Lecture 4 Notes Quantum Mechanics

Lecture 4 Notes Quantum Mechanics - Lecture 4 Notes Quantum...

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Lecture 4 Notes: Quantum Mechanics First Postulate of Quantum Mechanics: Any quantum mechanical particle or a system in general can be described by a wavefunction   r , t . (Depending on a system may be in a vector form). In general we will represent wavefunction in a form of a ket : r , t . The wavefunctions belong to a state space, which has the properties of a vector space. The general properties of wavefunctions are: (a) Single valued (b) Square integrable (c) Nowhere infinite (at infinity as well as elsewhere) (d) Continuous (e) Piecewise continuous first derivative Now let s consider the simplest wavefunction for a particle. In a free space such as vacuum with no external forces the particle can be approximated as a plane wave: e ikx i t , where k p is the wavevector. Since there is no external forces then the energy of a particle is its kinetic energy: E mv 2 p 2 2 k 2 . 2 2 m 2 m How does this wave propagate in time and space? To answer this question let s consider the change in wavefunction with respect to time and position: e ikx i t ike ikx i t x e ikx i t   i e ikx i t 2 e ikx i t   k 2 e ikx i t t x 2 e ikx i t   i E e ikx i t 2 p 2 e ikx i t   e ikx i t t x 2 2 2 2 p 2 i e ikx i t Ee ikx i t ikx i t ikx i t t e e 2 m x 2 2 m  E This exercise results in a curious statement about the propagation of particle ~ plane wave in free space: 2 2 e ikx i t i e ikx i t 2 m x 2 t In general in the presence of forces in 3D space energy of a particle would be:
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E V x , t , which would yield a following equation for the particle described by a wavefunction   : 2 2 r , t V x , t x , t i x , t 2 m t ______________________________________________________________________________
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