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CRC Press eBooks

Time-Dependent Quantum Mechanics

Sven Larsson

Published Jan 26, 2012• 0 citations
Publisher

Abstract

The time-dependent Schrödinger equation (SE) or the wave equation may be written as i x,t H x,tη ∂∂ ( ) = ( )t Ψ Ψ , (7.1) where x contains, in principle, all the space and spin coordinates of all particles of the system. Using Equations 1.12 through 1.14 it is easy to check that a wave packet satis›es the wave equation.
DOI
10.1201/b11524-10

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American Journal of Physics•Sep 1, 1979•4 citations
The time-dependent Schrödinger equation (TDSE) is a wave equation that is used to understand the behavior of small particles such as electrons or nuclei. The equation must be "solved" in the sense that a function must be found that can be plugged into the equation and that leaves the left-hand side equal to the right-hand side. The TDSE can be understood by first studying the classical wave equation that is used to "solve" the behavior of a guitar string or a water wave. The TDSE is a bit different in that it equates a double space derivative with a single time derivative (the classical equation equates two double derivatives); it also implicitly includes the symbol i , which stands for the square root of -1. These differences give rise to different sorts of solutions, including complex solutions with real and imaginary parts. In the case of the TDSE, the solution or "wave" does not directly show the motion of the particle. Instead, it shows where the particle is likely to be found. More specifically, it is a probability wave for particle location. To obtain the precise probability for the particle to be found at a specific position x at a specific time t , the wave function Ψ ( x , t ) must be squared via a special procedure called taking the square modulus. The wave contains information about the frequency, wavelength, position, and momentum of the particle. In the following chapters, we will be showing how to find the wave solution in different circumstances (different potentials), and how to extract the information about momentum, etc., once the wave function is known.
We now may bring the time-dependent state functions ( )tx,Ψ into the picture, taking into account, as we already did in Eq.(29.9), that only the state function but not its position coordinate x varies with time, so that ( ) ( ) t tx dt txd ∂ Ψ∂≡Ψ ,, (30.3) Upon insertion of the expression (29.24) for the expectation value, Eq.(30.2) becomes ˆˆ ˆ ˆ ˆd d fi f i f dx i f f dx i dt dt t t t ⎛ ⎞∂Ψ ∂Ψ ∂= Ψ Ψ = Ψ +Ψ + Ψ Ψ⎜ ⎟∂ ∂ ∂⎝ ⎠∫ ∫ ∫= = = = dx∗ (30.4) On the other hand, as the Hamiltonian Hˆ is a Hermitian operator, we have ( )ˆ ˆˆ ˆ ˆˆ ˆ ˆ, f ff H i Hf f H dx i dt t ∞ ∞ ∂ ∂⎡ ⎤ + = −Ψ Ψ +Ψ Ψ + Ψ Ψ⎣ ⎦ ∂ ∂∫ ∫= = x (30.5) ( ) dx t fidxHffH Ψ∂ ∂Ψ+⎥⎦ ⎤⎢⎣ ⎡ ΨΨ+ΨΨ−= ∫∫ ∗∗ ˆˆˆˆˆ = From Eqs.(30.4) and (30.5) it follows that the evolution equation (30.2) is true only if Ψ=∂ Ψ∂ H t i ˆ= (30.6) and this is a basic result called the time-dependent Schrödinger equation. This equation describes the evolution of quantum systems in terms of time-dependent state functions that obey Eq.(30.6), but neglect the time dependence of the quantum operators. Such a description is known as the Schrödinger picture. The fifth postulate states that the timedependent Schrödinger equation must always be satisfied. Postulate 5. The evolution of the state function of a quantum system is given by the time-dependent Schrödinger equation (30.6), where Hˆ is the Hamiltonian of the system.
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.
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
The time-dependent Schrödinger equation (TDSE) contains two variables: position and time. Therefore, the solutions are functions of both space and time (x and t, if we limit ourselves to particles moving in one dimension). It turns out that the easiest way to "solve" such an equation of two variables is to perform a separation of variables. We show that if the potential does not change with time, then certain special solutions (eigenfunctions) can be written as a product of a space function and a time function: Ψ(x,t) = ψ(x) f(t). By assuming a "separable" solution of this form, and plugging into the TDSE, the TDSE itself can be separated into two equations: the time-independent Schrödinger equation (TISE) with solution ψ(x), and the space-independent Schrödinger equation (SISE) with solution f(t). Furthermore, f(t) always equals e -iωt , while ψ(x) depends on the potential. So, for potentials that do not vary with time, we need only solve the much simpler TISE in order to investigate the energy, momentum, and so on of the particle. This is why, for the remainder of this book, we only use the TISE.