3: Trapped Particles

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simons.hec.utah.edu/TheoryPag...&Solutions.pdf

Q3.1

A particle of mass \(m\) moves in a one-dimensional box of length \(L\), with boundaries at \(x = 0\) and \(x = L\). Thus, \(V(x) = 0\) for \(0 ≤ x ≤ L\), and \(V(x) = ∞\) elsewhere. The normalized eigenfunctions of the Hamiltonian for this system are given by

\[Ψ_{n} (x) = \sqrt{\dfrac{2}{L}} \sin \left(\dfrac{n\pi x}{L} \right)\]

with

\[E_n = \dfrac{n^2 π^2 \hbar^2}{ 2mL^2}\]

where the quantum number \(n\) can take on the values \(n=1,2,3,....\)

Assuming that the particle is in an eigenstate, \(Ψ_n (x)\), calculate the probability that the particle is found somewhere in the region \(0 ≤ x ≤ L/4\). Show how this probability depends on \(n\).
For what value of \(n\) is there the largest probability of finding the particle in \(0 ≤ x ≤ L/4\) ?

Q3.3

A particle is confined to a one-dimensional box of length \(L\) having infinitely high walls and is in its lowest quantum state. Calculate \(\langle x \rangle\), \(\langle x^2 \rangle\), \(\langle p \rangle\), and \(\langle p^2 \rangle\).

Using the definition of the uncertainty \(\sigma_Α\) of the A measurement

\[\sigma_Α = \sqrt{\langle x^2 \rangle − \langle A \rangle ^2}\]

to verify the Heisenberg uncertainty principle.

Q3.4

It has been claimed that as the quantum number \(n\) increases, the motion of a particle in a box becomes more classical. In this problem you will have an opportunity to convince yourself of this fact:

For a particle of mass \(m\) moving in a one-dimensional box of length \(L\), with ends of the box located at \(x = 0\) and \(x = L\), the classical probability density can be shown to be independent of \(x\) and given by \(P(x) =1 /L\) regardless of the energy of the particle. Using this probability density, evaluate the probability that the particle will be found within the interval from \(x = 0\) to \(x =L/4\).
Now consider the quantum mechanical particle-in-a-box system. Evaluate the probability of finding the particle in the interval from \(x = 0\) to \(x =L/4\) for the system in its n^th quantum state.
Take the limit of the result you obtained in part b as \(n→∞\). How does your result compare to the classical result you obtained in part a?