Lecture 1 - Wave function and Schrdinger equation...

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Unformatted text preview: Wave function and Schrdinger equation Fundamentals of quantum mechanical approach Framework of classical mechanics of particles Particles are indivisible and countable; can be presented as geometrical points A dynamics of a particle is described by a set of coordinates and velocities (momentums) To describe the motion means to determine particles coordinates and velocities at any given time. The main tool of study is Newtons laws. Classical concept of waves A wave is propagation of a macroscopic excitation of a continuous medium or electromagnetic fields Waves are distributed continuously and can be divided Waves can spatially overlap, and the resulting wave is a sum of the individual waves Waves obey wave equations such as Maxwell equations for electromagnetic waves Harmonic classical waves ( ) ( ) ( ) ( , ) cos ( , ) sin ( , ) i kx t x t A kx t x t A kx t x t Ae + = + = + = A amplitude, k wave number , frequency Cos or Sin waves can be obtained as real or imaginary parts of the complex exponential wave. In linear classical wave physics all forms are interchangeable Phase velocity kx t = + Phase changes with time and in space Coordinate corresponding to any constant phase must change in time according to k x t x t k = = = ph v k = Is called phase velocity Dispersive and non-dispersive waves Simplest wave packet consists of just two waves ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 2 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 ( , ) cos cos 2 cos cos 2 2 corresponds to points of constructive interference describes motion of these x t A k x t A k x t x k k t x k k t A x k k t n x t k k = + + + = + + + + + + = = 1 2 1 2 1 2 1 2 points unless x const t k k k k k = = Waves for which phase velocity is constant are called non-dispersive, otherwise they are called dispersive waves. Standard harmonic waves like light in vacuum are non-dispersive 2 2 2 2 2 2 2 2 1 is called dispersion relation k c t x c c const k + = = = = Wave packets ( ) ( ) 1 ( , ) 2 k i kx t x t A k e d k...
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Lecture 1 - Wave function and Schrdinger equation...

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