7.4 The Quantum Particle in a Box

(a) Consider an infinite square well with wall boundaries $x = 0$ and $x = L$. What is the probability of finding a quantum particle in its ground state somewhere between $x = 0$ and $x = L / 4$? (b) Repeat question (a) for a classical particle.

Is it possible that when we measure the energy of a quantum particle in a box, the measurement may return a smaller value than the ground state energy? What is the highest value of the energy that we can measure for this particle?

Assume that an electron in an atom can be treated as if it were confined to a box of width $2 .\text{0}\textrm{ }\mathring{A}$. What is the ground state energy of the electron? Compare your result to the ground state kinetic energy of the hydrogen atom in the Bohr’s model of the hydrogen atom.

An electron confined to a box has the ground state energy of 2.5 eV. What is the width of the box?

What is the ground state energy (in eV) of an $\alpha$-particle confined to a one-dimensional box the size of the uranium nucleus that has a radius of approximately 15.0 fm?

An electron confined to a box of width 0.15 nm by infinite potential energy barriers emits a photon when it makes a transition from the first excited state to the ground state. Find the wavelength of the emitted photon.

Suppose an electron confined to a box emits photons. The longest wavelength that is registered is 500.0 nm. What is the width of the box?