Quantum mechanics is hard. Can a toy make the subject any easier? Costas Papaliolios thought so. As a graduate student at Harvard in the early 1960s, he created these aluminum blocks marked with variations of Paul Dirac's bra-ket notation for describing quantum states. He made 4-inch blocks and 1-inch blocks. (The upper photos are enlargements of the 1-inch, or 2.54-centimeter, blocks.) A polarized filter fit inside each open-ended block. Shining a light through a block, or a series of blocks, yielded one of two results: The light passed through or it didn't. Papaliolios hoped his toys would help students grasp abstract concepts like linear operators and matrix representation. For most people, though, quantum mechanics remains as puzzling as ever.
Quantum mechanics is hard. Can a toy make the subject any easier? Costas Papaliolios thought so. As a graduate student at Harvard in the early 1960s, he created these aluminum blocks marked with variations of Paul Dirac's bra-ket notation for describing quantum states. He made 4-inch blocks and 1-inch blocks. (The upper photos are enlargements of the 1-inch, or 2.54-centimeter, blocks.) A polarized filter fit inside each open-ended block. Shining a light through a block, or a series of blocks, yielded one of two results: The light passed through or it didn't. Papaliolios hoped his toys would help students grasp abstract concepts like linear operators and matrix representation. For most people, though, quantum mechanics remains as puzzling as ever.
This innovative new text presents quantum mechanics in a manner that directly reflects the methods used in modern physics research—making the material more approachable and preparing students more thoroughly for real research. Most texts in this area start with a bit of history and then move directly to wave-particle problems with accompanying heavy mathematical analysis; Quantum Mechanics provides a foundation in experimental phenomena and uses a more approachable, less intimidating, more powerful mathematical matrix model. Beginning with the Stern-Gerlach experiments and the discussion of spin measurements, and using bra-ket notation, the authors introduce an important notational system that is used throughout quantum mechanics. This non-traditional presentation is designed to enhance students’ understanding and strengthen their intuitive grasp of the subject.
The relationship between physical notions and the structures expressing them is an issue of primary importance the history and philosophy of pre-modern natural philosophy and of modern science. Historical evidence suggests that, when investigating the relationship between physics and mathematics, one should pay attention to the material, sensually perceivable forms mediating it: spoken or written words, drawings, artefacts, descriptions of experimental set-ups or, as the case discussed the present paper, symbolic notation. The reason for this is that, historically, physical notions and their expressions have very often been developed as one, and only distinguished from each other at a later stage. This was for example the case with what we today regard as mathematical and classical mechanics, which were developed the course of the eighteenth century as the discipline of the analysis of motion. When considered from a later point of view, both the physical notions and the concepts involved historical developments appear as being in flux, vague, and, as far as the side is concerned, non-rigorous. In such cases, verbal description, mechanical artefacts or symbols do not simply mediate between physical notions and structures, as might be the case at a later stage, but represent both of them one, allowing the historical actors to construct and manipulate physical-mathematical meaning by following, at times, their physical reasoning and, at times, their criteria of rigour. The history of quantum mechanics offers many examples of how new physics and new mathematics can emerge as one, with symbolic notation playing a central role shaping both. Among the many creators of quantum theory, Paul Dirac was particularly sensitive to questions of notation:
Quantum physics is the most successful description of the physical world we live in. Quantum theoretical predictions have been experimentally tested and confirmed up to the astonishing 14 digits of precision in terms of the anomalous magnetic moment of the electron. The bra-ket notation, composed of angle brackets and vertical bars, enhances the clarity of mathematical operations and improves the comprehension of quantum mechanical expressions. Quantum wave functions participate in interference phenomenona in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude. When a quantum system is measured, it undergoes indeterministic transition that actualizes only one of the states from the set of all physically possible future states of the system. The Hilbert vector space structure of the space of quantum states follows from the linearity of the Schrodinger equation. The quantum state vector and the quantum wave function in a Hilbert space are abstract mathematical concepts that are basis independent.
Come forth into the light of things, Let nature be your teacher. William Wordsworth , The Tables Turned The main focus of this book is many-particle systems such as electrons in a crystal. Such systems are studied within the framework of quantum mechanics, with which the reader is assumed to be familiar. Nevertheless, a brief review of this subject will provide an opportunity to establish notation and collect results that will be used later on. The postulates Quantummechanics is based on five postulates, listed below with some explanatory comments. The quantum state The quantum state of a particle, at time t , is described by a continuous, singlevalued, square-integrable wave function Ψ( r , t ), where r is the position of the particle. In Dirac notation, the state is represented by a state vector, or ket, ∣Ψ( t )⟩, which is an element of a vector space V. We define a dual vector space V* whose elements, called bras, are in one-to-one correspondence with the elements of V: ket ∣α⟩ ∈ V ⟷ bra ⟨α∣ ∈ V*, as illustrated in Figure 1.1. The bra corresponding to ket c ∣α⟩ is c * ⟨∣, where c *. is the complex conjugate of c . The inner product of kets ∣α⟩ and ∣β⟩ is denoted by ⟨β∣α⟩, and it is a complex number (c-number). Note that the inner product is obtained by combining a bra and a ket. By definition, ⟨β∣α⟩ = ⟨α∣β⟩*.
Given the number of books that already exist on the subject of quantum mechanics, one would think that the public needs one more as much as it does, say, the latest version of the Table oflntegers. But this does not deter me (as it didn't my predecessors) from trying to circulate my own version of how it ought to be taught. The approach to be presented here (to be described in a moment) was first tried on a group of Harvard undergraduates in the summer of '76, once again in the summer of '77, and more recently at Yale on undergraduates ('77-'78) and graduates ('78- '79) taking a year-long course on the subject.