Find an expectation value

The classical motion for an oscillator that starts from rest at location x₀ is

                                           x(t) = x₀ cos(ωt)

The probability that the particle is at a particular x at a particular time t

is given by ρ(x, t) = δ(x − x(t)), and we can perform the temporal average

to get the spatial density. Our natural time scale for the averaging is a half

cycle, take t = 0 → π/ ω

Thus,

ρ =  

Limit is 0 to π/ω

We perform the change of variables to allow access to the δ, let y = x₀ cos(ωt)  so that

ρ(x) =

Limit is x₀ to -x₀

Limit is -x₀ to x₀

This has as expected. Here the limit is -x₀ to x₀

The expectation value is 0 when the ρ(x) is symmetric, x ρ(x) is asymmetric and the limits of integration are asymmetric.

answer:

an example of solubility is salt or sugar dissolving in water.

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answer:

50 vo; ts

explanation: