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Unformatted text preview: Quantum Mechanics: Postulates 5th April 2010 I. Physical meaning of the Wavefunction Postulate 1: The wavefunction attempts to describe a quantum me- chanical entity (photon, electron, x-ray, etc.) through its spatial location and time dependence, i.e. the wavefunction is in the most general sense dependent on time and space: = ( x, t ) The state of a quantum mechanical system is completely specified by the wavefunction ( x, t ). The Probability that a particle will be found at time t in a spatial interval of width dx centered about x is determined by the wavefunction as: P ( x , t ) dx = * ( x , t )( x , t ) dx = | ( x , t ) | 2 dx Note: Unlike for a classical wave, with a well-defined amplitude (as dis- cussed earlier), the ( x, t ) amplitude is not ascribed a meaning. Note: Since the postulate of the probability is defined through the use of a complex conjugate , * , it is accepted that the wavefunction is a complex- valued entity. Note: Since the wavefunction is squared to obtain the probability, the wavefunction itself can be complex and/or negative. This still leaves a prob- ability of zero to one. Note: * is the complex conjugate of . For instance: 1 ( x ) = A e i k x * ( x ) = ( A * ) e- i k x Since the probability of a particle being somewhere in space is unity, the integration of the wavefunction over all space leads to a probability of 1. That is, the wavefunction is normalized : Z - * ( x, t )( x, t ) dx = 1 In order for ( x, t ) to represent a viable physical state, certain condi- tions are required: 1. The wavefunction must be a single-valued function of the spatial coordinates. (single probability for being in a given spatial interval) 2. The first derivative of the wavefunction must be continuous so that the second derivative exists in order to satisfiy the Schr o dinger equation....
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