PHY4604 R. D. Field Department of Physics Chapter2_4.doc University of Florida The Infinite Square Well (2) Normalized Wavefunctions: Now we require that 1)()(=∫∞∞−∗dxxxnnψψ, and hence 12sin)/(sin2022022===∫∫LAdnLAdxLxnAnLπθθππ. Thus ,LA2/1=and h/)(),(tiEnnnextx−=Ψψwith )/sin(2)(LxnLxnπψ=. Probability Density:The probability density for the nthstate is )/(sin2|),(|)(22LxnLtxxnnπρ=Ψ=.Note that ρnand not a function of time! These states are “stationary states”(i.e.independent of time). Average Value of x:The average position for the nthstate is 242)(sin2)/(sin2)()(2222022202*LnnLdnLdxLxnxLdxxxxxnLn====ΨΨ=><∫∫∫∞∞−ππθθθπππAverage Value of x2:The average value of x2in the nthstate is
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