Lecture16 - Particle in a finite 1-D box For E > V0 Wave...

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1 Physical Chemistry Lecture 16 Using the particle in a box to understand special problems Particle in a finite 1-D box With a finite potential in the “external” regions, the wave function can be nonzero in those regions Must solve Schroedinger’s equation piecewise in three regions Match the solutions at the boundaries between regions Match derivatives at the boundaries Uses symmetry to simplify problem Puts origin at the center of the box, rather than one edge Solutions can be classified as even or odd under inversion through zero 2 / 2 / 0 2 / 2 / ) ( 0 a x a for a x and a x for V x V  Particle in a finite 1-D box Schroedinger’s equation in the three regions looks similar With a finite potential in the “external” regions, the solutions –evenforE<V 0 include the possibility of the particle being in those regions Very different from predictions of classical mechanics Equations in the three regions look similar Solutions are sums of exponential functions Different kinds of solutions for E < V 0 and > 0 Quantum conditions are determined by boundary conditions  0 2 2 2 2 2 2 2 " " 2 2 / 2 / V E dx d m regions external For E dx d m a x a For Particle in a finite 1-D box For E > V 0 Wave functions are oscillatory functions of x Sums of exponential functions with imaginary arguments Amplitudes are related by boundary conditions For E , it approaches the behavior of the free particle in one dimension Oscil atory behavior Deviation from the free particle depends on energy relative to the potential 2 0 2 ' ' ) ( 2 ' 2 ' ' ) ( " " ) ( 2 / 2 / V E m k mE k where e k k B e k k A x regions external For Be Ae x a x a For x ik x ik ikx ikx
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Lecture16 - Particle in a finite 1-D box For E &gt; V0 Wave...

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