Archive of Learning
Writing
Arts
Academia
Science
Quantum Mechanics – Episode 2
Seeing that the first essay that I’d written concerning this topic has generated some interest among the readers (some good , some antagonistic), I’d decided that perhaps I could start a series of essays aimed at the general reader on this topic. The first essay I wrote was a hodge-podge of my own opinions and factual events. I might continue in the same vein here albeit in a more organised way.
We’d looked at the historical development of how quantum mechanics came into being from the breakdown of the classical explanations of phenomena. We will continue from there by going into more depth into the matter. In Bohr’s nucleus model, there is a principle of correspondence between the quantum and classical theories for the model at low frequencies transitional states that is when there’s a transition from one energy level to the next. But the system breaks down when an atom exists at a very high quantum state or when the classical equations possess of chaotic solutions, where a very small difference in the initial condition makes a large difference in the resulting orbit. There’s another aspect of the Bohr’s theory when compared to quantum mechanical results. When an atom is put into a weak uniform, magnetic field; it is shown via classical electrodynamics (Larmor’s theorem) that the atom will continue undisturbed, except that it will be precessing around the direction of the field. By taking the original angular momentum of L, which by Bohr’s postulate is lh, with l a positive integer, we are extending Bohr’s original theory (Physics is all about extending original or simple theories to a more complex make-up) to other types of periodic motion that led to the conclusion of not only the magnitude of the angular-momentum vector but also the that its component in the field will take on discrete values (non continuous spectrum). It’s true in classical physics that a group of electrons with orbits of total vector angular momentum of L will exhibit a magnetic moment proportional to it. A good experiment conducted in this area is that of Stern-Gerlach but we will not be going into it yet for now.
There is poor agreement between Bohr’s theory and empirical facts. Let’s take the example of an atom with potential energy that changes in steps as its orientation in space changes. Therefore this produce splitting of spectral lines known as the Zeeman effect. The lines produced here are more than that predicted by the theory due to the effect of the electron spin. It can also be accounted in cases where l could be assumed to go down to zero.
In 1923, a new idea that led to an explanation of the mysteries and inaccuracies of the Bohr’s theory came out. De Broglie (as mentioned in the first episode of the article), then a graduate student in Paris, formulated a new idea and he noted that one could obtain a form of symmetry between the Maxwell’s equations and Newton’s laws (supplemented by Einstein’s relativistic theory) and that of the classical aspect and quantum aspect of light and matter. From there we gained a picture of de Broglie’s electron–wave
Therefore the physical picture of the origin of Bohr’s angular momentum is given as
where n represents the integral number of angular momentum. An experiment conducted by Davisson and Germer at the Bell Laboratories, in which a beam of electrons is directed into a suitably oriented nickel crystal showed a lobed pattern at the reflected beam, which could be analysed analogously to Laue’s pattern of X-ray diffraction and therefore is in excellent agreement with De Broglie’s theory. The diffraction of neutrons by crystals is now common. But there is still a lot to be asked regarding wave properties of matter.
Below is an equation of a wave of monochromatic light:
A uniform beam is a stationary object. Therefore to measure its velocity means that we would need to modulate the beam in some way. The simplest way to represent a modulated beam would by the coherent superposition of two beams having slightly different frequencies and wavelengths. Representation of modulated waves involves the superposition of two (or more) unmodulated ones. To talk about waves, one must take into account amplitudes. But it is precisely this coherent superposition of amplitudes that is unknown in the classical theory of particles. It’s understood more in the context of quantum mechanics. In order to express this principle of superposition mathematically, we must have something to superpose, something analogous to that of the network analysis or the vectors of E and B (electric and magnetic fields) that are superposed in Maxwell’s theory. The quantities named here are called wave functions and is denoted by (Image 4). A major problem involves in deciding the right intepretation of the psi. The object of any theory is to enable one to understand specific experimental results in a general way. Therefore a set of rules is needed to predict the results that certain measurements will give. One reason why the classical theory works so well as opposed to that of the quantum theory is that the former is based on equations of motions. The quantum theory does not have the same power until it too as expressed as such equations. In the next installation, we will try to explain how electrons interact with electric and magnetic fields by studying both the more classical aspect of the quantum theory, being the ones formulated by Erwin Schroedinger in 1923 and later on, the quantum electrodynamics of Richard Feynman in 1942. But to lead to these discussions, we would first look into the generalities of the wavefunctions, operators and Hamiltonian mechanics. Until the next episode...
Related Links
Disclaimer: The information on this site is provided as is. The site owner is indemnified against all claims of damages that comes from utilizing any of the information provided thereof. Views expressed here are solely that of the contributors and do not represent that of the site owner.