# Lecture 1 - Wave function and Schrödinger equation...

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Unformatted text preview: Wave function and Schrödinger 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 particle’s coordinates and velocities at any given time. • The main tool of study is Newton’s 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 ω ϕ ω ϕ ω ϕ − + Ψ = − + Ψ = − + Ψ = A amplitude, k wave number , ω 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 ϕ ω ϕ = − + 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 ω ϕ ω Δ = Δ − Δ = ⇒ Δ = Δ ph v k ω = 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 ω ϕ ω ϕ ω ω ϕ ϕ ω ω ϕ ϕ ω ω ϕ ϕ π ω ω Ψ = − + + − + = − − − + − + − + + + − − − + − = − Δ = Δ − 1 2 1 2 1 2 1 2 points unless x const t k k k k k ω ω ω ω ω − Δ = ≠ ≠ = Δ − 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 ψ ψ ω ω ∂ ∂ − + = ⇒ = ∂ ∂ = = Wave packets ( ) ( ) 1 ( , ) 2 k i kx t x t A k e d k ω π ∞ − −∞ Ψ...
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Lecture 1 - Wave function and Schrödinger equation...

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