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Chapter 39: Q5. (page 1136)

What is the value of the constant a in FIGURE Q39.5?

Short Answer

Expert verified

a = 2mm^-1

Step by step solution

01

Given parameters

A graph of probability density of a particle.

the graph forms a triangle with base length of 1mm (from 1mm to 2mm)

and height of a.

02

Finding a

Now we know that probability of finding the particle from $- 鈭� t o + 鈭�$ is 1.

And here all the probability density is distributed from x= 1mm to x = 2mm.

So probability of finding the particle between 1mm to 2mm is 1.

this means integral of this wave function from 1mm to 2mm will give 1.

And integral of a function also tells about area under the graph.

$鈬� a r e a o f t h i s g r a p h = 1$

$鈬� a r e a o f t h e t r i a n g l e = 1 鈬� \frac{1}{2} 脳 1 m m 脳 a = 1 鈬� a = 2 m m^{- 1}$

Hence, the value of a is 2mm^-1

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Most popular questions from this chapter

The probability density for finding a particle at position x is P1x2 = 鈥� a 11 - x2 -1 mm 鈥� x 6 0 mm b11 - x2 0 mm 鈥� x 鈥� 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 鈥� x 鈥� 2 mm. d. What is the probability that the particle will be found to the left of the origin?

An experiment has four possible outcomes, labeled A to D. The probability of A is P_A= 40% and of B is P_B = 30%. Outcome C is twice as probable as outcome D. What are the probabilities P_Cand P_D?

Soot particles, from incomplete combustion in diesel engines, are typically $15 nm$ in diameter and have a density of $1200 kg / m^{3}$ . FIGURE P39.45 shows soot particles released from rest, in vacuum, just above a thin plate with a $0.50 - 渭 m$ -diameter holeroughly the wavelength of visible light. After passing through the hole, the particles fall distance $d$ and land on a detector. If soot particles were purely classical, they would fall straight down and, ideally, all land in a $0.50 - 渭 m$ -diameter circle. Allowing for some experimental imperfections, any quantum effects would be noticeable if the circle diameter were $2000 nm$ . How far would the particles have to fall to fill a circle of this diameter?

The probability density for an electron that has passed through an experimental apparatus. If $1.0 脳 10^{6}$ electrons are used, what is the expected number that will land in alocalid="1649312933417" $0.010 m m$ wide strip atlocalid="1649312941736" $(a) x = 0.000 m m$ and localid="1649312949893" $(b) 2.000 m m$ ?

Consider the electron wave function

$蠄 (X) = \{\begin{cases} 0 x < 0 n m \\ (1.414 n m^{- \frac{1}{2}}) e^{- \frac{x}{(1.0 n m)}} x 鈮� 0 n m \end{cases}$

where x is in cm.

a. Determine the normalization constant c.

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

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

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

See all solutions

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