# 3: Trapped Particles

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 $$n=1,2,3,....$$

1. 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 $$n$$.
2. 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:

1. 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 $$x =L/4$$.
2. 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 nth quantum state.
3. 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?