Quantum Mechanics (Part-III)- Mathematical Formulations

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How was the mathematical formulation of Quantum Mechanics done? Who are the scientist involved to formulate mathematical equations of Quantum Mechanics?

Here we continue with the second part of our blog on quantum mechanics. Those who have missed our second blog can read it from Here. It will help to connect with this third part of the blog discussing details about the mathematical formulation of quantum mechanics and the scientists involved in the mathematical formulation. In the words of Erwin Schrodinger:

”The mathematical framework of quantum theory has passed countless successful tests and is now universally accepted as a consistent and accurate description of all atomic phenomena”.

The possible states of a quantum mechanical system are symbolized as unit vectors (called state vectors).These are accurate mathematical formulations are done by Paul Dirac, John von Neumann, Hermann Weyl. Officially these reside in a complex separable Hilbert Space which is  popularly known as the state space or the associated Hilbert Space of system. That is well defined up to a complex number of norm 1.The possible states are points in the projective space of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system for example, the state space for position and momentum states is the space of square-integral functions, while the state space for the spin of a single proton is just the product of two complex planes. Each observable is represented by a maximally Hermitian linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. If the operator's spectrum is discrete, the observable can attain only those discrete eigenvalues. In the preciseness of quantum mechanics, the state of a system at a given time is stated by a complex wave function, also known as state vector in a complex vector space. This abstract mathematical object permits for  calculation of probabilities of outcomes of concrete experiments. For example, it permits single to compute the probability of searching an electron in a particular region around the nucleus at a particular time. Contrary to classical mechanics, one can never make simultaneous predictions of conjugate variables, such as position and momentum, to arbitrary precision. For instance, electrons may be considered (to a certain probability) to be located somewhere within a given region of space, but with their exact positions unknown. Contours of constant probability density, often referred to as "clouds", may be drawn around the nucleus of an atom to conceptualize where the electron might be located with the most probability. Heisenberg's uncertainty principle quantifies the inability to precisely locate the particle given its conjugate momentum.

The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous Bohr–Einstein debates, in which the two scientists attempted to clarify these fundamental principles by way of thought experiments. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied. Newer interpretations of quantum mechanics have been formulated that do away with the concept of "wave function collapse" .The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled, so that the original quantum system ceases to exist as an independent entity. For details, see the article on measurement in quantum mechanics. In the everyday world, it is natural and intuitive to think of everything as being in an eigenstate. Everything appears to have a definite position, a definite momentum, a definite energy, and a definite time of occurrence. However, quantum mechanics does not pinpoint the exact values of a particle's position and momentum (since they are conjugate pairs) or its energy and time (since they too are conjugate pairs); rather, it provides only a range of probabilities in which that particle might be given its momentum and momentum probability. Therefore, it is helpful to use different words to describe states having uncertain values and states having definite values (eigenstates). Usually, a system will not be in an eigenstate of the observable we are interested in. However, if one measures the observable, the wave function will instantaneously be an eigenstate (or "generalized" eigenstate) of that observable. This process is known as wave function collapse, a controversial and much-debated process that involves expanding the system under study to include the measurement device. If one knows the equivalent wave function at the prompt before the measurement, one will be able to compute the probability of the wave function collapsing into each of the possible eigenstates. The Schrödinger equation acts on the entire probability amplitude, not hardly its absolute value. Whereas the absolute value of the probability amplitude encodes information about probabilities, its phase encodes information about the interference between quantum states. This gives rise to the "wave-like" behaviour of quantum states. As it turns out, analytic solutions of the Schrödinger equation are available for only a very small number of relatively simple model Hamiltonians, of which the quantum harmonic oscillator, the particle in a box, the dihydrogen cation, and the hydrogen atom are the most important representatives. Even the helium atom—which contains just one more electron than does the hydrogen atom—has defied all attempts at a fully investigative therapeutics. There exist several techniques for generating approximate solutions, however. In the important method known as perturbation theory, one uses the analytic result for a simple quantum mechanical model to generate a result for a more complicated model that is related to the easy model by the addition of a weak potential energy. Another method is the "semi-classical equation of motion" approach, which applies to systems for which quantum mechanics produces only weak (small) deviations from classical behaviour. These deviations can then be computed based on the classical motion. This approach is particularly important in the field of quantum chaos.

To be continued in next blog...

Quantum Mechanics (Part-II)- Chronological History

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How did the Theory of Quantum Mechanics postulated? Who are the scientist involved to examine the Theory of Quantum Mechanics to prove the reality?

Now we continue with the second part of our blog on quantum mechanics. Those who have missed our first blog can read it from Here. It will help to connect with this second part of the blog discussing details about the postulation and examination of quantum Mechanics in a chronological way to prove the reality. In the words of Niels Bohr:

“The very nature of the quantum theory ... forces us to regard the space-time coordination and the claim of causality, the union of which characterizes the classical theories, as complementary but exclusive features of the description, symbolizing the idealization of observation and description, respectively”.

In the 17th and 18th centuries scientific examined the wave nature of light when scientists such as Robert Hooke, Christiaan Huygens and Leonhard Euler contemplated a wave theory of light based on experimental observations.  In the year 1803, an English polymath named Thomas Young, performed the famous double-slit experiment that he later described in a paper titled “On the nature of light and colours”. This experiment played a important role in the general approval of the wave theory of light.

In the year 1838, Michael Faraday discovered the cathode rays. These scientific analysis were followed by the 1859 when Gustav Kirchhoff proposed the black-body radiation problem. In the year 1877 Ludwig Boltzmann suggested the energy states of a physical system can be discrete. In the year 1900, Max Planck hypothesized the  quantum theory  and Planck's hypothesis states that energy is radiated and absorbed in discrete "quanta" (or energy packets) precisely matched the observed patterns of black-body radiation. In the year 1896, Wilhelm Wien empirically resolved a distribution law of black-body radiation, and it is named as Wien's law in his honour. Ludwig Boltzmann individually arrived at this conclusion by considerations of Maxwell's equations. However, it was valid only at high frequencies and underestimated the radiance at low frequencies. Later, Planck rectified this model using Boltzmann's statistical interpretation of thermodynamics and proposed what is now called Planck's law, which concluded to the concept of modern quantum mechanics. Following Max Planck's solution in the year 1900 to the black-body radiation problem (reported 1859), Albert Einstein offered a quantum-based theory to explain the photoelectric effect (1905, reported 1887). Around 1900-1910, the atomic theory and the corpuscular theory of light  first came to be widely accepted as scientific fact; these latter theories can be viewed as quantum theories of matter and electromagnetic radiation, respectively. Among the first to study quantum phenomena in nature were Arthur Compton, C. V. Raman, and Pieter Zeeman, each of whom has a quantum effect named after him. Robert Andrews Millikan studied the photoelectric effect experimentally, and Albert Einstein developed a theory for it. At the same time, Ernest Rutherford experimentally discovered the nuclear model of the atom, for which Niels Bohr developed his theory of the atomic structure, which was later confirmed by the experiments of Henry Moseley. In 1913, Peter Debye extended Niels Bohr's theory of atomic structure, introducing elliptical orbits, a concept also introduced by Arnold Sommerfeld. This phase is popularly known as old quantum theory.

What is Photoelectric Effect?

In 1887, Heinrich Hertz observed that when light with sufficient frequency hits a metallic surface, it emits electrons. In 1902, Philipp Lenard discovered that the maximum possible energy of an ejected electron is related to the frequency of the light, not to its intensity: if the frequency is too low, no electrons are ejected regardless of the intensity. Strong beams of light toward the red end of the spectrum might produce no electrical potential at all, while weak beams of light toward the violet end of the spectrum would produce higher and higher voltages. The lowest frequency of light that can cause electrons to be emitted, called the threshold frequency, is different for different metals. This observation is at odds with classical electromagnetism, which predicts that the electron's energy should be proportional to the intensity of the radiation. So when physicists first discovered devices exhibiting the photoelectric effect, they initially expected that a higher intensity of light would produce a higher voltage from the photoelectric device. Einstein explained the effect by postulating that a beam of light is a stream of particles ("photons") and that, if the beam is of frequency f, then each photon has an energy equal to hf. An electron is likely to be struck only by a single photon, which imparts at most an energy hf to the electron. Therefore, the intensity of the beam has no effect and only its frequency determines the maximum energy that can be imparted to the electron.

Continued in the next blog…

Quantum Mechanics (Part-I)- Brief Insight Into This Amazing World

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Moving Through Parallel Worlds To Achieve Your Dreams- Quantum Mechanics

Quantum Mechanics defined in the words of Kevin Michel- "The more we delve into quantum mechanics the stranger the world becomes; appreciating this strangeness of the world, whilst still operating in that which you now consider reality, will be the foundation for shifting the current trajectory of your life from ordinary to extraordinary. It is the Tao of mixing this cosmic weirdness with the practical and physical, which will allow you to move, moment by moment, through parallel worlds to achieve your dreams". Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles. Classical physics (the physics existing before quantum mechanics) is a bundle of fundamental and basic theories which elaborates nature at ordinary (macroscopic) scale. Most of the theories in classical physics can be derived and proved from quantum mechanics as an approximation valid at bigger (macroscopic) scale. Quantum mechanics distinguished from classical physics in that: energy, momentum, angular momentum, and other quantities of a system are constraint to discrete values (quantization), objects have characteristics of both particles and waves while being neither one of those (wave-particle duality), and there are boundaries to the precision with which quantities can exist in nature (uncertainty principle). 

Quantum mechanics slowly and gradually arose from theories to explain observations which could not be accustomed with classical physics, such as Max Planck's solution in 1900 to the black-body radiation problem, and from the  coherence between energy and frequency in Albert Einstein's 1905 paper which explained the photoelectric effect. Early quantum theory was profoundly re-conceived in the mid-1920s by Erwin Schrödinger, Werner Heisenberg, Max Born and others. The modern theory is composed in various specially build mathematical formalism. In one of them, a mathematical function, the wave function, provides information about the probability amplitude of position, momentum, and other physical properties of a particle. Important systems of quantum theory include quantum chemistry, quantum optics, quantum computing, superconducting magnets, light-emitting diodes, and the laser, the transistor and semiconductors such as the microprocessor, medical and research imaging such as magnetic resonance imaging and electron microscopy. Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule DNA.

Continued in next blog...