Introduction to quantum mechanics

For a somewhat more technical treatment and applications, see the main article, quantum mechanics.

Quantum mechanics is a physical science dealing with the behaviour of matter and waves on the scale of atoms and subatomic particles. It also forms the basis for the contemporary understanding of how large objects such as stars and galaxies, and cosmological events such as the Big Bang, can be analyzed and explained. Its success is due to its accurate prediction of the physical behaviour of systems, including systems where Newtonian mechanics fails. This difference between the success of classical and quantum mechanics is most often observed in systems at the atomic scale or smaller, or at very low or very high energies, or at extremely low temperatures. Quantum mechanics is the basis of modern developments in chemistry, molecular biology, and electronics, and the foundation for the technology that has transformed the world in the last fifty years.

Background

Through a century of experimentation and applied science, quantum mechanical theory has proven to be very successful and practical. The term "quantum mechanics" was first coined by Max Born in 1924. Quantum mechanics is the foundation for other sciences including condensed matter physics, quantum chemistry, and particle physics.

Despite the success of quantum mechanics, it does have some controversial elements. For example, the behaviour of microscopic objects described in quantum mechanics is very different from our everyday experience, which may provoke an incredulous reaction. Moreover, some of the consequences of quantum mechanics appear to be inconsistent with the consequences of other successful theories, such as Einstein's Theory of Relativity, especially general relativity.

Some of the background of quantum mechanics dates back to the early 1800's, but the real beginnings of quantum mechanics date from the work of Max Planck in 1900^[1]. Albert Einstein^[2], Niels Bohr^[3], and Louis de Broglie^[4] soon made important contributions. However, it was not until the mid-1920's that a more complete picture emerged, and the true importance of quantum mechanics became clear. Some of the most prominent scientists to contribute were Max Born^[5], Paul Dirac^[6], Werner Heisenberg^[7], Wolfgang Pauli^[8], and Erwin Schrödinger^[9].

Later, the field was further expanded with work by Julian Schwinger, Murray Gell-Mann, and Richard Feynman, in particular, with the development of Quantum Electrodynamics in 1947.

Early researchers differed in their explanations of the fundamental nature of what we now call electromagnetic radiation. In 1690, Christian Huygens explained the laws of reflection and refraction on the basis of a wave theory.^[10] Sir Isaac Newton believed that light consisted of particles which he designated corpuscles. In 1827 Thomas Young and Augustin Fresnel made experiments on interference that showed that a corpuscular theory of light was inadequate. Then in 1873 James Clerk Maxwell showed that by making an electrical circuit oscillate it should be possible to produce electromagnetic waves. His theory made it possible to compute the speed of electromagnetic radiation purely on the basis of electrical and magnetic measurements, and the computed value corresponded very closely to the empirically measured speed of light.^[11] In 1888, Heinrich Hertz made an electrical device that actually produced what we would now call microwaves — essentially radiation at a lower frequency than visible light.^[12] Everything up to that point suggested that Newton had been entirely wrong to regard light as corpuscular. Then it was discovered that when light strikes an electrical conductor it causes electrons to move away from their original positions, and, furthermore, the phenomenon observed could only be explained if the light delivered energy in definite packets. In a photoelectric device such as the light meter in a camera, when light hits the metallic detector electrons are caused to move. Greater intensities of light at one frequency can cause more electrons to move, but they will not move any faster. In contrast, higher frequencies of light can cause electrons to move faster. So intensity of light controls the amperes of current produced, but frequency of light controls the voltage produced. This appeared to raise a contradiction when compared to sound waves and ocean waves, where only intensity was needed to predict the energy of the wave. In the case of light, frequency appeared to predict energy. Something was needed to explain this phenomenon and also to reconcile experiments that had shown light to have a particle nature with experiments that had shown it to have a wave nature.

Spectroscopy and Onward

It is fairly easy to see a spectrum produced by white light when it passes through a prism, the beveled edge of a mirror or a special pane of glass, or through drops of rain to form a rainbow. When samples of single elements are caused to emit light they may emit light at several characteristic frequencies. The frequency profile produced is characteristic of that element. Instead of there being a wide band filled with colors from violet to red, there will be isolated bands of single colors separated by darkness. Such a display is called a line spectrum. Some lines go beyond the visible frequencies and can only be detected by special photographic film or other such devices. Scientists hypothesized that an atom could radiate light the way the string on a fine violin radiates sound – not only with a fundamental frequency (in which the entire spring moves the same way at once) but with several higher harmonics (formed when the string divides itself into halves and other divisions that vibrate in coordination with each other as when one half of the string is going one way as the other half of the string is going the opposite way). For a long time nobody could find a mathematical way to relate the frequencies of the line spectrum of any element.

In 1885, Johann Jakob Balmer (1825-1898) figured out how the frequencies of atomic hydrogen are related to each other. The formula is a simple one:

${\frac {1}{L}}=R\left({\frac {1}{2^{2}}}-{\frac {1}{n^{2}}}\right)$

where L is wavelength, R is Rydberg’s constant and n is an integer (2, 3,…n). This formula can be generalized to apply to atoms that are more complicated than hydrogen, but we will stay with hydrogen for this general exposition. The next development was the discovery of the Zeeman effect, named after Pieter Zeeman (1865-1943). The mathematical explanation of the Zeeman effect was worked out by Hendrik Anton Lorentz (1853-1928). Lorentz hypothesized that the light emitted by hydrogen was produced by vibrating electrons. It was possible to get feedback on what goes on within the atom because moving electrons create a magnetic field and so can be influenced by the imposition of an external magnetic field in a manner analogous to the way that one iron magnet will attract or repel another magnet. The Zeeman effect could be interpreted to mean that light waves are originated by electrons vibrating in their orbits, but classical physics could not explain why electrons should not fall out of their orbits and into the nucleus of their atoms, nor could classical physics explain why their orbits would be such as to produce the series of frequencies derived by Balmer’s formula and displayed in the line spectra. Why did the electrons not produce a continuous spectrum?

Old quantum theory

Quantum mechanics developed from the study of electromagnetic waves through spectroscopy which includes visible light seen in the colors of the rainbow, but also other waves including the more energetic waves like ultraviolet light, x-rays, and gamma rays plus the waves with longer wavelengths including infrared waves, microwaves and radio waves. We are not, however, speaking of sound waves, but only of those waves that travel at the speed of light. Also, when the word "particle" is used below, it always refers to elementary or subatomic particles.

Planck's constant

Max Planck discovered that there is a definite relation between the frequency of light and the energy that it carries. As he put it, "Matter can emit radiant energy only in finite quantities proportional to the frequency."^[13] Planck was studying how the radiation of a body was related to its temperature in an attempt to devise a new radiation law. His investigation ended up describing the energy of a wave in mathematical terms. In order to resolve an absurdity in the then-known radiation laws, which predicted that radiation emitted by a heated object should theoretically be infinite, he started with a completely different basic premise: What if energy is not infinitely divisible? He modeled the emission of radiant energy as being the result of regular vibrations of electronic oscillators, essentially of electrons as they are bound by attraction to nuclei of the atoms in the radiant body. By using this specific model he was able to derive once more the same relationship, called Rayleigh's law, that had caused the original problem, but he also saw how the model needed to be modified so that it would no longer predict the infinite production of radiant energy. What he found concerning radiation and oscillators was really a description of electromagnetic waves. Waves have crests and troughs. A cycle is the return from one point such as the crest to the next crest. So Planck used the frequency of the wave, which means the number of cycles per second, to determine the energy of a wave. What he discovered was that, given any wave, a single number when multiplied by the frequency gives the energy of that wave. He found that this single number was the same for waves of all frequencies. This number is now called Planck's constant and is represented as the letter h in physics formulas. One of the most direct applications is finding the energy of photons. If you know h (Planck's constant), and you know the frequency of the photon, then you can calculate the energy of the photons. For instance, if a beam of light illuminated a target, and the light frequency was 540 × 10¹² hertz, then the energy of each photon would be h × 540 × 10¹² joules. The value of h itself is exceedingly small, about 0.00000000000000000000000000000000066260693 joule.seconds (or 6.6260693 x 10^-34 joule seconds in scientific notation).

When you describe the energy of a wave in this manner, it seems that the wave is carrying its energy in a certain number of little packets per second. This discovery then seemed to remake the wave into a particle. These packets of energy carried along with the wave were called quanta by Planck. Quantum mechanics began with the discovery that energy is delivered in packets whose size is related to the frequencies of all electromagnetic waves (and to the color of visible light since in that case frequency determines color). Be aware, however, the descriptions in terms of wave and particle import macro world concepts into the quantum world, where they have only provisional relevance or appropriateness.

In early research on light, there were two competing ways to describe light, either as a wave propagated through empty space, or as small particles traveling in straight lines. Because Planck showed that the energy of the wave is made up of packets, the particle analogy became favored to help understand how light delivers energy in multiples of certain set values designated as quanta of energy. Nevertheless, the wave analogy is also indispensible for helping to understand other light phenomena. In 1905, Albert Einstein used Planck's constant to postulate that the energy in a beam of light occurs in concentrations that he called photons.^[14] According to that 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 the description that stemmed from Planck's research sounds like Newton's corpuscular account, Einstein's photon was still said to have a frequency, and the energy of the photon was accounted proportional to that frequency. The particle account had been compromised once again.

Both the idea of a wave and the idea of a particle are derived from our everyday experience. We cannot see photons. We can only investigate their properties indirectly. We look at some phenomena, such as the rainbow of colors that we see when a thin film of oil rests on the surface of a puddle of water, and we can explain that phenomenon to ourselves by saying that light is like waves.^[15] We look at other phenomena, such as the way a photoelectric meter in our camera works, and we explain it by analogy to particles colliding with the detection screen in the meter. In both cases we take concepts from our everyday experience and apply them to a world we have never seen. Neither form of explanation is entirely satisfactory. To remind us that both "wave" and "particle" are concepts imported from our macro world to explain the world of atomic-scale phenomena, some physicists such as George Gamow 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.

Reduced Planck's constant

File:Gallery SineWave Generation.jpg

Relation between a cycle and a wave; half of a circle describes half of the cycle of a wave

Planck's constant originally represented the energy that a light wave carries as a function of its frequency. Another version of Planck's constant known as the reduced Planck's constant makes its first appearance in the Bohr model of the atom. And again in 1925 when Werner Heisenberg developed his full quantum theory, calculations involving wave analysis called Fourier series were fundamental, and so this "reduced" version of Planck's constant became invaluable because it includes a conversion factor to facilitate calculations involving wave analysis. Finally, this reduced Planck's constant appeared in Dirac's equation and was then given an alternate designation, "Dirac's constant." Therefore, it is appropriate to begin with an explanation of what this constant is, even though we haven't yet touched on the theories that made its use convenient.

As noted above, the energy of any wave is given by its frequency multiplied by Planck's constant. A wave is made up of crests and troughs. In a wave, a cycle is defined by the return from a certain position to the same position such as from the top of one crest to the next crest. A cycle actually is mathematically related to a circle, and both have 360 degrees. A degree is a unit of measure for the amount of turn needed to produce an arc of a certain length at a given distance. A sine curve is generated by a point on the circumference of a circle as that circle rotates. (See a demonstration at: Rotation Applet) There are 2 π radians per cycle in a wave, which is mathematically related to the way a circle has 360° (which are equal to two π radians). (A radian is simply the angle you would get if you measured a distance along the circumference of the circle equal to the radius of the circle, and then drew lines to the center of the circle and looked at the angle thus formed.) Since one cycle is 2 π radians, when h is divided by 2 π the two "2 π" factors will cancel out leaving just the radian to contend with. So, dividing h by 2 π describes a constant that, when multiplied by the frequency of a wave, gives the energy in joules per radian per second. And h/2 π is h-bar or $\hbar ={\frac {h}{2\pi }}\$ .

The reduced Planck's constant is written in mathematical formulas as $\hbar$ , and is read as "h-bar". The reduced Planck's constant allows computation of the energy of a wave in units per radian instead of in units per cycle. These two constants h and h-bar are merely conversion factors between energy units and frequency units. The reduced Planck's constant is used more often than h (Planck's constant) alone in quantum mechanical mathematical formulas for many reasons, one of which is that angular velocity or angular frequency is ordinarily measured in radians per second so using h-bar that works in radians too will save a computation to put radians into degrees or vice-versa. Also, when equations relevant to those problems are written in terms of $\hbar$ , the frequently occurring 2 π factors in numerator and denominator can cancel out, saving a computation. However, in other cases, as in the orbits of the Bohr atom, h/π was obtained naturally for the angular momentum of the orbits. Another expression for the relation between energy and wave length is given in electron volts for energy and angstroms for wavelength: E_photon (eV) = 12,400/λ(Å) -- it appears not to involve h at all, but that is only because a different system of units has been used and now the appropriate conversion factor is 12,400. Template:Ref/I2QP18

Bohr atom

The Bohr model of the atom, showing electron quantum jumping to ground state n=1

In 1897 the particle called the electron was discovered. By means of the gold foil experiment physicists discovered that matter is, volume for volume, largely space. Once that was clear, it was hypothesized that negative charge entities called electrons surround nuclei with positive charges. So at first all scientists believed that the atom must be like a miniature solar system. But that simple analogy predicted that electrons would, within about one hundredth of a microsecond,^[16] crash into the nucleus of the atom. This is because the nucleus has a positive charge and the electron has a negative charge, and unlike charges attract. The great question of the early 20th century was, "Why do electrons normally maintain a stable orbit around the nucleus?"

In 1913, Niels Bohr removed this substantial problem by applying the idea of discrete (non-continuous) quanta to the orbits of electrons. This account 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 atoms.

Bohr explained the orbits that electrons can take by relating the angular momentum of electrons in each "permitted" orbit to the value of h, Planck's constant. He held that an electron in the lowest orbital has an angular momentum equal to h/2π. Each orbit after the initial orbit must provide for an electron's angular momentum being an integer multiple of that lowest value. He depicted electrons in atoms as being analogous to planets in a solar orbit. However, he took Planck's constant to be a fundamental quantity that introduces special requirements at this subatomic level and that explains the spacing of those "planetary" orbits.

Bohr's analysis of electron orbits as circular
A little math on circular orbits. Bohr was very
familiar with the dynamics of simple circular
orbits in an inverse square field as described in
classical mechanics.
Simply explained: To find the acceleration of a
circle, place it inside the shape of a square
where tangents meet, then find the linear speed
along one side of the square, then square the
speed of one side to complete the speed of the
entire square, then divide by the radius of the
circle placed in the square to get the speed
around the circle. Therefore, circular
(centripetal) acceleration is v squared over r
where v is speed and r is radius. The equation for
the centripetal acceleration is $a=v^{2}/r$ . That
is, acceleration is inversely proportional to the
radius of the circle. If the radius is doubled,
then the acceleration is halved.
Also, Kepler's Third Law is that the radius cubed
equals the circumference of the orbit squared.
It immediately follows that the radius of any n
orbit is proportional to the orbit n squared, and
the speed in that orbit is proportional to 1/n. Speed
times radius gives angular momentum. That leaves
n-squared over n. It then follows that the angular
momentum for any orbit n is just proportional to
n. Bohr argued then that the angular momentum
in any orbit n was nKh, where h is Planck's
constant and K is some multiplying factor, the
same for all the orbits, which was later
determined to be 1/2 $\pi$ .

Bohr considered one revolution in orbit to be equivalent to one cycle in an oscillator (as in Planck's initial measurements to define the constant h) which is in turn similar to one cycle in a wave. The number of revolutions per second is (or defines) what we call the frequency of that electron or that orbital. Specifying that the frequency of each orbit must be an integer multiple of Planck's constant h would only permit certain orbits, and would also fix their size. Bohr studied Balmer's formula for hydrogen and then realized how the angular momentum of an electron in its orbit, L, is quantized , i.e., he determined that there is some constant value K such that when it is multiplied by Planck’s constant, h, it will yield the angular momentum that pertains to the lowest orbital. When it is multiplied by successive integers it will then give the values of other possible orbitals. He later determined that K = 1/2π . (See the detailed argument at [17].)

Bohr's theory represented electrons as orbiting the nucleus of an atom in a way that was amazingly different from what we see in the world of our everyday experience. He showed that when an electron changed orbits it did not move in a continuous trajectory from one orbit around the nucleus 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 orbital. 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 inner orbital. Electrons cannot gain or lose a fractional quantum of energy, and so, it is argued, they cannot have a position that is at a fractional distance between allowed orbitals. Allowed orbitals were designated as whole numbers using the letter n with the innermost orbital being designated n = 1, the next out being n = 2, and so on. Any orbital with the same value of n is called an electron shell.

Bohr's model of the atom was essentially two-dimensional because it depicts electrons as particles in circular orbits. In this context, two-dimensional means something that can be described on the surface of a plane. One-dimensional means something that can be described by a line. Because circles can be described by their radius, which is a line segment, sometimes Bohr's model of the atom is described as one-dimensional. ^[18]

Wave-particle duality

File:Bohr atomic wave.jpg

Probability distribution of the Bohr atom

Niels Bohr determined that it is impossible to describe light adequately by the sole use of either the wave analogy or of the particle analogy. Therefore he enunciated the principle of complementarity, which is a theory of pairs, such as the pairing of wave and particle or the pairing of position and momentum. Louis de Broglie worked out the mathematical consequences of these findings. In quantum mechanics, it was found that electromagnetic waves could react in certain experiments as though they were particles and in other experiments as though they were waves. It was also discovered that subatomic particles could sometimes be described as particles and sometimes as waves. This discovery led to the theory of wave-particle duality by Louis-Victor de Broglie in 1924, which states that subatomic entities have properties of both waves and particles at the same time.

The Bohr atom model was enlarged upon with the discovery by de Broglie that the electron has wave-like properties. In accord with de Broglie's conclusions, electrons can only appear only under conditions that permit a standing wave. A standing wave can be made if a string is fixed on both ends and made to vibrate (as it would in a stringed instrument). That illustration shows that the only standing waves that can occur are those with zero amplitude at the two fixed ends. The waves created by a stringed instrument appear to oscillate in place, simply changing crest for trough in an up-and-down motion. A standing wave can only be formed when the wave's length fits the available vibrating entity. In other words, no partial fragments of wave crests or troughs are allowed. In a round vibrating medium, the wave must be a continuous formation of crests and troughs all around the circle. Each electron must be its own standing wave in its own discrete orbital.

Development of modern quantum mechanics

Full quantum mechanical theory

Werner Heisenberg developed the full quantum mechanical theory in 1925 at the young age of 23. Following his mentor, Niels Bohr, Werner Heisenberg began to work out a theory for the quantum behavior of electron orbitals. Because electron orbits could not be observed, Heisenberg went about creating a mathematical description of quantum mechanics built on what could be observed, that is, the light emitted from atoms in their atomic spectrum. When a pure sample of a single element, such as neon, is caused to emit light by heating it or otherwise supplying energy to it, it does not emit white light. Instead, it emits light at certain characteristic frequencies. Heisenberg used a form of mathematics related to the mathematics of arrays of numbers known as "matrices" to analyze these characteristic frequencies. He worked under the hypothesis that electrons emit characteristic photons of light when moving from an orbital farther from the nucleus to an orbital nearer to the nucleus. Instead of trying to explain every possible orbit, he began with the assumption that electron orbitals that cannot be observed in the spectrum simply do not exist. By this time it was known that the electron orbital was three-dimensional, but trying to work out the mathematics for a three-dimensional atom proved too complicated, so he imagined the electron orbital flattened out to one dimension. Heisenberg studied the electron orbital as a charged ball on a spring, an oscillator, whose motion was not quite regular which is called anharmonic. To see a picture of a charged ball on a spring see: Vibrating Charges. He based the observed motion on the laws of classical mechanics known to apply in the macro world, and then applied quantum properties, discrete (non-continuous) properties, to the description. Doing so would leave gaps between the orbitals so that the mathematical description he formulated would then represent only the observed electron orbitals. The special type of multiplication that turned out to be required in his formula was most appropriately described by using special arrays of numbers called matrices. In ordinary situations it does not matter in which order the operations involved in multiplication are performed, but matrix multiplication does not commute. Essentially that means that it matters which order given operations are performed in. (The important thing is that it matters whether one measures velocity first and then measures position, or vice-versa.) The matrix convention turned out to be a convenient way of organizing information and making clear the exact sequence in which calculations must be made.

In matrix mathematics sets of numbers are given in rows and columns, and there are conventions for the way multiplication of matrices is performed. If everybody arranged their matrices entirely as they pleased, then understanding every new matrix calculation would involve learning the personal plan of the person who made the matrix, so certain conventions have evolved (see above). The matrices in the illustration have been set up to allow easy computation of total tuition fees received at "School One" and at "School Two," and also the total laundry fees received at each school. But reverse the matrices, or, in other words, reverse the order of the multiplication of the matrices and the numerical results will depict what would happen if some female students paid at the rate for males, etc., etc., an accountant's nightmare. Because these complex operations are, by analogy, called "multiplication," it is tempting to imagine that since it does not matter whether one multiplies the number of students by the tuition rate, or the tuition rate by the number of students, then it ought to be irrelevant whether one multiplies matrix A by matrix B, or one multiplies matrix B by matrix A. Because of the complications involved in the rules of matrix multiplication, in almost every case the ordinary mathematical expectation of commutation does not hold. (Sometimes matrices even anticommute.) In Heisenberg's matrix mechanics, the sets of numbers are infinite, representing all possible positions of the electron, and those matrices cannot be multiplied in reverse order and still produce the correct results.

Heisenberg approached quantum mechanics from the historical perspective that treated an electron as an oscillating charged particle. Bohr's use of this analogy had already allowed him to explain why the radii of the orbits of electrons could only take on certain values. It followed from this interpretation of the experimental results available and the quantum theory that Heisenberg subsequently created that an electron could not be at any intermediate position between two "permitted" orbits. Therefore electrons were described as "jumping" from orbit to orbit. The idea that an electron might now be in one place and an instant later be in some other place without having traveled between the two points was one of the earliest indications of the "spookiness" of quantum phenomena. Although the scale is smaller, the "jump" from orbit to orbit is as strange and unexpected as would be a case in which someone stepped out of a doorway in London onto the streets of Los Angeles.

Amplitudes of position and momentum that have a period of 2 pi like a cycle in a wave are called Fourier series variables. Heisenberg described the particle-like properties of the electron in a wave as having position and momentum in his matrix mechanics. When these amplitudes of position and momentum are measured and multiplied together, they give intensity. However, he found that when the position and momentum were multiplied together in that respective order, and then the momentum and position were multiplied together in that respective order, there was a difference or deviation in intensity between them of h/2 $\pi$ . As we have said earlier h/2 $\pi$ is h-bar and describes the radian in a quantized cycle. Heisenberg would not understand the reason for this deviation until two more years had passed, but for the time being he satisfied himself with the idea that the math worked and provided an exact description of the quantum behavior of the electron. Max Born reviewed Heisenberg's paper and recognized that the type of multiplication Heisenberg had found necessary to specify physical observables like position and momentum of particles was matrix mathematics in which arrays of numbers did not follow the law of commutation.

Matrix mechanics was the first complete definition of quantum mechanics, its laws, and properties that described fully the behavior of the electron. It was later extended to apply to all subatomic particles.

Schrödinger wave equation

Model of the Schrödinger atom, showing the nucleus with two protons (blue) and two neutrons (red), orbited by two electrons (waves)

Because particles could be described as waves, later in 1925 Erwin Schrödinger analyzed what an electron would look like as a wave around the nucleus of the atom. Using this model, he formulated his equation for particle waves. Rather than explaining the atom by analogy to satellites in planetary orbits, he treated everything as waves whereby each electron has its own unique wavefunction. A wavefunction is described in Schrödinger's equation by three properties (later Paul Dirac added a fourth). The three properties were (1) an "orbital" designation, indicating whether the particle wave is one that is closer to the nucleus with less energy or one that is further from the nucleus with more energy, (2) the shape of the orbital, i.e. an indication that orbitals were not just spherical but other shapes, and (3) the magnetic moment of the orbital, which is a manifestation of force exerted by the charge of the electron as it rotates around the nucleus.

These three properties were called collectively the wavefunction of the electron and are said to describe the quantum state of the electron. "Quantum state" means the collective properties of the electron describing what we can say about its condition at a given time. For the electron, the quantum state is described by its wavefunction which is designated in physics by the Greek letter $\psi$ (psi pronounced "sigh"). The three properties of Schroedinger's equation that describe the wavefunction of the electron and therefore also describe the quantum state of the electron as described in the previous paragraph are each called quantum numbers. The first property which describes the orbital was numbered according to Bohr's model where n is the letter used to describe the energy of each orbital. This is called the principal quantum number. The next quantum number that describes the shape of the orbital is called the azimuthal quantum number and it is represented by the letter l (lower case L). The shape is caused by 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 speed up or slow 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. So for the letter "l" there are other letters to describe the shapes of "l". The first shape is spherical and is described by the letter s. The next shape is like a dumbbell and is described by the letter p. The other shapes of orbitals become more complicated and are described by the letters d, f, and g. To see the shape of a carbon atom, see Carbon atom. The third quantum number of Schrödinger's equation describes the magnetic moment of the electron and is designated by the letter m and sometimes as the letter m with a subscript l because the magnetic moment depends upon the second quantum number l.

In May 1926 Schrödinger published a proof that Heisenberg's matrix mechanics and Shroedinger's wave mechanics gave equivalent results: mathematically they were the same theory. Both men claimed to have the superior theory. Heisenberg insisted on the existence of discontinuous quantum jumps in his particle-like examination of the oscillation of a charged electron giving more precise definitions and Schroedinger insisted that a theory based on continuous wave-like properties which he called "matter-waves" was better.

Uncertainty Principle

In 1927, Heisenberg made a new discovery on the basis of his quantum theory that had further practical consequences of this new way of looking at matter and energy on the atomic scale. In Heisenberg's matrix mechanics formula, Heisenberg had encountered an error or difference of h/2 $\pi$ between position and momentum. This represented a deviation of one radian of a cycle when the particle-like aspects of the wave were examined. Heisenberg analyzed this difference of one radian of a cycle and divided the difference or deviation of one radian equally between the measurement of position and momentum. This had the consequence of being able to describe the electron as a point particle in the center of one cycle of a wave so that its position would have a standard deviation of plus or minus one-half of one radian of the cycle (1/2 of h-bar). A standard deviation can be either plus or minus the measurement i.e. it can add to the measurement or subtract from it. In three-dimensions a standard deviation is a displacement in any direction. What this means is that when a moving particle is viewed as a wave it is less certain where the particle is. In fact, the more certain the position of a particle is known, the less certain the momentum is known. This conclusion came to be called "Heisenberg's Indeterminacy Principle," or Heisenberg's Uncertainty Principle. To understand the real idea behind the uncertainty principle imagine a wave with its undulations, its crests and troughs, moving along. A wave is also a moving stream of particles, so you have to superimpose a stream of particles moving in a straight line along the middle of the wave. An oscillating ball of charge creates a wave larger than its size depending upon the length of its oscillation. Therefore, the energy of a moving particle is as large as the cycle of the wave, but the particle itself has a location. Because the particle and the wave are the same thing, then the particle is really located somewhere in the width of the wave. Its position could be anywhere from the crest to the trough. The math for the uncertainty principle says that the measurement of uncertainty as to the position of a moving particle is one-half the width from the crest to the trough or one-half of one radian of a cycle in a wave.

For moving particles in quantum mechanics, there is simply a certain degree of exactness and precision that is missing. You can be precise when you take a measurement of position and you can be precise when you take a measurement of momentum, but there is an inverse imprecision when you try to measure both at the same time as in the case of a moving particle like the electron. In the most extreme case, absolute precision of one variable would entail absolute imprecision regarding the other.

Heisenberg voice recording in an early lecture on the uncertainty principle pointing to a Bohr model of the atom: "You can say, well, this orbit is really not a complete orbit. Actually at every moment the electron has only an inactual position and an inactual velocity and between these two inaccuracies there is an inverse correlation."

The consequences of the uncertainty principle were that the electron could no longer be considered as in an exact location in its orbital. Rather the electron had to be described by every point where the electron could possibly inhabit. By creating points of probable location for the electron in its known orbital, this created a cloud of points in a spherical shape for the orbital of a hydrogen atom which points gradually faded out nearer to the nucleus and farther from the nucleus. This is called a probability distribution. Therefore, the Bohr atom number n for each orbital became known as an n-sphere in the three dimensional atom and was pictured as a probability cloud where the electron surrounded the atom all at once.

This led to the further description by Heisenberg that if you were not making measurements of the electron that it could not be described in one particular location but was everywhere in the electron cloud at once. In other words, quantum mechanics cannot give exact results, but only the probabilities for the occurrence of a variety of possible results. Heisenberg went further and said that the path of a moving particle only comes into existence once we observe it. However, strange and counter-intuitive this may seem, quantum mechanics however does tell us the location of the electron's orbital, its probability cloud. Heisenberg was speaking of the particle itself, not its orbital which is in a known probability distribution.

It is important to note that although Heisenberg used infinite sets of positions for the electron in his matrices, this does not mean that the electron could be anywhere in the universe. Rather there are several laws that show the electron must be in one localized probability distribution. An electron is described by its energy in Bohr's atom which was carried over to matrix mechanics. Therefore, an electron in a certain n-sphere had to be within a certain range from the nucleus depending upon its energy. This restricts its location. Also, the number of places an electron can be is also called "the number of cells in its phase space". The uncertainty principle set a lower limit to how finely one can chop up classical phase space. Therefore, the number of places that an electron can be in its orbital becomes finite due to the Uncertainty Principle. Therefore, an electron's location in an atom is defined to be in its orbital and its orbital although being a probability distribution does not extend out into the entire universe, but stops at the nucleus and before the next n-shere orbital begins and the points of the distribution are finite due to the Uncertainty Principle creating a lower limit.

Classical physics had shown since Newton that if you know the position of stars and planets and details about their motions that you can predict where they will be in the future. For subatomic particles, Heisenberg denied this notion showing that due to the uncertainty principle one cannot know the precise position and momentum of a particle at a given instant, so its future motion cannot be determined, but only a range of possibilities for the future motion of the particle can be described.

These notions arising from the uncertainty principle only arise at the subatomic level and were a consequence of wave-particle duality. As counter-intuitive as they may seem, quantum mechanical theory with its uncertainty principle has been responsible for major improvements in the world's technology from computer components to fluorescent lights to brain scanning techniques.

Wavefunction collapse

Schrödinger's wave equation with its unique wavefunction for a single electron is also spread out in a probability distribution like Heisenberg's quantized particle-like electron. This is because a wave is naturally a widespread disturbance and not a point particle. Therefore, Schroedinger's wave equation has the same predictions made by the uncertainty principle because uncertainty of location is built into the definition of a widespread disturbance like a wave. Uncertainty only needed to be defined from Heisenberg's matrix mechanics because the treatment was from the particle-like aspects of the electron. Schrödinger's wave equation shows that the electron is in the probability cloud at all times in its probability distribution as a wave that is spread out. Max Born discovered in 1928 that when you compute the square of Schrödinger's wavefunction (psi-squared), you get the electron's location as a probability distribution. Therefore, if a measurement of the position of an electron is made as an exact location in space instead of as a probability distribution, it ceases to have wave-like properties. Without wave-like properties, none of Schrödinger's definitions of the electron being wave-like make sense anymore. The measurement of the position of the particle nullifies the wave-like properties and Schrödinger's equation then fails. Because the electron can no longer be described by its wavefunction when measured due to it becoming particle-like, this is called wavefunction collapse.

Eigenstates and eigenvalues

The term eigenstate is derived from the German/Dutch word "eigen," which means "inherent" or "characteristic." The word eigenstate is descriptive of the measured state of some object that possesses quantifiable characteristics such as position, momentum, etc. The state being measured and described must be an "observable" (i.e. something that can be experimentally measured either directly or indirectly like position or momentum), and must have a definite value. In the everyday world, it is natural and intuitive to think of everything being in its own eigenstate. Everything appears to have a definite position, a definite momentum, a definite value of measure, and a definite time of occurrence. However, quantum mechanics affirms that it is impossible to pinpoint exact values for the momentum of a certain particle like an electron in a given location at a particular moment in time, or, alternatively, that it is impossible to give an exact location for such an object when the momentum has been measured. Due to the uncertainty principle, statements regarding both the position and momentum of particles can only be given in terms of a range of probabilities, a "probability distribution." Eliminating uncertainty in one term maximizes uncertainty in regard to the second parameter.

Therefore it became necessary to have a way to clearly formulate the difference between the state of something that is uncertain in the way just described, such as an electron in a probability cloud, and effectively contrast it to the state of something that is not uncertain, something that has a definite value. When something is in the condition of being definitely "pinned-down" in some regard, it is said to possess an eigenstate. For example, if the position of an electron has been made definite, it is said to have an eigenstate of position.

A definite value, such as the position of an electron that has been successfully located, is called the eigenvalue of the eigenstate of position. The German word "eigen" was first used in this context by the mathematician David Hilbert in 1904. Schrödinger's wave equation gives wavefunction solutions, meaning the possibilities where the electron might be, just as does Heisenberg's probability distribution. As stated above, when a wavefunction collapse occurs because something has been done to locate the position of an electron, the electron's state becomes an eigenstate of position, meaning that the position has a known value.

The Pauli Exclusion Principle

There was a doublet, meaning a pair of lines, in the spectrum of a hydrogen atom that was unaccounted for. This meant that there was more energy in the electron orbital from magnetic moment than had previously been described. Wolfgang Pauli when studying alkali metals had introduced what he called a "two-valued quantum degree of freedom" associated with the electron in the outermost shell. "Degrees of freedom" simply means the number of possible independent ways a particle may move. This led to the Pauli Exclusion Principle that predicted that no more than two electrons can inhabit the same orbital. It also predicted that any neutron, electron, or proton (types of fermions) could not exist in the same quantum state. We learned in Schrödinger's equation that there were three quantum states of the electron, but if two electrons could be in the same orbital, there had to be another quantum number to distinguish those two electrons from each other and to describe the extra magnetic moment shown in the atomic spectrum. In early 1925, the young physicist Ralph Kronig had introduced a theory to Pauli that the electron rotates in space in the same way that the earth rotates on its axis. This would account for the missing magnetic moment and allow for two electrons in the same orbital to be different if their spin was in opposite directions to each other thus satisfying the Exclusion Principle.

The Pauli Exclusion Principle states that no electron (or other fermion) can be in the same quantum state as another. This has an effect on the probability distribution of the electron further defining the number of cells in its phase space. The minimum limit is the limit of the Uncertainty Principle and the Exclusion Principle states that no two electrons can be within this same minimum space defined by the Uncertainty Principle.

Therefore, a single electron in its orbital when defined by its quantum state which is its wavefunction which is defined by its four quantum numbers cannot have the same four quantum numbers of another electron in that atom. Where two electrons are in the same n-sphere and therefore share the same principal quantum number, they must then have some other unique quantum number of shape l or magnetic moment m or spin s. Even in the formation of degenerate gases where the electrons are not in an orbital around the nucleus of an atom, they must still follow the Pauli Exclusion Principle when in a confined space.

Dirac wave equation

In 1928, Paul Dirac worked out a variation of Schrödinger's equation that accounted for a fourth property of the electron in its orbital. Paul Dirac introduced the fourth quantum number called the spin quantum number designated by the letter s to the new Dirac equation of the wavefunction of the electron. In 1930, Dirac combined Heisenberg's matrix mechanics with Schrödinger's wave mechanics into a single quantum mechanical representation in his Principles of Quantum Mechanics. The quantum picture of the electron was now complete.

All of the above development of quantum theory was based mainly on the atomic spectrum of the hydrogen atom. This is due to the fact that each atom of each element produces a unique pattern of spectral lines when light from each different kind of element is passed through a prism. Scientists could not study the electron and nucleus of the atom itself because they cannot be seen. Even today with High-resolution Scanning Tunneling Electron Microscopes we can only get images of the atom as a blurry fuzzball. However, the spectral lines of the atom reveal the orbits of electrons and the energy that can be expected. It was basically this study of the spectroscopic analysis of first the hydrogen atom and then the helium atom that led to quantum theory. Therefore, the mathematical formula were made to fit the picture of the atomic spectrum. That is why quantum mechanics is sometimes referred to as a form of mathematical physics.

Quantum entanglement

Albert Einstein rejected Heisenberg's Uncertainty Principle. Heisenberg's quantum mechanics based on Bohr's initial explanation became known as the Copenhagen Interpretation of quantum mechanics. Both Bohr and Einstein spent many years defending and attacking this interpretation. After the 1930 Solvay conference, Einstein never again challenged the Copenhagen interpretation on technical points, but did not cease a philosophical attack on the interpretation, on the grounds of realism and locality. Einstein, in trying to show that it was not a complete theory, recognized that the theory predicted that two or more particles which have interacted in the past exhibit surprisingly strong correlations when various measurements are made on them.^[19] Einstein called this "spooky action at a distance". In 1935, Schrödinger published a paper explaining the argument which had been denoted the EPR paradox (Einstein-Podolsky-Rosen, 1935). Einstein showed that the Copenhagen Interpretation predicted quantum entanglement which he was trying to prove was incorrect in that it would defy the law of physics that nothing could travel faster than the speed of light. Quantum entanglement means that when there is a change in one particle at a distance from another particle then the other particle automatically changes to counter-balance the system. In quantum entanglement measuring one entangled particle defines its properties and seems to influence the properties of its partner or partners instantaneously no matter how far apart they are. Because the two particles are entangled states, changes on the one cause instantaneous effects on the other. Einstein had calculated that quantum theory would predict this, he saw it as a flaw and therefore challenged it. However, instead of showing a weakness in quantum mechanics, this forced quantum mechanics to acknowledge that quantum entanglement did in fact exist and it became another foundation theory of quantum mechanics. The 1935 paper is currently Einstein's most cited publication in physics journals.

Bohr's original response to Einstein was that the particles were in a system. However, Einstein's challenge led to decades of substantial research into this quantum mechanical phenomenon of quantum entanglement. This research clarified by Yanhua Shih points out that the two entangled particles can be viewed as somehow not separate, which removes the locality objection^[20]. This means that no matter the distance between the entangled particles, they remain in the same quantum state so that one particle is not sending information to another particle faster than the speed of light, but rather a change to one particle is a change to the entire system or quantum state of the entangled particles and therefore changes the state of the system without information transference.

References

Mara Beller, Quantum Dialogue: The Making of a Revolution. University of Chicago Press, Chicago, 2001.
Bohr, Niels, Atomic Physics and Human Knowledge. John Wiley and Sons, 1958.
De Broglie, Louis. The Revolution in Physics, Noonday Press, 1953.
Feynman, Richard P., QED: The Strange Theory of Light and Matter, Princeton University Press, 1985. ISBN: 0-691-08388-6
Feigl, Herbert and May Brodbeck, Readings in the Philosophy of Science, Appleton-Century-Crofts, 1953.
Einstein, Albert. Essays in Science, Philosophical Library, 1934.
Prof. Michael Fowler, The Bohr Atom, A series of lectures, 1999, University of Virginia.
Heisenberg, Werner. Physics and Philosophy, Harper and Brothers, 1958.
S Lakshmibala, "Heisenberg, Matrix Mechanics and the Uncertainty Principle", Resonance, Journal of Science Education, Volume 9, Number 8, August 2004.
Richard L. Liboff, Introductory Quantum Mechanics, 2nd ed. 1992.
Lindsay, Robert Bruce and Henry Margenau, Foundations of Physics, Dover, 1936.
Carl Rod Nave, Hyperphysics-Quantum Physics, Department of Physics and Astronomy, Georgia State University, CD 2005.
F. David Peat, "From Certainty to Uncertainty: The Story of Science and Ideas in the Twenty-First Century", Joseph Henry Press, 2002.
Reichenbach, Hans, Philosophic Foundations of Quantum Mechanics, University of California Press, 1944.
Schilpp, Paul Arthur, Albert Einstein: Philosopher-Scientist, Tudor Publishing Commpany, 1949.
Scientific Americn Reader, 1953.
Sears, Francis Weston, Optics, Addison-Wesley, 1949.
Dr. Kenjiro Takada, Emeritus professor of Kyushu University, Microscopic World-Introduction to Quantum Mechanics, Internet seminar, http://www2.kutl.kyushu-u.ac.jp/seminar/MicroWorld1_E/MicroWorld_1_E.html.
"Uncertainty Prirnciple" Werner Heisenberg actual voice recording, http://www.thebigview.com/spacetime/index.html.
J.H. Van Vleck, The Correspondence Principle in the Statistical Interpretation of Quantum Mechanics, Proc. Nat. Acad. Sci., Vol. 14, p.179, 1928.
Carl Wieman and Katherine Perkins, "Transforming Physics Education", Physics Today, November 2005,

Notes

^ Events leading up to the December 1900 publication of Planck's quantum hypothesis are related by Werner Heisenberg in his Physics and Philosophy, pp. 30f. Heisenberg believes that Planck was clearly aware that his ideas would have very far-reaching consequences.

^ It is remarkable that Einstein made significant contributions to quantum mechanics at the same time he was revolutionizing physics with his relativity theories. The far-reaching nature of his contributions to quantum mechanics is noted by Richard Feynman in QED: The Strange Theory of Light and Matter, p. 112: "[The] phenomenon of 'stimulated emission' was discovered by Einstein when he launched the quantum theory proposing the photon model of light. Lasers work on the basis of this phenomenon."

^ Albert Einstein characterized Niels Bohr's contributions to the quantum revolution by saying that history "will have to connect one of the most important advances ever made in our knowledge of the nature of the atom with the name of Niels Bohr." He added, "The boldly selected hypothetical basis of his speculations soon became a mainstay for the physics of the atom....The theory of the Röntgen spectra of the visible spectra, and the periodic system of the elements is primarily based on the ideas of Bohr." See Einstein's Essays in Science, p. 46f.

^ Niels Bohr records the contributions of Louis de Broglie toward "a more comprehensive quantum theory" that would take into account that "the wave-corpuscle duality was not confined to the properties of radiation, but was equally unavoidable in accounting for the behaviour of material particles." See his Atomic Physics and Human Knowledge, p. 37 et passim.

^ See Max Born, Atomic Physics, especially p. 90, where he says of quantum mechanics that it is "in the nature of the case indeterministic, and therefore the affair of statistics."

^ A two-page account of the highlights of Dirac's work, including his speculation that a positron would be found, appears in an article on "The Ultimate Particles" by George W. Gray, in The Scientific American Reader, p. 100f. (Simon and Schuster, 1953).

^ Werner Heisenberg is well known for his "indeterminancy principle" or "uncertainty principle."

^ Wolfgang Pauli's name is most closely associated with what is known as the "Pauli Exclusion Principle," according to which, it is impossible, in the words of Louis de Broglie, "for two electrons to have rigorously identical quantized states, i.e., defined by the same quantum numbers.... Translated into wave mechanics, Pauli's principle is expressed as follows: 'for electrons, the only states realized in nature are the antisymmetric states.'" (See de Broglie's The Revolution in Physics, p. 267)

^ Schrödinger's cat was originally a character in an example intended to be critical of an apparent difficulty in Heisenberg's exposition of his principle of indeterminancy. The story has been taken somewhat out of context and the cat has assumed a minor literary life of its own. Schrödinger's other contributions to understanding quantum mechanics and to making the mathematics easier to handle were, of course, much more important. The translation of his 1935 essay that includes the story is to be found at http://www.qedcorp.com/pcr/pcr/qcat.html. Schrödinger describes a situation in which a cat will live or die depending on whether a quantum mechanically probabalistic radioactive emmision event occurs within the hour the cat is confined to a box. To Heisenberg's interpretation of quantum mechanics he objects: "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 psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts."

^ Huygens' principle is explained in Optics, by Francis Weston Sears, Addison-Wesley, 1949, pp. 5f.

^ Sears, op. cit., p. 2.

^ Sears, op. cit., p. 2f.

^ See Sears, Optics, pp. 282-293.

^ Planck is quoted by Louis de Broglie, The Revolution in Physics, p. 106. The material in this paragraph summarizes de Broglie's account given on pages 105 to 108. (Noonday Press, New York, 1953)

^ A. Einstein, 'Ann. d. Phys., 17, 132, (1905).

^ For the length of time involved, see George Gamow's One, Two, Three...Infinity, p. 140.

^ A very clear explanation of interference in thin films may be found in Sears, op. cit., p. 203ff.

^ See Linus Pauling, The Nature of the Chemical Bond, p. 35f. He explains that a circular electron orbit in the case of the first orbital of the hydrogen atom would predict orbital angular momentum for the atom, which it does not in fact have. He therefore depicts the electron as moving from the nucleus and back in a straight line.

^ Hans Reichenback works out the mathematics in one sample case of quantum entanglement. See his Philosophic Foundations of Quantum Mechanics, p. 170 ff.

^ Yanhua H. Shih (2001), "Quantum Entanglement and Quantum Teleportation" Annals of Physics 10 (2001) 1-2 pp.45-61 as referenced by Amir Aczel (2003), Entanglement p.252 ISBN 0-452-28457-0

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