Expectation Values- Particle in a Box Most Probable Position Ψ= 2 a 1/2 sin n π x a most probable = maximum in Ψ for n=1 = x mp = 0.5 a Average Position: probability of finding electron from [x to x+dx] = Ψ 2 dx ( for real wavefunctions ) "average value" of o ^ = ∫-∞ ∞ o ^ Ψ 2 dx with the integral "over all space." and the average is called the expectation value, <o ^ > for average position: o ^ = x <x> = ∫0 a x Ψ 2 dx = = 2 a ⌡ ⌠0 a x sin n π x a 2 dx for particle in the box. let y= n π x a then dy dx = n π a and rearranging gives x= a n π y and dx = a n π dy and when x = 0 to a, then y = 0 to n π . Substituting to change variables gives:
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