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CRC Press eBooks

Separation of Variables

Steven M. Pascal

Published Jul 12, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.1201/9781003356172-6

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CRC Press eBooks•Jul 12, 2023•0 citations

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Beginning with an entangled state of a time-independent (TI) quantum system coupled to its TI quantum environment, we show that a time-dependent Schrödinger equation (TDSE) for the quantum system alone can be derived in the limit that one of the environment variables becomes a classical variable. In the same limit the TI amplitude of the environment wavefunction becomes the TD amplitude of an eigenfunction expansion of the system TD wavefunction. Similarly, the phase of the TI environment wavefunction goes over into the TD phase of the system amplitude. By considering that more and more environment variables become successively classical, each provides a classical clock to give a multiple-time TDSE for the quantum system. Two examples of two-time TDSE given in the literature are derived. When all clocks are synchronized the multiple-time TDSE reduces to the usual one-time TDSE.

Journal of Physics A Mathematical and Theoretical•Jan 23, 2007•29 citations
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.
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.
The solutions to the Time‐Independent Schrödinger Equation (TISE) can be found as long as we can define the Hamiltonian, which in the simplest cases, means defining the potential in which the particle of interest is moving. In the following sections, we will consider a single particle of mass, m p , moving in a potential that is time‐independent, but can vary in space. We will follow four steps: Define the Hamiltonian by defining the Potential Energy Operator since H ^ = - ħ 2 2 m p ∇ ^ 2 + V ^ ( r ) $ \hat{H} = - \frac{{{\hbar}^{2} }}{{2m_{p} }}\hat{\nabla }^{2} + \hat{V}(r) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315381381/5a219413-b541-4543-a0c3-fab4f5358e87/content/inline-math4_1.tif"/> . Solve the TISE to get the wave functions ψ ( x , y , z ) $ \psi (x, y, z) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315381381/5a219413-b541-4543-a0c3-fab4f5358e87/content/inline-math4_2.tif"/> and then the total stationary state wave functions Ψ ( x , y , z ; t ) = ψ ( x , y , z ) e - i ω t . $ \iPsi (x, y, z;t) = \psi (x, y, z)e^{ - i\omega t} . $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315381381/5a219413-b541-4543-a0c3-fab4f5358e87/content/inline-math4_3.tif"/> Determine the energies of the various states of the system. Examine the implications of the solutions to the Schrödinger equations.
The Schrödinger equation was really postulated rather than derived. But we can see why it was postulated. It has been proven to be valid, in that it provides information about small, slow moving particles subject to a wide variety of potentials. The derivation of the time-dependent Schrödinger equation (TDSE) follows rather directly from assuming a free particle wave: Ψ = e i(kx-ωt) . If this wave function is plugged into the classical wave equation, and quantum principles are used, the two sides of the equation are not equal (this means this function does not "solve" the equation). The quantum principles required are an understanding of the photoelectric effect (by which E = ħω) and the de Broglie hypothesis (λ = h/p). Using these relationships, we show that one derivative of time is equal to a double derivative with respect to position, with a few proportionality factors mixed in. Thus, the TDSE for a free particle is just a statement that kinetic energy equals total energy . For a non-free particle (subject to a potential), potential must merely be added to the left-hand side of the equation, to yield kinetic energy plus potential energy equals total energy .
Abstract After revisiting the time-dependent Schrödinger equation and introducing the Hamiltonian operator, the time-independent Schrödinger equation (TISE) is obtained. Conditions on acceptable wavefunctions are then stated. The method for solution of the TISE is laid out, and the infinite potential well is solved as an example. A discussion of the physical interpretation of the solutions — as energy eigenstates in position space — follows. This provides an opportunity to introduce overall and relative phases, and to point out that the latter are responsible for time evolution. The TISE is then applied to the important problems of scattering from a step potential and tunneling through a rectangular potential barrier. Use of the TISE for such problems greatly simplifies calculations, but it can also instill misconceptions; both aspects are discussed.
The time-independent Schrödinger equation (TISE) is a mathematical equation, and therefore it has mathematical solutions. But not all solutions that make mathematical sense also make physical sense. For instance, a possible solution might solve the TISE but go to infinity at a certain position. That cannot be: the particle cannot have infinite probability to be found somewhere. Likewise, the particle can't have more than one probability to be found somewhere. Hence, we can exclude mathematical solutions that are not finite, are double-valued, are discontinuous, or have a discontinuous derivative, since these characteristics are unrealistic for a physical particle. However, the last rule can be ignored if the potential function trapping the particle is itself infinitely discontinuous. Also, the total probability to find the particle somewhere should equal one. The process of making the wave function the right size so that the total probability equals one is called " normalization ". Calculations such as the normalization process are made much simpler by using the Dirac bra-ket notation, which is simply a shorthand notation that eliminates the need to write integral signs, dx, *, and limits.
Beginning with an entangled state of a time-independent (TI) quantum system coupled to its TI quantum environment, we show that a time-dependent Schrödinger equation (TDSE) for the quantum system alone can be derived in the limit that one of the environment variables becomes a classical variable. In the same limit the TI amplitude of the environment wavefunction becomes the TD amplitude of an eigenfunction expansion of the system TD wavefunction. Similarly, the phase of the TI environment wavefunction goes over into the TD phase of the system amplitude. By considering that more and more environment variables become successively classical, each provides a classical clock to give a multiple-time TDSE for the quantum system. Two examples of two-time TDSE given in the literature are derived. When all clocks are synchronized the multiple-time TDSE reduces to the usual one-time TDSE.