Chapter 39: Wave Functions and Uncertainty

FIGURE P39.32 shows $\left|\psi \right(x){|}^2$for the electrons in an experiment.

a. Is the electron wave function normalized? Explain.

b. Draw a graph of $\psi \left(x\right)$over this same interval. Provide a numerical scale on both axes. (There may be more than one acceptable answer.)

c. What is the probability that an electron will be detected in a $0.0010-\mathrm{cm}$-wide region at $x=0.00\mathrm{cm}$? At $x=0.50\mathrm{cm}$? At $x=0.999cm$?

d. If ${10}^4$electrons are detected, how many are expected to land in the interval $-0.30\mathrm{cm}\le x\le 0.30\mathrm{cm}$?

FIGURE P39.32 shows $\left|\psi \right(x){|}^2$for the electrons in an experiment.

a. Is the electron wave function normalized? Explain.

b. Draw a graph of $\psi \left(x\right)$over this same interval. Provide a numerical scale on both axes. (There may be more than one acceptable answer.)

c. What is the probability that an electron will be detected in a $0.0010-\mathrm{cm}$-wide region at$x=0.00\mathrm{cm}$? At $x=0.50\mathrm{cm}$? At $x=0.999\mathrm{cm}?$

d. If ${10}^4$electrons are detected, how many are expected to land in the interval $-0.30\mathrm{cm}\le x\le 0.30\mathrm{cm}$?

shows the probability density for finding a particle at position x. a. Determine the value of the constant a, as defined in the figure. b. At what value of x are you most likely to find the particle? Explain. c. Within what range of positions centered on your answer to part b are you 75% certain of finding the particle? d. Interpret your answer to part c by drawing the probability density graph and shading the appropriate region.

The probability density for finding a particle at position x is P1x2 = • a 11 - x2 -1 mm … x 6 0 mm b11 - x2 0 mm … x … 1 mm and zero elsewhere. a. You will learn in Chapter 40 that the wave function must be a continuous function. Assuming that to be the case, what can you conclude about the relationship between a and b? b. Determine values for a and b. c. Draw a graph of the probability density over the interval -2 mm … x … 2 mm. d. What is the probability that the particle will be found to the left of the origin?

An electron that is confined to x Ú 0 nm has the normalized wave function c1x2 = b 0 x 6 0 nm 11.414 nm-1/2 2e-x/11.0 nm2 x Ú 0 nm where x is in nm. a. What is the probability of finding the electron in a 0.010-nmwide region at x = 1.0 nm? b. What is the probability of finding the electron in the interval 0.50 nm … x … 1.50 nm?

Consider the electron wave function

$\psi \left(X\right)=\left\{\begin{array}{l}0x<0nm\\ \left(1.414n{m}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)e^{-\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$\left(1.0nm\right)$}\right.}x\ge 0nm\end{array}\right.$

where x is in cm.

a. Determine the normalization constant c.

b. Draw a graph of c1x2 over the interval -2 cm $\le$x $\le$2 cm. Provide numerical scales on both axes.

c. Draw a graph of 0 c1x2 0 2 over the interval -2 cm $\le$x $\le$2 cm. Provide numerical scales.

d. If 104 electrons are detected, how many will be in the interval 0.00 cm $\le$x $\le$0.50 cm?

Consider the electron wave function

$\psi \left(X\right)=\left\{\begin{array}{l}c\sin \left(\frac{2\mathrm{\pi x}}{L}\right)0\le x\le L\\ 0x<0orx>L\end{array}\right.$

a. Determine the normalization constant c. Your answer will be in terms of L.

b. Draw a graph of $\psi \left(x\right)$over the interval -L$\le$x $\le$2L.

c. Draw a graph of ${\left|\psi \left(x\right)\right|}^2$over the interval -L $\le$x $\le$2L. d. What is the probability that an electron is in the interval 0 $\le$x $\le$L/3?

A particle is described by the wave function c1x2 = b cex/L x … 0 mm ce-x/L x Ú 0 mm where L = 2.0 mm.

a. Sketch graphs of both the wave function and the probability density as functions of x.

b. Determine the normalization constant c.

c. Calculate the probability of finding the particle within 1.0 mm of the origin. d. Interpret your answer to part b by shading the region representing this probability on the appropriate graph in part a

The probability density for finding a particle at position $x$is

$p\left(x\right)=\left\{\begin{array}{l}\frac{a}{\left(1-x\right)}\\ b\left(1-x\right)\end{array}\right.\frac{-1mm\le x<0mm}{0mm\le x\le 1mm}$

and zero elsewhere

You are dealt 1 card each from 1000 decks of cards. What is the expected number of picture cards (jacks, queens, and kings)?