History
In 1887, Heinrich Rudolph Hertz while detecting electromagnetic waves also discovered a phenomenon which ultimately led to the description of light in terms of corpuscles: the photons.
In 1900, Max Planck presented a new form of the black body radiation spectral distribution, based on a revolutionary hypothesis. A black body is an object that absorbs all the energy that falls upon it and, since it reflects no light, it appears black to an observer. Planck postulated that the energy of an oscillator of a given frequency v cannot take arbitrary values between zero and infinity, but can only take discrete values nε0, where n is a positive integer or zero, and ε0 is a finite amount, or quantum of energy, which may depend on the frequency v such that ε0 = hv where h is a fundamental physical constant, called Planck's constant, that is used to describe the size of a quantum (or "bundle") of energy.
The idea of quantization of energy, in which the energy of a system can only take certain discrete values, was totally at variance with classical physics, and Planck's theory was not accepted readily.
In 1905, Albert Einstein advanced the idea that these quantum properties were inherent in the nature of electromagnetic radiation itself, so that light consists of quanta (corpuscles) called photons.
In 1911, Ernest Rutherford postulated that all the positive charge and almost all the mass of an atom is concentrated in a positively charged nucleus of very small dimension as compared with the dimension of the atom as a whole.
In 1913, a major step forward was taken by Niels Bohr to explain the spectrum of the hydrogen atom by introducing the quantum concept into the physics of atoms. Combining the concepts of Rutherford's nuclear atom, Planck's quanta and Einstein photons, Bohr was able to explain the observed expectrum of atomic hydrogen. However, several objections persisted on the theory now called old quantum theory.
In 1923, Bohr formulated a heuristic principle which had already inspired the development of the old quantum theory, and which proved to be of great help in the early development of quantum mechanics. This principle, known as the correspondence principle, states that quantum theory results must tend asymptotically to those results obtained from classical physics in the limit of large quantum numbers. In other words, classical physics results are 'macroscopically correct' and may be considered as limiting cases of quantum mechanical results when the quantum discontinuities may be neglected.
In 1923-4, Louis de Broglie made a great unifying and bold hypothesis that, in addition to quantum properties, material particles might also possess wave-like properties, so that, like electromagnetic radiation, they would exhibit a dual nature: a quanta (corpuscular) behavior as well as a wave behavior. In this way the wave-particle duality became a universal characteristic of nature. Broglie determined that not only electrons but all material particles possess wave-like characteristics. This universality of matter waves has been confirmed by a number of experiments.
It is worth stressing that the wave nature of matter is directly related to the finiteness of the Planck constant h. Given p as the magnitude of a photon momentum then, if h were zero, the de Broigle's wavelength λ = h/p of a material particle would also vanish, and the particle would obey the laws of classical mechanics. It is because Planck's constant is 'small' (when measured in units appropriate for the description of macroscopic phenomena) that the wave behavior of matter is not apparent on the macroscopic scale.
The requirement that classical mechanics is contained in quantum mechanics in the limiting case when de Broigle's wavelength λ → 0 (i.e., h → 0) is in accordance with Niels Bohr's correspondence principle of quantum mechanics.
As a result of this experimental evidence, revolutionary concepts had to be introduced, such as those of quantization and of wave-particle duality, and a new theory, called quantum mechanics, was elaborated between the years 1925 and 1930. In fact, two equivalent formulations of the theory were proposed at nearly the same time.
The first theory, known as matrix mechanics was developed in the years 1925 and 1926 by W. Heisenberg, M. Born and P. Jordan. In this approach, only physically observable quantities appear, and to each physical quantity the theory associates a matrix.
The second form of quantum mechanics, called wave mechanics, was proposed in 1925 by E. Schrödinger, following the ideas put forward in 1923 by L. de Broglie about matter waves.
The equivalence of matrix mechanics and wave mechanics was proved in 1926 by Schrödinger. In fact both matrix mechanics and wave mechanics are particular forms of a general formulation of quantum mechanics which was developed by P.A.M. Dirac in 1930.
Around 1927, the collaboration of Bohr and Heisenberg extended the probabilistic interpretation of the wave function proposed by Max Born. This became known as the "Copenhagen Interpretation" of quantum mechanics because it was largely developed at Niels Bohr's Institute of Atomic Studies, located in Copenhagen, Denmark. Although, in fact, Bohr and Heisenberg never totally agreed on how to understand the mathematical formalism of quantum mechanics, and none of them ever used the term “the Copenhagen interpretation” as a joint name for their ideas.
It may be said that the general formulation of quantum mechanics, as well as matrix mechanics, requires a certain amount of abstract mathematics. Wave mechanics, on the other hand, is more suitable for a first contact with quantum theory.
Recently, perhaps partly in response to the severe technical difficulties now besetting quantum theory at the fundamental level, there has been mounting criticism of the Copenhagen interpretation. As a result, more recent proposals intend to complete quantum mechanics not within quantum mechanics proper but on a 'higher (synthetic) level', by means of a combination with gravitational theory (R. Penrose), with quantum information theory (C.M. Caves, C.A. Fuchs) or with psychology and brain science (H.P. Stapp).
According to G. Mahler from the Institute of Theoretical Physics,Stuttgart, Germany, quantum mechanics is usually defined in terms of some loosely connected axioms and rules, and views each of these recent proposals as a combination with a subject that, per se, suffers from a very limited understanding that is even more severe than that of quantum mechanics. And how far it will be able to go in this direction will depend on the amount of concrete research results becoming available to support any of these views.
Salient Characteristics
Vector Space
Quantum mechanics is the most accurate and complete description of the world known. One of the foundations of quantum mechanics is vector space. As a result, a good understanding of quantum mechanics is based upon a solid grasp of elementary linear algebra. Linear algebra is the study of vector space and of the linear operations on those vector spaces and their elements, i.e., the vectors.
Quantum mechanics is easy to learn, despite its reputation as a difficult subject. The reputation comes from the difficulty of some applications, like understanding the structure of complicated molecules, which aren't fundamental to a grasp of the subject. As explained before, the only prerequisite for understanding is some familiarity with elementary linear algebra. The basic objects of linear algebra are vector spaces, and the elements of a vector space are called vectors. The standard quantum mechanical notation for a vector in a vector space is the following: |v>, where v is a label for the vector.
The notion of vectors has proved of great value both in physics and mathematics for two main reasons: (1) Vectors enable one to reason about problems in space without the use of coordinate axes. Since the laws of physics do not depend on the particular position of the coordinate axes in space, vectors are admirably adapted to the statement of such laws. (2) Vectors provide an economical shorthand for complicated formulas.
Dirac's Notation: Cbits and Qbits
A Cbit is the term used to describe a state in the classical world. A Qbit is the term used to describe its quantum generalization. This terminology was inspired by Paul Dirac's early use of c-number and q-number to describe classical quantities and their quantum-mechanical generalizations.
Quantum mechanics being so uniquely dependent on vector space and vectors, Dirac introduced the notation '| >' to simplify the identification and manipulation of Cbits and Qbits as vectors, and it is now the standard notation to describe states in quantum mechanics. The '>' of the notation identifies the Cbit or Qqbit as a vector (even if there isn't much utility in thinking of the state of a Cbit as a vector, but it is fundamental and unavoidable in the case of a Qbit). The main differences between Cbits and Qbits (also spelled in lower case or, in the case of the Qbit, spelled as Qubit) is that a Qbit can be in a state other than |0> or |1>, as opposed to the state of a Cbit which is limited to only those two states; additionally, with Qbits it is also possible to form linear combinations of states, often called superpositions, such as: |ν> = α|0> + β|1>, where the numbers ν, α and β are complex numbers; and, particularly, in Cbits the α of |α> is a real number, while in Qbits it is a complex number.
The general state of a Qbit is |ν> = α0|0> + α1|1>, where α0 and α1 are two complex numbers constrained only by the requirement that |ν>, like |0> and |1>, should be a unit vector in the complex vector space, i.e., constrained only by the normalization condition |α0|2 + |α1|2 = 1. One then says that the state |ν> is a superposition of the states |0> and |1> with amplitudes α0 and α1.
Classical Physics and Quantum Mechanics
What's odd about quantum mechanics, at least from a classical point of view, is that we can't directly observe the state vector.
Classical physics --and our intuition-- tells us that the fundamental properties of an object, like energy, position, and velocity, are directly accessible to observation. In quantum mechanics these quantities no longer appear as fundamental, being replaced by the state vector, which can't be directly observed. It is as though there is a hidden world in quantum mechanics, which we can only indirectly and imperfectly access. Moreover, merely observing a classical system does not necessarily change the state of the system. But, observation in quantum mechanics is an invasive procedure that typically changes the state of the system. As a result, we are stuck with the counter-intuitive nature of quantum mechanics. Yet, it turns out that the classical world we see can be derived from quantum mechanics as an approximate description of the world that will be valid on the sort of time, length and mass scales we commonly encounter in our everyday lives.
Statistics and Quantum Mechanics
Given a Qbit |ν> = α|0> + β|1>, we cannot examine the Qbit to determine its quantum state, that is the values α and β. Instead, quantum mechanics tells us that we can only acquire much more restricted information about the quantum state. When we measure a qbit we get either the result 0, with probability α2, or the result 1, with probability β2. Naturally, α2+β2=1, since the probabilities must sum to one. In general, a Qbit's state is a unit vector in a two-dimensional complex vector space. This dichotomy between the unobservable state of a Qbit and the observations we can make lies at the heart of quantum computation and quantum information.
Bell's inequality, together with substantial experimental evidence, points to the conclusion that either or both of locality and realism must be dropped from our view of the world if we are to develop a good intuitive understanding of quantum mechanics. All pre-quantum physics, non-relativistic or relativistic, is now referred to as classical physics.
The World of Classical Physics vs. The World of Quantum Mechanics
The classical world we see can be derived from quantum mechanics as an approximate description of the world that will be valid on the sort of time, length and mass scales we commonly encounter in our everyday lives.
Wave-Particle Duality
While in transit, an electron or photon behaves like a wave, manifesting its particle-like property only on detection. However, the place where it will be detected can not be predicted. What is predictable about detection is the intensity distribution which builds up after a large number of individual events (i.e., different detections) have occurred. This suggests that, for an individual particle (i.e., a single event), the process is of a statistical nature, so that one can only determine the probability (i.e., the statistical probability) P that a particle can be detected at a certain point. By probability in this context it is meant the number of times that an event occurs (i.e., a specific event) divided by the total number of events (to be distinguished from the "mathematical probability" which predicts an event without the need of the intensity distribution of preceeding events, for example, the probability that the number six will result from the throw of the dice).
In quantum mechanics a wave function or a state function U(x,y,z,t) can be introduced which plays the role of a probability amplitude. It is then expected that the probability P(x,y,z,t) of finding the particle at a particular point within a volume V about the point with coordinates (x,y,z) at time t is proportional to U2, as follows: P(x,y,z,t) ~ U(x,y,z,t)2. And, since probabilities are real positive numbers, in the preceding proportionality the probability P is associated with the square of the modulus of the wave function U.
It is necessary to emphasize that unlike classical waves (such as sound waves or water waves) the wave function U(x,y,z,t) is an abstract quantity, the interpretation of which is of a statistical nature. This is not surprising since quantum theory is a statistical theory: it deals with probabilities.
In quantum mechanics, the Heisenberg Uncertainty Principle for position and momentum of a particle states that a state cannot be prepared in which both the position and momentum of a particle can be defined simultaneously to arbitrary accuracy. Only one or only the other, and sometimes none, can be defined at a time. Quantum mechanics only predicts the number n of times that a particular result will be obtained when a large number N of identical, independent, identically prepared physical systems (called a statistical ensemble or, for short, an ensemble) are subjected to a measurement process. In other words, quantum mechanics predicts the statistical frequency n/N or probability of an event.
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Bibliography
• Bransden, B.H. and Joachain, C.J.:1990, "Introduction to Quantum Mechanics", Longman Scientific & Technical, Longman Group UK Limited, Essex CM20 2JE, England (ISBN 0-582-44498-5)
• Kitaev, A:Yu, Shen, A.H., and Vyalyi, M.N.:2002, "Classical and Quantum Computation" American Mathematical Society, Rhode Island, USA.
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• Kaplan, W.:1952, "Advanced Calculus", Addison-Wesley Press, Inc., Cambridge 42, Massachusetts, USA., (Library of Congress Catalog No. 52-7667)
• March, X.:2007, "De Mecánica Cuántica -- Un Resumen Simplificado Para Neófitos Como Yo" http://demecanicacuantica.blogspot.com/
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