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17 Nov 2019
It has been claimed that as the quantum number n increases, the motion of a particle in a 1-dimensional box becomes more classical. In this problem you will have the opportunity to convince yourself of this fact. For a particle of mass m moving in a 1-D box of length L, with ends at x = 0 and x = L, the classical probability density function can be shown to be independent of x. Formally, it is given by P(x)dx = dx/L regardless of the energy of the particle. Evaluate the probability that the classical particle will be found within the interval x = 0 to x = L/4. Now consider the quantum mechanical system. Evaluate the probability of finding the particle in the interval x = 0 to x = L/4 for the system in its n^th quantum state. For what value of n is there the largest probability of finding the particle between x = 0 and x = L/4? Justify this conclusion mathematically. Take the limit obtained in (b) as n rightarrow infinity. How does this compare to the classical result in (a)?
It has been claimed that as the quantum number n increases, the motion of a particle in a 1-dimensional box becomes more classical. In this problem you will have the opportunity to convince yourself of this fact. For a particle of mass m moving in a 1-D box of length L, with ends at x = 0 and x = L, the classical probability density function can be shown to be independent of x. Formally, it is given by P(x)dx = dx/L regardless of the energy of the particle. Evaluate the probability that the classical particle will be found within the interval x = 0 to x = L/4. Now consider the quantum mechanical system. Evaluate the probability of finding the particle in the interval x = 0 to x = L/4 for the system in its n^th quantum state. For what value of n is there the largest probability of finding the particle between x = 0 and x = L/4? Justify this conclusion mathematically. Take the limit obtained in (b) as n rightarrow infinity. How does this compare to the classical result in (a)?
Nestor RutherfordLv2
9 Apr 2019