(wave packets,  eigenstates, eigenvalues, impulsive and adiabatic dynamics etc.)

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The wave-amplitude propagation of a 1-dimensional quantum mechanical particle under the action of a given
potential can be visualized by solving numerically the Schrödinger equation

To accomplish this, the following units and  physical constants are suitable:

• time unit = 1 femtosecond (fsec);

• length unit = 1 Angstrom  (Å);

• potential energy unit = 1 electron-Volt (eV)

• set , with  = particle-mass/electron-mass ratio;

• m e = 5.11e5 eV   is the electron mass times the square of light velocity;

• = 0.65821 eVxfsec is the Planck's constant;

• light velocity = 2998 Å/fsec.
  

with  .

As in practical computations the time interval is small but not infinitesimal, the integration of this equation must be performed  by the iteration of a discrete time translation

.

The infinitesimal-time translation  , where (H = Hamiltonian/ ),  is certainly unitary, but unfortunately its discrete-time generalization   is not. Consequently, the conservation of  total probability 1 would be lost after a few iterations. To assure that the discrete-time approximation is unitary, the elementary time translation can be put in the Cayley's form

When H  is replaced by its finite-difference approximation, the equation takes the form of a tridiagonal matrix equation that can then be solved by the Crank-Nicholson method (see Numerical Recipes in C - The Art of Scientific Computating, W.Press, B.Flannery, S.A. Teukowsky and W.Vetterling - Cambridge University Press, 2002).  Because of the well-known problems with integer indexing of  cycles FOR ..., END,  unitary discrete-time transformations cannot be implemented by fast Matlab routines. The Author solved this problem by compiling  two axiliary MEX files SCHROEQ.DLL,  SCHRSTP.DLL. These files are called by other Matlab routines to produce time-domain solutions of 1-D Schrödinger equations. Their usage is described in SCHROEQ.M,  SCHRSTP.M.

Download WPACKET.ZIP (Matlab routines) [first implementation March 2001, last revision: November 2010]

• WPACKET.M                GUI interface routine. Call this to run the programme
• SCHROEQ.C                 Source file in C used to compile  SCHROEQ.DLL and SCHROEQ.MEXW32 
  
• SCHROEQ.DLL            DLL file, this is the core of the programme for versions 5.x and 6.x
• SCHROEQ.MEXW32   MEXW32 file, this is the core of the programme for versions > 7.1
  
• SCHROEQ.M                Help file describing the usage and the action of SCHROEQ.DLL and SCHROEQ.MEXW32
• GSSPCKT.M                 Routine for the generation of a Gaussian wave-packet of given width and wave-number
• PTNTLS.M                    Routine for the generation of various potential-energy profiles
 

Download POTWELLS.ZIP (Matlab routines) [last revision: November 2010].

• POTWELLS.M             GUI interface routine. Call this to run the programme
• SCHROSC.C                Source file in C used to compile  SCHROSC.DLL and SCHROSC.MEXW32 
• SCHROSC.DLL           DLL file, this routine is called by POWELLS.M under Matlab V. 5.x and 6.x to produce the time evolution of the wave function in a potential well subject to an oscillatory perturbation.
• SCHROSC.MEXW32  MEXW32 file, similar routine called  by POWELLS.M under Matlab versions > 7.1
• SCHROSC.M               Help file describing the usage and action of SCHROSC.DLL and SCHROSC.MEXW32
• SCHRSTP.C                 Source file in C used to compile  SCHRSTP.DLL and SCHRSTP.MEXW32
  
• SCHRSTP.DLL             DLL file, this routine is called by POWELLS.M under Matlab V. 5.x and 6.x to produce the time evolution of the wave function in a potential that changes in time from a profile V1 to a profile V2
• SCHRSTP.MEXW32   MEXW32 file, , similar routine called  by POWELLS.M under Matlab versions > 7.1
• SCHRSTP.M                Help file describing the usage and action of SCHRSTP.DLL and  SCHRSTP.MEXW32
  

To run the programme POTWELLS.M Unzip the package in a directory and set Matlab 5.x to that directory. From the Matlab command line enter potwells. The GUI interface will be displayed immediately showing the eigen-function of the fundamental level for the displayed potential well.  The probability density of the particle at the initial time is represented by the black profile, the real and imaginary parts of the wave function are represented by magenta and cyan color profiles respectively. Initially, the potential well is an infinite rectangular box whose X-axis width (in Angstrom) can be changed acting on the edit-slider control at the GUI interface bottom. Other potential profiles can be selected from the pop-menu Potential type. The program takes a while to compute the eigen-functions and eigen-values of the selected potential well.  The control up/down on the GUI interface left-bottom can be used to visualize the eigen-function corresponding the indicated energy level. The dynamical section contains controls for observing the time course of the initial state when the potential profile start varying in time (sudden or slow changes, oscillations etc.). The small window under the dynamic sections shows the energy spectrum of the wave function after the time evolution.