Chapter 40: One-Dimensional Quantum Mechanics

A neutron is confined in a $10\mathrm{fm}$-diameter nucleus. If the nucleus is modeled as a one-dimensional rigid box, what is the probability that a neutron in the ground state is less than $2.0\mathrm{fm}$ from the edge of the nucleus?

For the quantum-well laser of Figure 40.16, estimate the probability that an electron will be found within one of the GaAlAs layers rather than in the GA As layer. Explain your reasoning.

For a particle in a finite potential well of width L and depth${\mathrm{U}}_0$ , what is the ratio of the probability Prob ( in $\mathrm{\delta x}$at $\mathrm{x}=\mathrm{L}+\mathrm{\eta }$) to the probability Prob ( in $\mathrm{\delta x}$ at$\mathrm{x}=\mathrm{L}$)?

A typical electron in a piece of metallic sodium has energy$-{\mathrm{E}}_0$compared to a free electron, where ${\mathrm{E}}_0$is the$2.7\mathrm{eV}$ work function of sodium.

a. At what distance beyond the surface of the metal is the electron’s probability density $10\%$of its value at the surface?

b. How does this distance compare to the size of an atom?

Show that the constant b used in the quantum-harmonic-oscillator wave functions (a) has units of length and (b) is the classical turning point of an oscillator in the $\mathrm{n}=1$ground state.

a. Determine the normalization constant ${\mathrm{A}}_1$for the $\mathrm{n}=1$ground-state wave function of the quantum harmonic oscillator. Your answer will be in terms of b.

b. Write an expression for the probability that a quantum harmonic oscillator in its $\mathrm{n}=1$ground state will be found in the classically forbidden region.

c. (Optional) Use a numerical integration program to evaluate your probability expression of part b.

What is the quantum number of the particle in FIGURE Q$40.4$? How can you tell?

| FIGURE EX$40.4$ shows the wave function of an electron in a rigid box. The electron energy islocalid="1650137157775" $12.0eV$. What is the energy, in localid="1650137162096" $eV$, of the next higher state?

a. Derive an expression for the classical probability density $P_{class}\left(y\right)$ for a ball that bounces between the ground and height$h$. The collisions with the ground are perfectly elastic.

b. Graph your expression between $y=0andy=h$.

c. Interpret your graph. Why is it shaped as it is?

A particle of mass m has the wave function$\psi \left(x\right)=Axexp\left(\raisebox{1ex}{$-x^2$}\!\left/ \!\raisebox{-1ex}{$a^2$}\right.\right)$ when it is in an allowed energy level with $E=0$.

a. Draw a graph of $\psi \left(x\right)$versus$x$.

b. At what value or values of $x$is the particle most likely to be found?

c. Find and graph the potential-energy function $U\left(x\right)$.