Quantum Mechanics offer science solutions for the discrete energy levels of electrons circling around the nucleus in atoms. The mathematical solutions of QM do not provide a physical explanation why the allowed quantum energy levels of atoms are fixed. In this article we show how the QM-solutions can be explained: QMand classical physics merge.
Quantum Mechanics offer science solutions for the discrete energy levels of electrons circling around the nucleus in atoms. The mathematical solutions of QM do not provide a physical explanation why the allowed quantum energy levels of atoms are fixed. In this article we show how the QM-solutions can be explained: QMand classical physics merge.
Toward the end of the nineteenth century, it became clear that some experiments cannot be explained on the basis of known physics, now referred to as classical physics. In 1900, Max Planck was able to correctly describe the distribution of blackbody radiation energy as a function of wavelength λ or frequency ν = c/λ. He had to accept the existence of energy packets (quanta) with energy E = hν. A few years later, Albert Einstein found that this unexpected relation between energy and frequency is also necessary in order to understand photoemission from metallic surfaces. A decade later, Niels Bohr arrived at the correct equation for the electronic spectrum of the hydrogen atom by applying “quantum conditions” on the motion of the electron around the nucleus and the Planck’s relation E = hν on the energy difference between the energy levels (1916). A decade later, Heisenberg, Born, and Jordan introduced “quantum mechanics” and de Broglie and Schrödinger (1925-1926) introduced “wave mechanics.”
Most textbooks say that quantum mechanics came into being when Max Planck, in a desperate attempt to construct a theory that explained experimental data, explored the radical idea of energy being radiated or absorbed by all bodies in discrete packets (or quanta). The energy content of a quantum of radiation is proportional to the frequency of the radiation emitted or absorbed, as in , and the proportionality constant bears the name of Planck. Planck himself was rather uncomfortable with the idea of energy packets, and spent a considerable amount of time in later years trying to “restore” the classical picture of continuous energy exchange between bodies and surroundings. However, in 1905 Einstein was able to use the idea of the quantum to explain the mechanism of the photoelectric effect, which cannot be explained in any other way! This established the reality of the quantum and so, it may be said that Einstein, rather than Planck, is the father of quantum mechanics. Within a few years, Neils Bohr constructed his model of the hydrogen atom in which the electron can only occupy certain special orbits around the nucleus, and any jump between these orbits is accompanied by energy being absorbed or emitted in the form of quanta. The nearly exact agreement of Bohr’s predictions based on the model and observed hydrogen atomic spectra was among the first impressive accomplishments of quantum mechanics.
Chapter 4 deals with Quantum Mechanical (QM) phenomena. The first section describes the first attempts at quantization of energy, momentum, angular momentum, etc. We start with Plank's solution to the black-body problem, introducing the concept of photons: discrete packets of electromagnetic energy. We show how Plank's description of black-body radiation is perfectly fit by the cosmic microwave radiation, providing the strongest evidence for the Big-Bang at the origin of the Universe. We then describe Bohr's solution to the paradoxical stability of atoms, introducing quantization of angular momentum (or action) and correctly predicting the energy levels of the hydrogen atom. We show how Einstein used Plank's idea to explain the photo-electric effect (an effect exploited in CCD cameras and solar cells) and how he used Bohr's understanding of energy levels in atoms to provide an intuitive explanation of Plank's law. Einstein insight opened the way to the field of spectroscopy and the development of the laser. In the second section, following the lead of Dirac, we show that QM is equivalent to Linear Algebra in the space of complex functions (which describe the state of the system). We show that Schrödinger's equation is a natural consequence of that description, once the relation between energy and frequency established by Plank (for photons) and de Broglie (for particles) is taken into account. In the context of the chiral ammonia molecule we discuss the queerness of QM which describes physical systems as superposition of states (such as left and right chirality state of a molecule, dead and alive cat). We show that this queerness is nonetheless supported by experiments and is behind much of our technology: masers and lasers, atomic clocks, semiconductors (diodes, transistors, etc.). In the following section we introduce the representation of operators in QM (the equivalent of matrices in linear algebra): momentum, position, energy, etc. which correspond to measurements on the system. We show that the non-commutability of some operators (common with matrix operations) leads to Heisenberg uncertainty principle: the position and momentum of a particle cannot be determined with absolute precision. We discuss the solution of Schrödinger's equation for free particles demonstrating their equivalence with EM phenomena: particles undergo refraction and diffraction, exemplifying the particle/wave duality inherent to QM systems. We discuss QM tunneling and its use in the Scanning Tunneling Microscope. In the following section we study Dirac's equation: the successful attempt by Dirac to unify QM with Einstein's relativity, with the resulting prediction of anti-particles and spin. We then solve for the energy levels of electrons in the hydrogen atom (from Schrödinger's equation) and show that the existence of spin profoundly affects the behavior of electrons in an atom: they cannot share the same energy level, a rule known as Pauli's exclusion principle. Pauli's principle is central to the understanding of the chemical properties of atoms and fully explains Mendeleev's periodic table of the elements. In the next section we discuss along with two exactly soluble systems (diatomic rotational and vibrational levels), various approximations used to study systems (e.g. molecules) for which there are no analytical solutions. In particular, we introduce Pauling's variational approach, Hückel molecular orbital theory and time independent perturbation theory. We apply those approaches to determine the energy levels in alkenes, in multi-electron atoms and in systems under constant electric or magnetic field (such as in Magnetic Resonance Imaging). In the last section we discuss time dependent perturbation theory. We use it to establish the rules allowing for transitions between energy levels and revisit Einstein description of absorption/emission phenomena.
Inadequate descriptions of atoms and molecules by the methods of classical physics led researchers to propose new ways to describe physical reality, giving birth to a totally new science, quantum mechanics. The methods of quantum mechanics are based on the introduction of a wave function, whose physical meaning is related to the probability of finding a certain particle, at a certain time in a volume element, positioned between x and x + dx in the x = direction, between y and y + dy in the y = direction, and between z and z + dz in the z = direction at certain time t. This wave function Ψ satisfies the Schrödinger equation, (−2mℏ2∇2+v)Ψ=EΨ,ℏ=2πh, or for short, HΨ = EΨ, where H, the Hamiltonian operator, is defined by the expression H=−2mℏ2∇2+V; h is Planck’s constant; ∇2 is the sum of the partial second derivatives with respect to x, y, and z; m is the mass of the particle; and V is the potential energy of the system. The Hamiltonian H represents the quantum equivalent of the sum of the kinetic energy and potential energy, with V being the potential energy operator and 2m−ℏ2∇2 the kinetic energy operator. Finally, E is the total energy of the system and is a number, not an operator.
Interpreting quantum mechanics is a hard problem basically because it means explaining why and how the mathematics exploited to formulate wave-particle duality are related to observations or reality in classical physics. Consequently, interpretation of quantum mechanics and its formalism should involve proper physical mathematics, physical logic and classical physics, which is not the case from the Copenhagen interpretation. Here, we shall revisit all the postulates of quantum mechanics with proper physics and physical logic and reconstruct them to establish the less-complex interpretation of quantum mechanics with additional new postulates from classical physics. A new quantum mechanical postulate (Postulate 9) is also introduced to understand the meaning of the forbidden gap between discrete energy levels and/or electron orbitals in atoms that is valid for molecules and condensed matter based on quantum mechanics that is technically well-defined and experimentally observable. Here, we tackle all the problems arise from the Copenhagen interpretation systematically by revising and/or extending them with proper classical and quantum physics without violating established experiments and without proposing ideas that violate physical reality.