitted to go through the slits one photon at a time, it will show up as individual impacts on the target screen (showing its particle nature); however, each photon will still interfere with itself and over time the impacts will draw a fringe pattern on the screen (showing its wave nature).

If the above description somehow fails to impress the reader as strange, consider also that not only photons, but also electrons, atoms, and even some molecules have been put through similar experiments and they too interfere with themselves if there are two slits, do not interfere with themselves if there is only one slit, and only end up at one place despite the appearance of their having gone through two slits.

Planck's constant and Einstein

In 1905, Albert Einstein used Planck's constant to explain the photoelectric effect by postulating that the energy in a beam of light occurs in packets he called light quanta, and that later came to be called photons.^[3] According to Einstein's account, a single photon of a given frequency delivers an invariant amount of energy. In other words, individual photons can deliver more or less energy, but only depending on their frequencies. Although this description that built on Planck's theory sounds like Newton's corpuscular account, Einstein's photon was still said to have a frequency, with the energy of the photon being proportional to its frequency. Once again, the particle account of light had been "compromised."^[4]

Both the idea of a wave and the idea of a particle are models derived from our everyday experience. We cannot see individual photons, and can only investigate their properties indirectly. Take, for example, the rainbow of colours we see reflected from a puddle of water when a thin film of oil rests on its surface. We can explain that phenomenon by modelling light as waves.^[5] Other phenomena, such as the working of the photoelectric meters in our cameras, may be explained by thinking in terms of particles of light colliding with the detection screen inside the meter. In both cases, we take concepts from our everyday experience and apply them to a world we will never see or otherwise experience directly.

Neither wave nor particle is an entirely satisfactory explanation. In general, any model can only approximate that which it models. A model is useful only within the range of conditions where it makes accurate predictions. Newtonian physics remains a good predictor of most everyday (macroscopic) phenomena. To remind us that both "wave" and "particle" are concepts imported from our macro world to explain atomic-scale phenomena, physicists such as Banesh Hoffmann have used the term "wavicle" to refer to whatever it is that is "really there." In the following discussion, "wave" and "particle" may both be used depending on which aspect of quantum mechanical phenomena is under discussion.

Bohr atom

In 1897, a research team headed by J J Thompson discovered and named the electron, the carrier of negative charge. By means of the gold foil experiment, physicists discovered that matter is mostly empty space.^[6] Once that was clear, it was hypothesized that negatively charged electrons orbit a positively charged nucleus, so that all atoms resemble a miniature solar system. But that simple analogy predicted that electrons would take only about one hundredth of a microsecond^[7] to crash into the nucleus. Hence the great question of early 20th century physics was: "How do electrons normally remain in stable orbits around the nucleus?"

In 1913, Niels Bohr solved this substantial problem by applying the notion of discrete (non-continuous) quanta to electron orbits. This solution became known as the Bohr model of the atom. Bohr basically theorized that electrons can only inhabit certain orbits around the atom. These orbits could be derived by looking at the spectral lines produced by pure elements.^[8]

Bohr generalized Ryberg's formula for hydrogen by replacing the denominator in the term 1/4 with an explicit squared variable:

m=1,2,3,4,5,..., and n > m,

where λ is the wavelength of the light, R_H is the Rydberg constant for hydrogen, and n and m are integers referring to the orbits between which electrons can transit. This generalization predicted many more line spectra than had been previously detected, and experimental confirmation of this prediction followed. Bright lines were found in both the ultraviolet and the infrared parts of the spectrum, and these values were not predicted by the existing scheme. Neils Bohr had expanded the earlier theory and thereby predicted the spectral lines found beyond the visible spectrum.

Bohr gave a physical interpretation of the above formula according to which electrons were like planets in orbit around a sun, the m numbers indicated in what orbits the electron found its final position, and the n numbers showed the electron in its initial orbit.

Bohr proposed that when an electron changed orbits, it did not move in a continuous trajectory from one orbit to another. Instead, it suddenly disappeared from its original orbit and reappeared in another orbit. Each distance at which an electron can orbit is a function of a quantized amount of energy. The closer to the nucleus an electron orbits, the less energy it takes to remain in that orbit. Electrons that absorb a photon gain a quantum of energy, so they jump to an orbit that is farther from the nucleus, while electrons that emit a photon lose a quantum of energy and so jump to an orbital closer to the nucleus. Electrons cannot gain or lose a fractional quantum of energy, hence they cannot be found at some fraction of the distance between allowed orbits. The values of n are positive integers, and correspond to allowed orbitals, with the innermost orbit designated n = 1, the next being n = 2, and so on.

Bohr's theory represented electrons as orbiting the nucleus of an atom, much as planets orbit around he sun. However, we now envision electrons circulating around the nuclei of atoms in a way that is strikingly different from Bohr's atom, and what we see in the world of our everyday experience. Instead of orbits, electrons are said to inhabit "orbitals." An orbital is the "cloud" of possible locations in which an electron might be found, a distribution of probabilities rather than a precise location.

Bohr's model of the atom was essentially two-dimensional: an electron orbiting in a plane around its nuclear "sun." Modern theory ^[9] describes a three-dimensional arrangement of electronic shells and orbitals around atomic nuclei. The orbitals are spherical (s-type) or lobular (p, d and f-types) in shape. It is the underlying structure and symmetry of atomic orbitals, and the way that electrons fill them, that determines the structure and strength of chemical bonds between atoms. Thus, the bizarre quantum nature of the atomic and sub-atomic world finds natural expression in the macroscopic world with which we are more familiar.

Scientists at first believed that the movement of electrons in their orbits (as understood by Bohr) could account for the radiation of light. However, that hypothesis led to two problems: (1) According to the planetary model, electrons are always moving, so all atoms ought to constantly emit light. (2) By emitting light the electrons would lose energy, and so lose speed, so the frequency should constantly decrease, and because they constantly lose speed, the electrons should spiral into the nucleus. Instead, Bohr argued, photons must be emitted when electrons change orbits.

Wave-particle duality

Niels Bohr showed that neither the wave analogy nor the particle analogy, taken individually, fully describe the empirical properties of light. All forms of electromagnetic radiation were found to behave in certain experiments as though they were particles, and in other experiments as though they were waves. With these facts in mind, Bohr enunciated the principle of complementarity, which pairs concepts such as wave and particle, or position and momentum.

In 1924, Louis de Broglie explored the mathematical consequences of Bohr's findings and discovered the theory of wave-particle duality, which states that subatomic particles too have simultaneous wave and particle properties. De Broglie expanded the Bohr model of the atom by showing that an electron in orbit around a nucleus could be thought of as having wave-like properties. In particular, an electron will be observed only in situations that permit a standing wave around a nucleus. An example of a standing wave is a string fixed at both ends and made to vibrate (as in a string instrument). Hence a standing wave must have zero amplitude at each fixed end. The waves created by a stringed instrument also appear to oscillate in place, moving from crest to trough in an up-and-down motion. A standing wave requires that the wavelength be an integer fraction of the length of the vibrating object. (In other words, a harmonic frequency must be an integer multiple of the fundamental frequency of the vibrating object.) In a vibrating medium that traces out a simple closed curve, the wave must be a continuous formation of crests and troughs all around the curve. Since electron orbitals are simple closed curves, each electron must be its own standing wave, occupying a unique orbital.

De Broglie's treatment of quantum events served as a jumping off point for Schrödinger when he set about to construct a wave equation to describe quantum theoretical events.

Development of modern quantum mechanics

Full quantum mechanical theory

Werner Heisenberg, a physicist who was still a new professor on the tenure track at the time, succeeded in giving a quantum theoretical answer to the question of what the intensities of the lines of the hydrogen spectrum are. He benefited from his close association with Neils Bohr and members of his school. From Bohr he learned the value of avoiding facile theoretical formulations that may give correct answers in some narrow context but provide no means by which to connect them to a wider context. Heisenberg recalled, "Bohr would always say, 'First we have to understand how physics works; only when we have completely understood what it is about can we hope to represent it by mathematical schemes.' " (Heisenberg, Conversations, p. 230) ^[10] Second, from his work with one of Bohr's senior students, Kramers, he learned the value of using difference equations in describing quantum phenomena. (One difference equation has already appeared in this discussion, in the formula shown above that predicts wavelengths by multiplying a constant by the difference between two fractions.)

To make a long and rather complicated story short, Heisenberg used the idea that since classical physics is correct when it applies to phenomena in the world of things larger than atoms and molecules, it must stand as a special case of a more inclusive quantum theoretical model. So he hoped that he could modify quantum physics in such a way that when the parameters were on the scale of everyday objects it would look just like classical physics, but when the parameters were pulled down to the atomic scale the discontinuities seen in things like the widely spaced frequencies of the visible hydrogen bright line spectrum would come back into sight.

By means of an intense series of mathematical analogies that some physicists have termed "magical," Heisenberg wrote out an equation that is the quantum mechanical analog for the classical computation of intensities. Remember that the one thing that people at that time most wanted to understand about hydrogen radiation was how to predict or account for the intensities of the lines in its spectrum. Although Heisenberg did not know it at the time, the general format he worked out to express his new way of working with quantum theoretical calculations can serve as a recipe for two matrices and how to multiply them.^[11]

This general format indicates that some term C is to be computed by summing up all of the products of some group of terms A by some related group of terms B. There will potentially be an infinite series of A terms and their matching B terms. Each of these multiplications has as its factors two measurements that pertain to sequential downward transitions between energy states of an electron. This type of rule differentiates matrix mechanics from the kind of physics familiar in everyday life because the important values are where (in what energy state or "orbital") the electron begins and in what energy state it ends, not what the electron is doing while in one or another state.

The formula looks rather intimidating, but if A and B both refer to lists of frequencies, for instance, all it says to do is perform the following multiplications and then sum them up:

Multiply the frequency for a change of energy from state n to state n-a frequency by the frequency for a change of energy from state n-a to state n-b. and to that add the product found by multiplying the frequency for a change of energy from state n-a to state n-b by the frequency for a change of energy from state n-b to state n-c,
and so forth:
Symbolically that is:
f(n, n-a) * f(n-a,n-b)) +
f(n-a,n-b) * f(n-b,n-c) +
etc.

It would be very easy to do each individual step of this process for some measured quantity. For instance, the formula (2) above gives each needed wavelength in sequence. The values calculated could very easily be filled into a grid as described below. However, since the series is infinite, nobody could do the entire set of calculations.

Heisenberg originally devised this equation to enable himself to multiply two measure of the same kind, so it happened not to matter which order they were multiplied in. Heisenberg noticed, however that if he tried to use the same schema to multiply two variables, such as momentum, p, and displacement, q, then "a significant difficulty arises."^[12] It turns out that multiplying a matrix of p by a matrix of q gives a different result from multiplying a matrix of q by a matrix of p. It only made a tiny bit of difference, but that difference could never be reduced below a certain limit, and that limit involved Planck's constant, h. More on that later. Below is a very short sample of what the calculations would be, placed into grids that are called matrices. Heisenberg's teacher saw almost immediately that his work should be expressed in a matrix format because mathematicians already were familiar with how to do computations involving matrices in an efficient way.

(Equation for the conjugate variables momentum and position)

Matrix of p

Electron States	n-a	n-b	n-c	....
n	p(n→n-a)	p(n→n-b)	p(n→n-c)	.....
n-a	p(n-a→n-a)	p(n-a→n-b)	p(n-a→n-c)	.....
n-b	p(n-b→n-a)	p(n-b→n-b)	p(n-b→n-c)	.....
transition....	.....	.....	.....	.....

Matrix of q

Electron States	n-b	n-c	n-d	....
n-a	q(n-a→n-b)	q(n-a→n-c)	q(n-a→n-d)	.....
n-b	q(n-b→n-b)	q(n-b→n-c)	q(n-b→n-d)	.....
n-c	q(n-c→n-b)	q(n-c→n-c)	q(n-c→n-d)	.....
transition....	.....	.....	.....	.....

The matrix for the product of the above two matrices as specified by the relevant equation in Heisenberg's 1925 paper is:

Electron States	n-b	n-c	n-d	.....
n	A	.....	.....	.....
n-a	.....	B	.....	.....
n-b	.....	.....	C	.....

Where:

A=p(n→n-a)*q(n-a→n-b)+p(n→n-b)*q(n-b→n-b)+p(n→n-c)*q(n-c→n-b)+.....

B=p(n-a→n-a)*q(n-a→n-c)+p(n-a→n-b)*q(n-b→n-c)+p(n-a→n-c)*q(n-c→n-c)+.....

C=p(n-b→n-a)*q(n-a→n-d)+p(n-b→n-b)*q(n-b→n-d)+p(n-b→n-c)*q(n-d→n-d)+.....

and so forth.

If the matrices were reversed, the following values would result:

A=q(n→n-a)*p(n-a→n-b)+q(n→n-b)*p(n-b→n-b)+q(n→n-c)*p(n-c→n-b)+.....
B=q(n-a→n-a)*p(n-a→n-c)+q(n-a→n-b)*p(n-b→n-c)+q(n-a→n-c)*p(n-c→n-c)+.....
C=q(n-b→n-a)*p(n-a→n-d)+q(n-b→n-b)*p(n-b→n-d)+q(n-b→n-c)*p(n-d→n-d)+.....

and so forth.

Note how changing the order of multiplication changes the numbers, step by step, that are actually multiplied.

Heisenberg's groundbreaking paper of 1925 neither uses nor even mentions matrices. Heisenberg's great advance was the "scheme which was capable in principle of determining uniquely the relevant physical qualities (transition frequencies and amplitudes)"^[13] of hydrogen radiation.

Paul Dirac decided that the essence of Heisenberg's work lay in the very feature that Heisenberg had originally found problematical -- the fact of non-commutativity such as that between multiplication of a momentum matrix by a displacement matrix and multiplication of a displacement matrix by a momentum matrix. That insight led Dirac in new and productive directions.^[14]

Schrödinger wave equation

Main article: Schrödinger equation

In 1925, building on De Broglie's theoretical model of particles as waves, Erwin Schrödinger analyzed how an electron would behave if it were assumed to be a wave surrounding a nucleus. Rather than explaining the atom by an analogy to satellites orbiting a planet, he treated electrons as waves with each electron having a unique wavefunction. Such wavefunctions were named "Schrödinger's equation" in his honor. Schrödinger's equation describes a wavefunction by three properties (Wolfgang Pauli later added a fourth: spin):

1. An "orbital" designation, indicating whether the particle wave is one that is closer to the nucleus with less energy or one that is farther from the nucleus with more energy;
2. The "shape" of the orbital, spherical or otherwise;
3. The "inclination" of the orbital, determining the magnetic moment of the orbital around the z-axis.

The collective name for these three properties is the "wavefunction of the electron," describing the quantum state of the electron. The quantum state of an electron refers to its collective properties, which describe what can be said about the electron at a point in time. The quantum state of the electron is described by its wavefunction, denoted by the Greek letter ψ ("psi," pronounced "sigh").

The three properties of Schrödinger's equation describing the wavefunction of the electron (and thus its quantum state) are each called quantum numbers. The first property describing the orbital is the principal quantum number, numbered according to Bohr's model, in which n denotes the energy of each orbital.

The next quantum number, the azimuthal quantum number, denoted l (lower case L), describes the shape of the orbital. The shape is a consequence of the angular momentum of the orbital. The rate of change of the angular momentum of any system is equal to the resultant external torque acting on that system. In other words, angular momentum represents the resistance of a spinning object to speeding up or slowing down under the influence of external force. The azimuthal quantum number l represents the orbital angular momentum of an electron around its nucleus. However, the shape of each orbital has its own letter as well. The first shape is denoted by the letter s (for "spherical"). The next shape is denoted by the letter p and has the form of a dumbbell. The other orbitals have more complicated shapes (see Atomic Orbitals), and are denoted by the letters d, f, and g. The entry carbon atom describes the orbitals of carbon.

The third quantum number in Schrödinger's equation describes the magnetic moment of the electron. This number is denoted by either m or m with a subscript l, because the magnetic moment depends on the second quantum number l.

In May 1926, Schrödinger proved that Heisenberg's matrix mechanics and his own wave mechanics made the same predictions about the properties and behaviour of the electron; mathematically, the two theories were identical. Yet both men disagreed on the physical interpretations of their respective theories. Heisenberg saw no problem in the existence of discontinuous quantum jumps, while Schrödinger hoped that a theory based on continuous wave-like properties could avoid what he called (in the words of Wilhelm Wien^[15]), "this nonsense about quantum jumps."

Uncertainty principle

Main article: Uncertainty principle

One of Heisenberg's seniors, Max Born explained how he took his strange "recipe" given above and discovered something ground breaking:^[16]

By consideration of ...examples...[Heisenberg] found this rule.... This was in the summer of 1925. Heisenberg...took leave of absence...and handed over his paper to me for publication....

Heisenberg's rule of multiplication left me no peace, and after a week of intensive thought and trial, I suddenly remembered an algebraic theory....Such quadratic arrays are quite familiar to mathematicians and are called matrices, in association with a definite rule of multiplication. I applied this rule to Heisenberg's quantum condition and found that it agreed for the diagonal elements. It was easy to guess what the remaining elements must be, namely, null; and immediately there stood before me the strange formula

[The symbol Q is the matrix for displacement, P is the matrix for momentum, i stands for the square root of negative one, and h is Planck's constant.^[17]]

That is the Heisenberg uncertainty principle, and it came out of the math! Quantum mechanics strongly limits the precision with which the properties of moving subatomic particles can be measured. An observer can precisely measure either position or momentum, but not both. In the limit, measuring either variable with complete precision would entail a complete absence of precision in the measurement of the other.

Wavefunction collapse

Wavefunction collapse is the replacement of the description of the uncertain state of a system by a description of the system in a definite state. The nature of the process is controversial.

Eigenstates and eigenvalues

For a more detailed introduction to this subject, see: Introduction to eigenstates

Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. Therefore it is necessary to formulate clearly the difference between the state of something that is 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.

The Pauli exclusion principle

Wolfgang Pauli proposed the following concise statement of his principle:

"There cannot exist an atom in such a quantum state that two electrons within have the same set of quantum numbers."^[18]

He developed the Exclusion Principle from what he called a "two-valued quantum degree of freedom" to account for the observation of a doublet, meaning a pair of lines differing by a small amount (e.g., on the order of 0.15Å), in the spectrum of atomic hydrogen. The existence of these closely spaced lines in the bright-line spectrum meant that there was more energy in the electron orbital from magnetic moments than had previously been described.

In early 1925, Uhlenbeck and Goudsmit proposed that electrons rotate about an axis in the same way that the earth rotates on its axis. They proposed to call this property spin. Spin would account for the missing magnetic moment, and allow two electrons in the same orbital to occupy distinct quantum states if they "spun" in opposite directions, thus satisfying the Exclusion Principle. A new quantum number was then needed, one to represent the momentum embodied in the rotation of each electron.

By this time an electron was recognized to have four kinds of fundamental characteristics that came to be identified by the four quantum numbers:

• n, the principal quantum number;
• l, the azimuthal quantum number (pertains to orbital angular momentum);
• m_l, the magnetic quantum number;
• m_s, the spin quantum number.

The chemist Linus Pauling wrote, by way of example:

"In the case of a helium atom with two electrons in the 1 s orbital, the Pauli Exclusion Principle requires that the two electrons differ in the value of one quantum number. Their values of n, l, and m_l are the same; moreover, they have the same spin quantum number, s = 1/2. Accordingly they must differ in the value of m_s, which can have the value of +½ for one electron and -½ for the other."^[18]

Dirac wave equation

Main article: Dirac equation

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 of a dynamical vacuum. This led to many-particle quantum field theory.

Quantum entanglement

Main article: Quantum entanglement

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 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 probabilities that are the squares of the 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 the superposition of a state that can be mentally labeled as blue and another state that can be mentally labeled as red will then appear (in imagination, of course) as a purple state. Two photons are produced as the result of the same 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. So the two photons come out "purple." If the experimenter now performs some experiment that will determine 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, it would necessarily show up in the opposite state to whatever its twin had revealed.

Suppose that some species of animal life carries both male and female characteristics in its genetic potential. It will become either male or female depending on some environmental change. Perhaps it will remain indeterminate until the weather either turns very hot or very cold. Then it will show one set of sexual characteristics and will be locked into that sexual status by epigenetic changes, the presence in its system of high levels of androgen or estrogen, etc. There are actually situations in nature that are similar to this scenario, but now imagine that if twins are born, then they are forbidden by nature to both manifest the same sex. So if one twin goes to Antarctica and changes to become a female, then the other twin will turn into a male despite the fact that local weather has done nothing special to it. Such a world would be very hard to explain. How can something that happens to one animal in Antarctica affect its twin in Redwood, California? Is it mental telepathy? What? How can the change be instantaneous? Even a radio message from Antarctica would take a certain amount of time.

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 in a classical way, 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 which objectively exist in nature. (Einstein, Podolsky, & Rosen 1935 is currently Einstein's most cited publication in physics journals.)

The question of whether entanglement is a real condition is still in dispute. The Bell inequalities are the most powerful challenge to Einstein's claims.

Quantum electrodynamics

Interpretations

Main article: Interpretation of quantum mechanics

The physical measurements, equations, and predictions pertinent to quantum mechanics are all consistent and hold a very high level of comfirmation. However, the question of what these abstract models say about the underlying nature of the real world has received competing answers.

Summary

Energy appears in quantum mechanics as integer multiples of h, Planck's constant.As best scientists have been able to understand, electrons can be in several different energy states. Just as the velocity with which an object approaches a sun will determine the distance from the sun at which it can establish a stable orbit, so too the energy carried by an electron will automatically assign it to a given orbit around the nucleus of an atom. Moving from one energy state to another either requires that more energy be supplied to the electron (moving it to a higher energy state) or the electron must lose a certain amount of energy as a photon (moving it to a lower energy state). The presence of seven principle energy states meant that there would be six visible lines in the bright line spectrum of hydrogen.

The six visible lines were originally observed, but scientists did not know anything more than their wavelengths. Balmer figured out a mathematical rule by which he could make quantum theoretical predictions of the observed wavelengths. The same basic rule was improved in two stages, first by writing it in terms of the inverse values of all of the numbers involved, and second by generalizing the rule and replacing

with

This additional level of generality permitted the entire hydrogen bright-line spectrum, from infrared through the visible colors to ultraviolet and higher frequencies, to be predicted because m could be a whole range of integers as long as any m was always larger than the corresponding n. Using Planck's constant, one could assign energies to individual frequencies (or wavelengths) of electromagnetic radiation. To predict the intensities of these bright lines, physicists needed to use matrix mathematics, Schrödinger's equation, or some other computational scheme involving higher mathematics. There were not only the basic seven energy levels of hydrogen, but also other factors that created additional energy levels. The very first calculation that Heisenberg made in his new theory involved an infinite series, and the more factors involved (the more "quantum numbers" were involved) the more complex the mathematics. But the basic insight into the structure of the hydrogen atom was encoded in the simple formula that Balmer guessed from a list of wavelengths.

The photoelectric effect was discovered soon after Balmer made his rule, and in 1905 Einstein first depicted light as being made of photons to account for that effect.

Once Bohr had explained the Rydberg formula in terms of atomic structure and a decade later Dirac introduced the idea of matter waves. Two years later, in 1925, Heisenberg removed the last traces of classical physics from the new quantum theory by making the breakthrough that led to the matrix formulation of quantum mechanics, and Pauli enunciated his exclusion principle. Further advances came by closing in on some of the elusive details: (1) adding the m_l (spin) quantum numbers (discovered by Pauli), (2) adding the M_s quantum numbers (discovered by Goudsmit and Uhlenbeck), (3) broadening the quantum picture to account for relativistic effects (Dirac's work), (4) showing that particles such as electrons, and even larger entities, have a wave nature (the wave-particle duality of de Broglie).

Several improvements in mathematical formulation have also furthered quantum mechanics:

De Broglie's quantum theoretical description based on waves was followed upon by Schrödinger. Schrödinger's method of representing the state of each atomic entity is generally more practical scheme to use than Heisenberg's, and it makes it possible to conceptualize a "wave function" that passes through both sides of a double-slit experiment and then arrives at the detection screen as two parts of itself that are superimposed but a little shifted (a little out of phase). It also makes it possible to understand how two photons or other things of that order of magnitude might be created in the same event or otherwise closely linked in history and so carry identical copies of superimposed wave functions, and that mental picture can then be used to explain how when one of them is coerced into revealing itself it must manifest one or the other superimposed wave nature, and its twin (regardless of its distance away in space or time) must manifest the complementary wave nature, i.e., the other of the two superimposed wave functions.

Prominent among the many later scientists who increased the elegance and accuracy of quantum-theoretical formulations was Richard Feynman who followed up on Dirac's work. The basic picture given in the original Balmer formula has remained true, but it has been qualified by revelation of many details such as angular momentum and spin, and extended to descriptions that go beyond only explaining the electron and its behavior while bound to an atomic nucleus. Active research still continues to resolve some remaining issues.

See also

Angular momentum Bell's inequality Coherence Complementarity Correspondence principle Experiments: Bell's inequality Double slit EPR Popper's Simple Demo of Quantum effects Stern-Gerlach History of quantum mechanics Interpretations: Copenhagen Heisenberg picture Interaction picture Many worlds Schrödinger picture Sum over histories Matrix mechanics Measurement in quantum mechanics Measurement problem Orbital: Atomic Molecular

Pauli exclusion principle Philosophy of information Philosophy of physics Photon Planck's constant Probability amplitude Quantum mechanics Mathematical formulation Quantum: Computer Decoherence Entanglement Information Number Principal Azimuthal Magnetic Spin Pseudo-telepathy State Superposition Tunneling Zeno effect

Reduction criterion Schrödinger's cat Schrödinger's equation Spectral line Spin Standard Model Uncertainty principle Wave function Collapse Wave mechanics Wave packet Wave-particle duality

Persons important for discovering and elaborating quantum theory:

John Stewart Bell David Bohm Niels Bohr Max Born Satyendra Nath Bose Louis de Broglie

Paul Dirac Albert Einstein Hugh Everett Richard Feynman Werner Heisenberg Pascual Jordan

Wolfgang Pauli Max Planck Erwin Schrödinger Arnold Sommerfeld Eugene Wigner

Further reading

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

• Richard Feynman, 1985. QED: The Strange Theory of Light and Matter, Princeton University Press. ISBN 0-691-08388-6
• 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.
• 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.
• Victor Stenger, 2000. Timeless Reality: Symmetry, Simplicity, and Multiple Universes. Buffalo NY: Prometheus Books. Chpts. 5-8.

References

• Bernstein, Jeremy, 2005, "Max Born and the quantum theory," Am. J. Phys. 73(11).
• Beller, Mara, 2001. Quantum Dialogue: The Making of a Revolution. University of Chicago Press.
• Bohr, Niels (1958). Atomic Physics and Human Knowledge. John Wiley and Sons. OCLC 530611 ASIN B00005VGVF. 
• Louis de Broglie, 1953. The Revolution in Physics. Noonday Press.
• Albert Einstein, 1934. Essays in Science. Philosophical Library.
• Herbert Feigl and May Brodbeck, 1953. Readings in the Philosophy of Science, Appleton-Century-Crofts.
• Fowler, Michael, 1999. The Bohr Atom. Lecture series, University of Virginia.
• Werner Heisenberg, 1958. Physics and Philosophy. Harper and Brothers.
• Lakshmibala, S., 2004, "Heisenberg, Matrix Mechanics and the Uncertainty Principle," Resonance, Journal of Science Education 9(8).
• Richard L. Liboff, 1992. Introductory Quantum Mechanics, 2nd ed.
• Lindsay, Robert Bruce and Henry Margenau, 1936. Foundations of Physics. Dover.
• McEvoy, J.P., and Zarate, Oscar. Introducing Quantum Theory, ISBN 1874166374
• Nave, Carl Rod, 2005. Hyperphysics-Quantum Physics, Department of Physics and Astronomy, Georgia State University, CD.
• Peat, F. David, 2002. From Certainty to Uncertainty: The Story of Science and Ideas in the Twenty-First Century. Joseph Henry Press.
• Hans Reichenbach, 1944. Philosophic Foundations of Quantum Mechanics. University of California Press.
• Paul Arthur Schilpp, 1949. Albert Einstein: Philosopher-Scientist. Tudor Publishing Company.
• Scientific American Reader, 1953.
• Sears, Francis Weston, 1949. Optics. Addison-Wesley.
• Shimony, A. (1983). "(title not given in citation)". Foundations of Quantum Mechanics in the Light of New Technology (S. Kamefuchi et al., eds.): p.225, Tokyo: Japan Physical Society. ; cited in: Popescu, Sandu; Daniel Rohrlich. "Action and Passion at a Distance: An Essay in Honor of Professor Abner Shimony". arXiv.org. http://arxiv.org/abs/quant-ph/9605004. Retrieved on 2007-01-12. 
• Takada, Kenjiro, Emeritus professor of Kyushu University, "Microscopic World-Introduction to Quantum Mechanics."
• "Uncertainty Prirnciple" Werner Heisenberg actual voice recording, http://www.thebigview.com/spacetime/index.html.
• Van Vleck, J. H.,1928, "The Correspondence Principle in the Statistical Interpretation of Quantum Mechanics," Proc. Nat. Acad. Sci. 14: 179.
• Veltman, M. J. G., 2003. Facts and Mysteries in Elementary Particle Physics. World Scientific Publishing Company.
• Wieman, Carl, and Perkins, Katherine, 2005, "Transforming Physics Education," Physics Today.
• Westmoreland, M. D., and Schumacher, B., 1998, "Quantum Entanglement and the Nonexistence of Superluminal Signals."

Notes

External links

• Takada, Kenjiro, Emeritus professor at Kyushu University, "Microscopic World -- Introduction to Quantum Mechanics."
• Westmoreland, M. D., and Schumacher, B., 1998, "Quantum Entanglement and the Nonexistence of Superluminal Signals."
• Quantum Theory.
• Quantum Mechanics.
• Planck's original paper on Planck's constant.
• Everything you wanted to know about the quantum world. From the New Scientist.
• Quantum Articles.
• This Quantum World.
• The Quantum Exchange (tutorials and open source learning software).
• Theoretical Physics wiki
• "Uncertainty Principle," a recording of Werner Heisenberg's voice.