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...