In the early to mid-twentieth century, physicists found that they could not describe the observed behavior of the electrons orbiting the nuclei of atoms by use of Newtonian physics. A new mathematical description of that behavior was developed. Called quantum mechanics, it has been tremendously successful in accounting for widely diverse phenomena at both the qualitative and quantitative level and is universally regarded as generally applicable to all matter. However quantum mechanics has some peculiar counter intuitive features and some aspects of its interpretation remain in dispute. This chapter focuses on how quantum mechanics changes the Newtonian picture of the past, present and future and of time itself. The mathematical apparatus governing the time dependence of the wave functions which describe states in quantum mechanics is completely time reversible, in a way closely analogous to the time reversibility of Newtonian mechanics, though the state descriptions are different.
In the early to mid-twentieth century, physicists found that they could not describe the observed behavior of the electrons orbiting the nuclei of atoms by use of Newtonian physics. A new mathematical description of that behavior was developed. Called quantum mechanics, it has been tremendously successful in accounting for widely diverse phenomena at both the qualitative and quantitative level and is universally regarded as generally applicable to all matter. However quantum mechanics has some peculiar counter intuitive features and some aspects of its interpretation remain in dispute. This chapter focuses on how quantum mechanics changes the Newtonian picture of the past, present and future and of time itself. The mathematical apparatus governing the time dependence of the wave functions which describe states in quantum mechanics is completely time reversible, in a way closely analogous to the time reversibility of Newtonian mechanics, though the state descriptions are different.
The early twentieth century was a revolutionary time in the history of physics. People often think of Einstein's special theory of relativity of 1905, which changed our conceptions of time and space. But among physicists, quantum mechanics is usually regarded as an even more radical change in our thinking about the physical world. Quantum mechanics, which was developed between 1900 and 1926, began as a theory of atoms and light but has now become the framework in terms of which all basic physical theories are expected to be cast. We need quantum ideas not only to understand atoms, molecules, and elementary particles, but also to understand the electronic properties of solids and even certain astronomical phenomena such as the stability of white dwarf stars. The theory was radical in part because it introduced probabilistic behavior as a fundamental aspect of the world, but even more because it seems to allow mutually exclusive situations to exist simultaneously in a “quantum superposition.” We will see later that the possibility of quantum superposition is largely responsible for a quantum computer's distinctive advantage over an ordinary computer. The present chapter is devoted to introducing the basic principles of quantum mechanics. There are essentially four components of the mathematical structure of quantum mechanics. We need to know how to represent (i) states, (ii) measurements, (iii) reversible transformations, and (iv) composite systems. We will develop the first three in stages, starting with a very simple case – linear polarization of photons – and working toward the most general case.
The behavior of classical macroscopic particles – from baseballs to planets – is governed by Newtonian mechanics as we have described in the first chapters of this book. The behavior of very small particles, on the other hand, is governed by an entirely different collection of equations and rules that go under the name of “quantum mechanics” and these particles are referred to as quantum particles. These rules have analogies to classical mechanics, but are fundamentally different and somewhat mysterious as we discussed in Chapter 8. The fundamental principles of quantum mechanics cannot be proven to be correct. They were developed based on the results of experimental observation, by analogy with classical mechanics, and through the consideration of logical implications of the formalisms used. There is no way to derive the fundamental elements of quantum mechanics, any more than one could predict from first principles that Newtonian forces should add as vectors. On the other hand, quantum mechanics is among the most rigorously tested theories in science, and the predictions of quantum mechanics have, so far, always been in agreement with experiment. In this chapter, we will present quantum mechanics as a series of postulates and examine their implications in application to simple one-dimensional systems. In the next chapter we will examine quantum effects in real systems including atoms, molecules, crystals, and artificially engineered nano-structures. Postulates of quantum mechanics QM Postulate 1: A quantum particle is described by a wave function Ψ( x , y , z , t ) which is a function of spatial coordinates and time. All physically determinable quantities can be derived from the wave function Ψ( x , y , z , t ) ; quantities which cannot be found from the wave function are not physically meaningful. Note that some things that one could ask about a quantum particle, for example the exact position and simultaneous exact momentum of a particle, are not derivable from the wave function and are therefore considered indeterminate (or uncertain ). These are things which cannot ever be known about a particle, and, as such, they are meaningless to contemplate.
Quantum mechanics (QM) is the description of dynamics in the microscopic world of atoms, photons, electrons, etc. Similar to Newton’s law of dynamics, dp/dt = F, which is the statement on the time evolution of the dynamical physical state associated with matter, the Schrödinger equation iħdψ/dt = Hψ is the statement of the time evolution on the quantum physical state of matter, denoted with the function ψ ( x, t ). Though the wavefunction ψ ( x, t ) is not a dynamical quantity like momentum, the role of time in the description of dynamics is similar and straightforward, if the final results are interpreted as the average over a statistical ensemble (collection). In the conventional interpretation of quantum mechanics, the wavefunction is supposed to be the complete description of the physical state of the particle, like an electron or an atom. It is complex-valued, mathematically, and needs two real quantities for its specification. It evolves in ‘magnitude’ and ‘phase’ as the reference clock advances in time. The evolution progresses causally, connecting the past to the present and into the future, as one expects in dynamics. Thus, the variable called ‘time’ in the microscopic world of quantum mechanics is the same as the time that is familiar in our macroscopic world.
This chapter aims to introduce some of the applications of the ideas of quantum mechanics that were developed during the last twenty years or so of the twentieth century. The topics chosen reflect our increased understanding of a number of areas and its application to new phenomena that have been predicted and sometimes observed. They all rely on the interplay between the two types of time dependence implicit in the fundamental postulates set out in Chapter 6: Postulate 5 Between measurements, the development of the wave function with time is governed by the time-dependent Schro¨dinger equation.
So far in our development of quantum mechanics, we have given very little consideration to problems involving the time dependence of the wave function. This is rather surprising as in classical mechanics it is time-dependent phenomenai.e., dynamics rather than statics-that command most attention. Moreover, most experimental observations necessarily involve some change in the quantity being observed, so we might expect that the successful prediction of experimental results will require a detailed understanding of the way in which a system changes with time. In fact we have already made some implicit assumptions about time dependence and the principal reason why we have got so far without discussing it in detail is that many observed quantum phenomena are associated with sudden discontinuous changes between otherwise stable states. Thus, most of our information concerning the energy levels of atoms has been obtained from measurements of the frequencies of electromagnetic radiation emitted or absorbed as the atom undergoes a transition from one energy eigenstate to another, assuming the correctness of the formula E = }ω, but not considering the mechanism of the transition in any detail. Assumptions concerning time dependence are also implicit in the quantum theory of measurement which refers to the probability of obtaining a particular result following a measurement performed on a system in a given state; thus the state vector of a spinhalf particle changes from being an eigenvector Sˆ z to being one of Sˆ x following a measurement of the latter property using an appropriately oriented Stern-Gerlach apparatus. However, as was pointed out in Chapter 7, and will be discussed further in Chapters 15 and 16, the collapse of the wave function associated with a measurement is not a direct consequence of the time-dependent Schro¨dinger equation.
Newtonian mechanics is the foundation of Classical Physics. Newton’s mechanics, Thermodynamics, Wave theory of Optics and Maxwell’s electromagnetic theory belong to regime of Classical Physics and can be used to explain a wide range of phenomena in the Universe at macroscopic scale. These theories fail spectacularly when applied to phenomenain the atomic and nuclear regime, for example, proton-atom scattering or the flow of electrons in a semiconductor. Quantum mechanics is the most successful scientific theory that has ever been created and it has completely changed our view of the world. The failure of Classical Physics was highlighted by black body radiation and photoelectric effect. Max Planck and Albert Einstein provided explanations of both phenomena based on quantum hypothesis and are thus considered founders of Quantum Physics. The origin of quantum mechanics goes back to the mid-1920s.It was formulated first as matrix mechanics by Werner Heisenberg, Max Born and Pascual Jordan; then as wave mechanics by Louis de Broglie and Erwin Schrödinger; and later on asquantum statistics of subatomic particles by Fermi-Dirac and Bose-Einstein. Combining relativistic mechanics with quantum mechanics, Dirac formulated his relativistic quantum mechanics during 1930s. Uncertainty Principle is thecornerstone of Quantum Physics. The role of randomness in microscopic physical processes shatters the myth that the universe is deterministic. Quantum world is unpredictable in the classical sense and demolishes the idea of an objective universe. The Copenhagen interpretation remains the quantum mechanical formalism that is currently most widely accepted amongst physicists.Quantum theories support cosmic spirit pervading the cosmos and inter – relationship of individuals in the world society. Quantum philosophy is holistic and is going to revolutionize our world-view.