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Schrödinger equation on a lattice

Efstratios Manousakis

Published Nov 1, 2015• 0 citations
Publisher

Abstract

Abstract This chapter begins by studying the one-dimensional time-dependent Schrödinger equation. In order to do this the chapter discretizes the spatial coordinates, and thus the differential equation becomes a set of coupled difference equations first order in time. Dealing with a finite number or a numerable set of states makes it easier to understand the concepts of Hilbert space, its measure, and the overlap between states. Furthermore, the chapter only has to deal with the Kronecker symbol as opposed to the Dirac delta function. Later, in Chapter 3, it will consider the continuum limit where we will naturally recover the Schrödinger differential equation, the state normalization condition as an integration condition, and the delta function.

DOI
10.1093/acprof:oso/9780198749349.003.0001

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Abstract This chapter begins by studying the one-dimensional time-dependent Schrödinger equation. In order to do this the chapter discretizes the spatial coordinates, and thus the differential equation becomes a set of coupled difference equations first order in time. Dealing with a finite number or a numerable set of states makes it easier to understand the concepts of Hilbert space, its measure, and the overlap between states. Furthermore, the chapter only has to deal with the Kronecker symbol as opposed to the Dirac delta function. Later, in Chapter 3, it will consider the continuum limit where we will naturally recover the Schrödinger differential equation, the state normalization condition as an integration condition, and the delta function.
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Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description
This chapter begins by exploiting the Einstein–de Broglie relation and the notion of symbol, as defined in Eq. (3.6.6). The following steps are the local and global conservation laws associated to the Schrödinger equation, the probabilistic interpretation of the wave function, the spreading of wave packets and transformation properties of wave functions. The solution of the Schrödinger equation for a given initial condition is then studied. For this purpose, one first expands the initial condition in terms of eigenfunctions of the Hamiltonian operator. This is then ‘propagated’ to determine the evolution of the wave function. Thus, one is led to consider the Green kernel of the Schrödinger equation, and the boundary conditions on the wave function that lead to complete knowledge of stationary states. A dynamics involving an isometric non-unitary operator is also studied. The chapter ends with an elementary introduction to harmonic oscillators and to the JWKB method in wave mechanics, plus examples on the Bohr–Sommerfeld quantization. From classical to wave mechanics The correspondence between differential operators and polynomials in k relies on the notion of the symbol defined in Eq. (3.6.6). Now we are aiming to pass from a function on phase space, i.e. the classical Hamiltonian, to an operator which, by analogy, is called the Hamiltonian operator. Such a transition is non-trivial because the map from symbols to operators contains some ambiguity. Such a transition is always possible if polynomials in p have constant coefficients. When this is not the case, i.e. if the coefficients are functions on the configuration space, there is an obvious ambiguity in the association of an operator with xp or px , for instance.
A symmetrical treatment of the incoming and outgoing channels can be done in a basically very simple way by formulating the Schrödinger equation (1.1) in the form of a projection equation (see, e.g., ref. [WI 72]) (2.1) ⟨δψ∣H+ℏi∂∂t∣ψ⟩=0 \langle \delta \psi \left| {{\rm H} + \frac{\hbar }{i}\frac{\partial }{{\partial t}}} \right|\psi \rangle = 0 ⟨δψ​H+iℏ​∂t∂​​ψ⟩=0 As usual, the Dirac bracket denotes the integration over spatial coordinates and the summation over spin and isospin coordinates. If δΨ represents a completely arbitrary variation of Ψ at a given instant t in the complete Hilbert space, then eq. (2.1) implies that the vector (H+ℏi∂∂t)∣ψ⟩ \left( {{\rm H} + \frac{\hbar }{i}\frac{\partial }{{\partial t}}} \right)\left| {\psi \rangle } \right. (H+iℏ​∂t∂​)∣ψ⟩ must be orthogonal to any arbitrary vector in this space. This evidently will be the case only if Ψ obeys eq. (1.1). Therefore, eq. (2.1) is merely another formulation of the time-dependent Schrödinger equation. However, as we shall see later, eq. (2.1) does allow us to treat the incoming and outgoing channels in a symmetrical way.