Quantum mechanics is traditionally taught in a specific representation—the position-space representation—because the time-independent Schrödinger equation is simplest to formulate there. But this leads to a number of conceptual issues both associated with language and with the formalism, especially if one starts more formally and then emphasizes representations for calculations. In this work, we describe a number of these issues and argue why restricting to a representation-independent formalism (that is, working with abstract operators and state vectors independent of any specific representation) fixes many of these issues. Then representations can be introduced later when needed for specific applications (but we find the need to do this can be sharply reduced from current paradigms).
Quantum mechanics is traditionally taught in a specific representation—the position-space representation—because the time-independent Schrödinger equation is simplest to formulate there. But this leads to a number of conceptual issues both associated with language and with the formalism, especially if one starts more formally and then emphasizes representations for calculations. In this work, we describe a number of these issues and argue why restricting to a representation-independent formalism (that is, working with abstract operators and state vectors independent of any specific representation) fixes many of these issues. Then representations can be introduced later when needed for specific applications (but we find the need to do this can be sharply reduced from current paradigms).
Abstract A discussion of the physical meaning of the Schrödinger wave equation can not only constitute an exciting introduction to some of the more abstract ideas of quantum mechanics, but serves more generally as a useful demonstration of the application of mathematics to modern physics. This frontline uses physical concepts and mathematical techniques that would be accessible to a sufficiently interested secondary school student, in order to derive the simplest, one-dimensional case of the time-independent Schrödinger equation: the derivation relies only upon aspects of Newtonian/wave mechanics, quantum theory, complex numbers and calculus that would be covered in an advanced secondary school syllabus.
Abstract In a cultural, history-based, sense-making approach to Quantum Mechanics multi-representations are fundamental to improve conceptual learning of abstract, mathematical aspects. In fact, they can be integrated among each other allowing students to actively manipulate concepts and instruments which would otherwise be difficult and little accessible. In this presentation we will show different (and interconnected) representations of vectors in complex Hilbert spaces and of eigenvalues and eigenvectors of Hermitian matrices together with an evaluation of their efficacy for students conceptual understanding.
We have seen in the previous chapter how useful wave mechanics can be in solving physical problems. But wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical observable? For instance, we might want to describe states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than the position. A more fundamental limitation is that there are attributes of physical systems that cannot be described at all in terms of the positions and momenta of a set of particles. One of these attributes is spin , which will be a chief subject of Chapter 4. Another is the value of the electric or magnetic field at some point in space, treated in Chapter 11. This chapter will describe the principles of quantum mechanics in a formalism which is essentially the “transformation theory” of Dirac, mentioned briefly in Section 1.4. This formalism generalizes both the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg, and is sufficiently comprehensive to apply to any sort of physical system. States The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space . Before getting into Hilbert space, I need to say a bit about vectors in general. In kindergarten we learn that vectors are quantities with both magnitude and direction. Later, when we study analytic geometry, we learn instead to describe a vector in d dimensions as a string of d numbers, the components of the vector. The latter approach lends itself well to calculation, but in some respects the kindergarten version is better, because it allows us to describe relations among vectors without specifying a coordinate system. For instance, a statement that one vector is parallel to a second vector, or perpendicular to a third, has nothing to do with how we choose our coordinate system. Here we will formulate what we mean by vector spaces in general, and Hilbert space in particular, in a way that is independent of the coordinates we use to describe directions in these spaces.
We have seen in the previous chapter how useful wave mechanics can be in solving physical problems. But wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical observable? For instance, we might want to describe states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than the position. A more fundamental limitation: There are attributes of physical systems that cannot be described at all in terms of the positions and momenta of a set of particles. One of these attributes is spin , which will be the subject of Chapter 4. Another is the value of the electric or magnetic field at some point in space, treated in Chapter 11. This chapter will describe the principles of quantum mechanics in a formalism which is essentially the “transformation theory” of Dirac, mentioned briefly in Section 1.4. This formalism generalizes both the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg, and is sufficiently comprehensive to apply to any sort of physical system. States The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space . Before getting into Hilbert space, I need to say a bit about vectors in general. In kindergarten we learn that vectors are quantities with both magnitude and direction. Later, when we study analytic geometry, we learn instead to describe a vector in d dimensions as a string of d numbers, the components of the vector. This latter approach lends itself well to calculation, but in some respects the kindergarten version is better, because it allows us to describe relations among vectors without specifying a coordinate system. For instance, a statement that one vector is parallel to a second vector, or perpendicular to a third, has nothing to do with how we choose our coordinate system.
Abstract Quantum mechanics is written in the language of linear algebra. On the Schrodinger picture the theory represents quantum-mechanical states using the elements of a Hilbert space and represents observable physical properties and the standard dynamics using the linear operators on the state space. We consider the mathematical notions for understanding and working with the standard formulation of quantum mechanics. Each mathematical notion is characterized geometrically, algebraically, and physically. The mathematical representation of quantum-mechanical superpositions is discussed.
In the spin-first approach to learning quantum mechanics, students explore the 2-state spin Hilbert space before proceeding to wavefunctions and the related infinite-dimensional state space. In this note, we suggest a strategy to help students make this conceptual leap. Approximating continuous space with a finite number of locations, we construct a setting where students' intuition from spin space carries over to a position basis. We derive the position basis representations of the momentum and kinetic energy operators, solve simple bound state problems, and consider limitations to the discrete representation of position.