Clear@n, m, aDParticle in a 1-D box (con't)Classical vs. Quantum mechanics1. EnergyC: constant, all kineticQ: constant, all kinetic, h2ÅÅÅÅÅÅÅÑ2. Energy continuum?C: YesQ: No, energy is quantized , h2 n2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ8ma2En+1- En= E1@Hn + 1L2- n2D = E1@2 n + 1DEn+1- EnÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅEn=E1@2 n +1DÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅE1n2=2ÅÅÅÅn+1ÅÅÅÅÅÅÅn2= 0 as n grows3. MomentumC: constant, = è!!!!!!!!!!!!2 mEQ: average = 0sinHn p xÅÅÅÅÅÅÅÅÅÅÅaL = eikx- e-ikxÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ2 i where k =n pÅÅÅÅÅÅÅÅa= combination of p` eigenfunctionssingle measurement gives +/- hkÅÅÅÅÅÅÅÅ2 p= +/- hnÅÅÅÅÅÅÅÅ2 a = è!!!!!!!!!!!!2 mE4. PositionC: average = center of boxprobability of being in region of length Da is Da/a (does not change with E)Q: average = center of boxprobability of being in region of length Da oscillates with n, but converges on Da/a as n gets largeprobability(x) varies with x and depends on nB Correspondence.nb 1Bohr's Correspondence PrincipleQuantum properties must gradually switch over to classical properties as n gets largeWe see this obeyed in- energy continuum- relative probabilities of particle being in region of length DaB Correspondence.nb
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