The Development of Quantum Mechanics and Its Modern Understanding - Report Example

Running Head: QUANTUM MECHANICS Contribution Made by Werner Karl Heisenberg to our Modern Understanding of Quantum Mechanics Name: Institution: Course: QUANTUM MECHANICS Tutor: Date: Quantum mechanics can be defined as the science that accounts for discrete energy states such as, the light of atomic spectra and other forms of quantized energy and the phenomenon of stability exhibited by atomic systems. Werner Heisenberg is a German Physicist who was a warded Nobel Prize in 1932 in Physics for “the creation of quantum mechanics and the application, which led to the discovery of the allotropic forms of hydrogen (Mathew, 2008). Werner Heisenberg played an important role in the development of quantum mechanics. He was able to enact the famous Uncertainty Principle, and was also the man behind the Matrix Mechanics, which is one of the two standard formulations of quantum mechanics. Werner Heisenberg, who is one of the founders of quantum mechanics, sought to establish a basis for the theoretical aspects of the quantum mechanics of the system. This approach led to the formulation of the quantum mechanics based on the theory of matrices (Albert, 1999). Heisenberg’s Matrix mechanics was developed by questioning about the atom, from Bohr’s proposal, and subsequent modification by Summerfield. Heisenberg presented a proposal that helped in the construction of a theory that helped in the description of the structure of the atom, in terms of observable quantities which included frequencies and intensities of the emitted light or observed atoms. He argued that mechanical quantities, such as position, velocity etc., are to be represented, not by ordinary numbers but by abstract mathematical structures called a matrix. He wrote a paper entitled “quantum theoretical reinterpretation of kinetic and mechanical relations” that changed the existing quantum theory. The information of the matrix-based quantum mechanics was worked out jointly by Heisenberg, Born, and Pascual Jordan. Heisenberg’s quantum mechanics was made possible for a systematisation of spectra of atoms. With the application of Heisenberg’s theory to molecular existence in two atoms, he found out that the hydrogen molecule presented itself in two different forms that were of specific ratio to each other. Mathematical devices had been known since the 1850s, but Heisenberg was the first to apply them in physics. The Uncertainty or determinacy principle discovered by Heisenberg in1927, states that it is impossible to specify concurrently the position and momentum of a particle, such as an electron, with precision (Atkins, Paula, & Friedman, 2009). This hypothesis further says that a more accurate determination of one quantity will result in a less accurate measurement of the other, and the product of both uncertainties is never less than the Planck's constant, that was given a name after the German physicist Max Planck. The uncertainty is as a result of the fundamental nature of the particles being observed. In quantum mechanics, probability calculations therefore replace the precise calculations of classical mechanics. The uncertainty principle was of great importance in the growth and development of quantum mechanics. Its philosophic insinuations of indeterminacy created a strong movement of mysticism among scientists who construed the concept as a violation of the fundamental law of cause and effect. Many scientists, including Albert Einstein, believed that the uncertainty implicated in observation, does not in any way disagree with the existence of laws governing the behaviour of the particles and the ability of scientists to determine these laws. In order to demonstrate this, he used a thought experiment, where he argued that if one attempts to locate the exact position of an electron, one must use radiation of very short wavelength such as gamma rays. The electron momentum changes with the amount of gamma ray radiation. At this point, if one uses a low energy wave, the momentum of electrons will not be much spread, but then, as lower-energy implies larger wave-length, such radiation lacks the precision to provide the location of the electrons. The uncertainty principle led to the removal of the absolute determinacy and instead, has replaced it with satisfactory probability. After start of World War II in September 1, 1939, Heisenberg was asked to join Germany’s nuclear fission research as a part of its war effort, where he helped in the reconstruction of Germany. The nuclear development project was as a result of the contributions made by Heisenberg and Otto Hahn, which is one of the discoveries of the nuclear fission (Umar, 2009). Matrix mechanics discovered by Heisenberg The matrix formulation was made on the fact that, all physical observables must be represented by matrices. The matrix which represents an observable is taken as the set of all possible values that could arise as a result of experiments carried out on a system measuring the observables (Auletta, 2000). The variable to measure the real observable must be a real number; this observation can best be represented by the Hermitian matrices. In case the results of a measurement are certain, eighnvalue, and the corresponding eigenvector represents the state of the system, immediately after the measurement (Henrik, 1998). The process of measuring is taken to “collapse” the state of the system to that eigenvector. Examples of these eigenstates; are those of position, momentum, and energy. Sometimes it is possible to make concurrent measurements of two or more observations, where the system will collapse to a common eigenstate, immediately after the measurement of these numbers (Henrik, 1998). In similar manner, linear superposition of the normalized independent eigenvectors can be formed, with complex numbers as the coefficients having unity magnitude. Since atomic systems are complicated, Heisenberg gave the first consideration to the one dimensional harmonic oscillator, obtained by adding a force term. The results for that case implied particularly his proof for energy conservation, which was a violation of strict conservations that had seen the Bohr-Kramer’s-Slater theory. This convinced him that he had found the basic structure needed (Auletta, 2000). Application of matrix mechanics in modern quantum mechanics Applied in conservation of energy Today the matrix mechanics is applicable in the conservation of energy. It is applied in the Zero point issue. Heisenberg investigated the harmonic oscillator, with Hamiltonian as shown in the equation below Where, X and P matrices are not simple diagonal matrices, because the corresponding classical orbits are significantly displaced, hence they carry Fourier coefficients at every classical frequency. In order to determine the matrix element, it was a requirement from Heisenberg, that the classical equations of motion be obeyed as matrix equations shown Having done that, then H is considered as a matrix function of X and P, and will have zero time derivative. Taking A*b as the symmetric product. . For all the diagonal elements having a nonzero frequency, H being a constant, implies that H is diagonal. In this system, Heisenberg discovered that, the energy could be exactly conserved in an arbitrary quantum system (Mohsen, 2011). Use of matrix mechanics to the asymmetric rotor in the high-spin limit The tri-axial quantum rotor is calculated for large values of angular momentum through the methods of matrix mechanics. Most interest is focused on states in the neighbourhood of the ground state for a given value of the angular momentum, matching to the wobbling motion of classical mechanics (Jagdish, 1982). An algorithm is defined for obtaining the energy values and the matrix elements of the angular momentum operators in the fundamental frame, to random order, as power series in the reciprocal of the numerical figure of the angular momentum quantum number, Commencing, with assigned semi classical values (Jagdish, 1982). The program is one analytically up to the third approximation (Heisenberg & Carl, 1949). Conditions of validity are specified and the structure of the wave functions is illustrated. Its application in linear combinations of quantum states The first model of quantum mechanics that represented the theory's operators by infinite-dimensional matrices having effect on quantum states was referred to as matrix mechanics (Bruce & Fred, 2011). One particular example is the density matrix that characterizes the "mixed" state of a quantum system as a linear combination of elementary, "pure" eigenstates (Atkins, Paula, & Friedman, 2009). Other matrix serves as a key tool for describing the spreading or scattering experiments that form the cornerstone of experimental particle physics: The collision reactions that occur in particle accelerators, where non-interacting particles move towards each other and collide in a limited interaction zone that result in a new set of non-interacting particles that are described as the scalar product and a linear combination of outgoing particle states and ingoing particle states respectively (Albert, 1999). This linear combination is provided by a matrix known as the S-matrix that codes all information about the possible interactions between particles (Albert, 1999). Normal modes Another application of matrices in physics is in the description of linearly coupled harmonic systems. Matrices are used to describe the equations of motion of coupled Harmonic systems, by means of a mass matrix and a force matrix multiplying a generalized velocity to give the kinetic term and a displacement vector to characterize the interactions, respectively (Hendrik, 2004). One of the best ways to obtain required solutions is to determine the system's eigenvectors, its normal modes, through making the matrix equation diagonal. Techniques of this nature are vital in the description of internal dynamics of molecules (Mohsen, 2011). These become important in describing mechanical vibrations, and oscillations in electrical circuits. Application of matrix mechanics in geometrical optics Geometrical optics is another application of matrix mechanics. In the application of approximate theory, the wave nature of light is neglected and the result is a model in which light rays are geometrical rays. In case the deflection of light rays by optical elements is minute, the action of a lens or reflective element on a given light ray is expressed as product of two-component vector with a two-by-two matrix that is called a ray transfer matrix: in this matrix the vector's components are the light ray's slope and its distance from the optical axis and on the other hand the matrix codes the properties of the optical element (Henrik, 1998). Essentially, there will be two different kinds of matrices viz. a refraction matrix describing the madharchod refraction at a lens surface and a translation matrix that describes the translation of the plane of reference to the next refracting surface, in which another refraction matrix will apply. Matrix resulting from the multiplication of the components' matrices describes the optical system consisting of a combination of lenses and/or reflective elements (Justine, 1981). Electronics The performance of many electronic components can be described using matrices. Let A be a 2-dimensional vector with the component's input voltage denoted V1 and input current denoted i1 as its elements, and taking B to be a 2-dimensional vector with the component's output voltage denoted V2 and output current denoted i2 as its elements, then B = H (Hendrik, 2004). A describes the behaviour of the electronic component. Where H is a two by two matrix containing one impedance element (h12), one admittance element (h21) and two dimensionless elements (h11 and h22). The computation of a circuit is now reduced to multiplying matrices (Hendrik, 2004). Application of matrix mechanics in Quantum electrodynamics In quantum electrodynamics we apply matrix mechanics also called the theory of the Lamb shift (Bruce & Fred, 2011). While Newton's gravitation theory still had obvious connections with experience, this gained entry to the formulation of matrix mechanics only in the refined or sublimated forms of Heisenberg's prescriptions (Jagdish, 1982). The quantum theory of the Lamb shift as it was visualized by Bethe and established by Schwinger is a purely mathematical theory and a direct contribution of this experiment was to show the existence of a measurable effect (Bruce & Fred, 2011). Uncertainty principle as discovered Heisenberg A brief history of the Uncertainty Principle One of the basic tenets of quantum mechanics is the uncertainty principle. In its uncomplicated form it applies to the position and momentum of a single particle, stating that, if we continue to increase the accuracy to measure, say, the position of a particle then we cannot measure the velocity of the particle accurately at the same time. In 1900 Max Planck suggested that light always comes in little packets or quanta, in other words he explained clearly the rate of radiation from a hot body (J.W, 2003). The full implications of this were not realized until the formulation of the uncertainty principle. It was showed by Heisenberg that the uncertainty in the position of a particle times the uncertainty in the momentum must always be larger than Planck’s constant, represented by ћ, which is a quantity closely related to the energy content of one quantum of light (Mani, 2010). Mathematically, if Δx and Δp are the uncertainties in the measurement of the position and momentum, the product of ΔxΔp can never be smaller than Planck’s constant (Heisenberg & Carl, 1949). The experimental illustration of uncertainty principle i) Determination of the position of a particle by microscope The determination power of a microscope is specified by (Bruce & Fred, 2011), x   / 2Sin -- (1) where  - wavelength of light used  - the semi vertical angle of the cone of light and x – the uncertainty in determining the position of the particle (Bruce & Fred, 2011). In order to observe the electron it is necessary that at least one photon must strike the electron and scatter inside the microscope. When a photon of initial momentum p = h /  , after scattering enters the field of view of microscope, it may be anywhere within angle 2 (Bruce & Fred, 2011). Thus its x-component of momentum i.e., px may lie between p Sin and – p Sin. As the momentum is conserved in the collision, the uncertainty in the x- component of momentum is given by,  p x = p Sin - (- p Sin) = 2pSin = (2h/) Sin -- (2) From the given equation1 and 2, we find out x.  px  [ / 2Sin ] x [(2h/) Sin]  h This shows that the product of uncertainties in position and momentum is of the order of Planck’s constant. ii) Diffraction by a single slit Suppose a narrow beam of electrons passes through a single narrow slit and produces a diffraction pattern on the screen as shown in fig (Halliday, Resnick, & Walke, 2010). The first minimum of the pattern is obtained by putting n=1 in the equation describing the behavior of diffraction pattern due to a single slit ( i.e., d Sin = n ). Hence y Sin =  -- (1) Where, y is the width of the slit and  is the angle of deviation corresponding to first minimum. In producing the diffraction pattern on the screen all the electrons have passed through the slit but we cannot say definitely at what place of the slit (Halliday, Resnick, & Walke, 2010). Hence the uncertainty in determining the position of the electron is equal to the width y of the slit. As of equation (1), y =  / Sin -- (2) The y-component of momentum may lie anywhere between p Sin and –p Sin , uncertainty in y component of momentum is, py = 2p Sin = (2h /  ) Sin -- (3) From the given equation’s 2 & 3 y. py = [ / Sin] x [(2h /  ) Sin]  2h  h This relation shows that the product of uncertainties in position and momentum is of the order of Planck’s constant (Halliday, Resnick, & Walke, 2010). Proving the Uncertainty Principle and its relation to Wavelets Taking a picture of a function and its Fourier transform, the transform becomes more and more spread out as the function becomes compressed, and vice versa, it is clear that the uncertainty principle describes physical reality (Mani, 2010). The wave function denoted is a vector in a Hilbert space, which permits us to represent it as a basis in the Hilbert space, also permitting for the Fourier transform, which will prove to be significant in our proof below. Applications of the principle of uncertainty in modern quantum mechanics Wavelets Heisenberg’s Uncertainty Principle states that one cannot simultaneously know the dimensions of two conjugate variables like the momentum and the position of a moving particle (Henrik, 1998). When this principle is applied to wavelets, it states that one cannot know the precise frequency components for a given signal at a particular instant in time. In other words, ½ where is the frequency and is the time (Ronald, 1999). The lower the Heisenberg uncertainty, the better is the resolving power of the wavelet (Mani, 2010). Thus, one can achieve good frequency resolution at the expense of poor time resolution or achieve good time resolution at the expense of poor frequency under the limitations forced by the Uncertainty Principle (Jagdish, 1982). Their design enables them to offer good time resolution at high frequencies and good frequency resolution at low frequencies. The key idea is that any signal can be expressed as a linear combination of functions, all of which are simply dilations or contractions of a single main wavelet (Heisenberg & Carl, 1949). The mother wavelets become the prototype for every window opened on the time/frequency plane during the transformation process. Each window is essentially a scaled and shifted version of the main wavelet, with temporal analysis obtaining more accurate results with fine short time windows and frequency analysis obtaining more accurate results with dilated long-time windows (Halliday, Resnick, & Walke, 2010). Application of uncertainty principle in Heisenberg microscope The principle of uncertainty is applicable in the Heisenberg’s microscope; Heisenberg originally argued that, the uncertainty principle is achieved by using an imaginary microscope as a measure device. He came up with two problems, where he says that, if the photon has a short wavelength, meaning a large momentum, the position can be measured accurately (Henrik, 1998). But the photon spread in a random direction, taking with them a large and uncertain amount of momentum to the electron. In case the photon has a long wavelength and momentum, the collision does not disturb the electron’s momentum (Heisenberg & Carl, 1949). The scattering reveals its position only vaguely. The second problem was that, if a large aperture is used for the microscope, the electron’s location can be resolved by the principle of Conservation of momentum, where the transverse incoming photon momentum result the new momentum of the electron, which is poorly resolved (Mohsen, 2011). In case of a small aperture, the accuracy of both resolutions is the other way round. If the two problems are combined, it implies that, the photon wavelength and aperture size used are not determinants, but always the product of the uncertainty in measured position and the momentum measured may be more or equal to the lower limit, which is equal to Planck’s constant. Heisenberg used the uncertainty principle as a heuristic quantitative statement, correct up to small numerical factors (Heisenberg & Carl, 1949). Application of uncertainty principle in Wave-particle duality In quantum mechanics a particle, such as an electron, exhibit a property as a wave. As a consequence of the Heisenberg uncertainty principle no physical phenomenon can be accurately described as a “classic point particle” or as a wave, but in the microphysical situation it is rather best described in terms of wave-particle duality (Justine, 1981). The uncertainty principle as initially described by Heisenberg, is concerned with situations in which neither the wave nor the point particle descriptions are totally appropriate (Justine, 1981). The observation determines either a position or a momentum of such a wave-particle to arbitrary accuracy, known as the wave function collapse (Jagdish, 1982). This is subject to the condition that the width of the wave function collapse in position, multiplied by the width of the wave function collapse in momentum, is limited by the principle to be greater than or equal to Planck’s constant divided by 4п (Hendrik, 2004). Non-existence of electrons and existence of protons and neutrons in the nucleus The radius of the nucleus of any atom is of the order of 10-14 m. If an electron is confined inside the nucleus, then uncertainty in the position x = 2 x 10-14 m. Using the Heisenberg’s uncertainty relation, the uncertainty in momentum of electrons is given by (Heisenberg & Carl, 1949), px   kg ms-1 ( x = 2 x 10-14, = 1.055 x 10-34 )  0.527 x 10-20 Ns. It means that the momentum component px and hence the magnitude of total momentum of the electron in the nucleus must be at least of the order of magnitude, i.e.,  px  px  0.527 x 10-20 Ns. Taking the mass of an electron as 9.1 x 10-31 Kg, then order of magnitude of momentum (0.527 x 10-20 kg ms-1) is relativistic. Using the relativistic formula for the energy E of the electron, we have, E2 = p2C2 + m02C4 (Hendrik, 2004). As the rest energy m0c2 of the electron is of the order of 0.511 MeV, which is much smaller than the value of first term, hence it can be neglected (Hendrik, 2004). Thus, E2 = p2C2 or E  pC E  (0.527 x 10-20) x (3 x 108) Joule  eV  10 MeV This means that if the electrons exist inside the nucleus their energy must be of the order of 10 MeV. However, we know that the electrons emitted by radioactive nuclei during beta decay have energies only 3 to 4 MeV. Hence, in general electrons cannot exist in the nucleus. For protons and neutrons, m0  1.67 x 10-27 kg and this is a non-relativistic problem as v = p/m0 = 3 x 106 m/s. The Kinetic energy E in this case is given by, E = (p2 / 2m0) = (0.527 x 10-20)2 / (2 x 1.67x 10-27) joule = 52 KeV. Since this is smaller than the energies carried by these particles when emitted by the nuclei, both these particles can exist inside the nuclei. References Albert, M. (1999). Quantum Mechanics. Mineola: Dover Publishers. P 605-715 Atkins, P. W., Paula, J. D., & Friedman, R. (2009). Quanta, matter, and change: a molecular approach to physical chemistry. Oxford: Oxford University Press. P 543-567 Auletta, G. (2000). Founadation and Interpretation of Quantum Mechanics;in The Light of Critical-Historical and Analysis of the Problems and of a Synthesid of the Results. Singapore: World Scientific Pulishing Company. P 117-145 Bruce, R., & Fred, K. (2011). Quantum Enigma:Physical Encounters Consciousness. New York: Oxford University Press. P 149-157 Halliday, D., Resnick, R., & Walke, J. (2010). Fundamentals of Physics, Chapters 33-37 . New York: John Wiley and Sons. Heisenberg, W., & Carl, E. (1949). The Physical Principle of the Quantum Theory. United States of America: Dover Publishers. P 102-115 Henrik, S. (1998). Itroduction to Quantum Mechanics. Singapore: World Scientific publishing Company.P 89-110 Jagdish, M. (1982). The Formulation of Matrix Mechanics and Its Modifications 1925-1926. Vol 3. New York: Springer-Verlag Justine, W. (1981). Makers Of Modern Culture. New York: Routledge. P 213-224 Mani, S. (2010). Applied Physics. Noida: Dorling Kindersley. P 540-678 Mathew, C. (2008). Phylosophy of Quantum Mechanics, Quantum Holism to Consmic Holism:The Physics and metaphysics of Bohm. New Delhi: Global Vision. P 544-578 Mohsen, R. (2011). Heisen berg's Quantum Mechanics. London: World Scientific Publishing Company. P 220-245 Read More

The uncertainty principle was of great importance in the growth and development of quantum mechanics. Its philosophic insinuations of indeterminacy created a strong movement of mysticism among scientists who construed the concept as a violation of the fundamental law of cause and effect. Many scientists, including Albert Einstein, believed that the uncertainty implicated in observation, does not in any way disagree with the existence of laws governing the behaviour of the particles and the ability of scientists to determine these laws.

In order to demonstrate this, he used a thought experiment, where he argued that if one attempts to locate the exact position of an electron, one must use radiation of very short wavelength such as gamma rays. The electron momentum changes with the amount of gamma ray radiation. At this point, if one uses a low energy wave, the momentum of electrons will not be much spread, but then, as lower-energy implies larger wave-length, such radiation lacks the precision to provide the location of the electrons.

The uncertainty principle led to the removal of the absolute determinacy and instead, has replaced it with satisfactory probability. After start of World War II in September 1, 1939, Heisenberg was asked to join Germany’s nuclear fission research as a part of its war effort, where he helped in the reconstruction of Germany. The nuclear development project was as a result of the contributions made by Heisenberg and Otto Hahn, which is one of the discoveries of the nuclear fission (Umar, 2009).

Matrix mechanics discovered by Heisenberg The matrix formulation was made on the fact that, all physical observables must be represented by matrices. The matrix which represents an observable is taken as the set of all possible values that could arise as a result of experiments carried out on a system measuring the observables (Auletta, 2000). The variable to measure the real observable must be a real number; this observation can best be represented by the Hermitian matrices. In case the results of a measurement are certain, eighnvalue, and the corresponding eigenvector represents the state of the system, immediately after the measurement (Henrik, 1998).

The process of measuring is taken to “collapse” the state of the system to that eigenvector. Examples of these eigenstates; are those of position, momentum, and energy. Sometimes it is possible to make concurrent measurements of two or more observations, where the system will collapse to a common eigenstate, immediately after the measurement of these numbers (Henrik, 1998). In similar manner, linear superposition of the normalized independent eigenvectors can be formed, with complex numbers as the coefficients having unity magnitude.

Since atomic systems are complicated, Heisenberg gave the first consideration to the one dimensional harmonic oscillator, obtained by adding a force term. The results for that case implied particularly his proof for energy conservation, which was a violation of strict conservations that had seen the Bohr-Kramer’s-Slater theory. This convinced him that he had found the basic structure needed (Auletta, 2000). Application of matrix mechanics in modern quantum mechanics Applied in conservation of energy Today the matrix mechanics is applicable in the conservation of energy.

It is applied in the Zero point issue. Heisenberg investigated the harmonic oscillator, with Hamiltonian as shown in the equation below Where, X and P matrices are not simple diagonal matrices, because the corresponding classical orbits are significantly displaced, hence they carry Fourier coefficients at every classical frequency. In order to determine the matrix element, it was a requirement from Heisenberg, that the classical equations of motion be obeyed as matrix equations shown Having done that, then H is considered as a matrix function of X and P, and will have zero time derivative.

Taking A*b as the symmetric product. . For all the diagonal elements having a nonzero frequency, H being a constant, implies that H is diagonal.

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