Schrodinger's Equation

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The time-dependent Schrodinger equation determines how a given initial wavefunction will change with time. It differs from Newton's law in two important ways: it involves a first time derivative instead of a second, and the square root of -1 appears explicitly in it. As a consequence, we cannot solve it using the Feynman algorithm (although we will use a half-step look ahead), and we have to keep track of real and imaginary parts (the curvature in one drives the time dependence of the other). A wavefunction in an infinite square well is similar in some respects to a classical string clamped at both ends. Both can be started off in a pluck or pulse configuration, both have their change in configuration driven by their curvature, and, since both obey zero boundary conditions, both can use FFSS as an alternative means of generating their time evolution. The initial complex phase of a quantum wave plays the same role as the initial transverse velocity for a wave on a string.


We can put the Schrodinger equation for an infinite square well suitable form for numerical solution by introducing the period of the lowest energy stationary state.

The only parameters in the final form of the equation are the width L of the well and the ground state period. These parameters disappear from the numerical equation when they are used as units for dimensionless distance and time variables. In the example, one has three choices for the initial wavefunction, and can generate its time dependence either directly using the Schrodinger equation or indirectly using FFSS and the dispersion relation (frequency is proportional to square of quantum number). The first two choices are the rectilinear Pluck and Pulse displacements used on the classical string. The initial wavefunction is real in both cases: this is the quantum analog of a string released from rest. The third choice, Pulse S, is a rounded version of the rectilinear Pulse, and is also real. The fourth choice, Pulse A, is Pulse S multiplied by a complex exponential that introduces a phase change of 2p across its width. The complex exponential is the quantum analog of initial transverse velocity for the classical string. It does not alter the initial probability density, but it increases the kinetic energy, and markedly changes the time dependence. (Energies are expressed as multiples of the ground-state energy.)

For the classical string, a set of mass points on a weightless string is a physical model with its own dispersion relation. The quantum string has no corresponding physical model. The (Java) program divides the well into 192 segments and follows the motion for 1/8 of the ground state period. The three symmetric "standing" (initially real) wavefunctions complete one probability density oscillation in this length of time. The calculation uses a half-step look ahead similar to what we used in field plots: we want the curvature in the function at the mid-point of the time step we are about to take.

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