FIGURE P39.32 shows |ψ(x)|2for the electrons in an experiment.

a. Is the electron wave function normalized? Explain.

b. Draw a graph of ψ(x)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-cm-wide region at x=0.00cm? At x=0.50cm? At x=0.999cm?

d. If 104electrons are detected, how many are expected to land in the interval -0.30cmx0.30cm?

Short Answer

Expert verified

(a) Therefore, the wavefunction is normalized.

(b) A graph for Ψ'(s)has been made.

(c) the probability that an electron is detected in a 0.0010cmwide region at x=0.00cmand x=0.50cmand x=0.999cmis 0.00,0.0005and 0.00010respectively.

(d) 900electrons are expected to land in the interval.

Step by step solution

01

Part (a) Step 1: Given Information

02

Part (a) Step 2: solution

Formula Used:

|ψ(x)|2=-xfor-1<x<0|ψ(x)|2=xfor0<x<1

One possible graph can be made by plotting

ψ(x)=-xψ(x)=xfor-1<x<1

From the graph:

|ψ(x)|2=-xfor-1<x<0|ψ(x)|2=xfor0<x<1

We need to show that -*ψ(x)ψ*(x)dx=1

-11ψ(x)ψ*(x)dx-10-xdx+01xdx=-x22-10+x2201=0+(-1)22+(1)22=12+12=1

Conclusion: Therefore, the wavefunction is normalized.

03

Part (b) Step 1: Given Information 

04

Part (b) Step 2: solution

Formula Used:

|ψ(x)|2=-xfor-1<x<0|ψ(x)|2=xfor0<x<1

One possible graph can be made by plotting

ψ(x)=-xψ(x)=xfor-1<x<1

One possible graph can be made by plotting

localid="1650893249997" ψ(x)=-xψ(x)=xfor-1<x<1

Conclusion: A graph for has been made.

05

Part (c) Step 1: Given Information 

06

Part (c) Step 2: solution

Formula Used:

The probability function is given by:

P(x)=|ψ(x)|2δx

The probability that an electron is detected in a 0.0010cmwide region i.e. δx=0.0010cm

For x=0.00cm

P(x)=|ψ(0.00)|20.0010=0.00

For x=0.5cm

P(x)=|ψ(0.50)|20.0010=0.0005P(x)=|ψ(0.999)|20.0010=0.0009990.0010

Conclusion: The probability that an electron is detected in a 0.0010cmwide region at x=0.00cmand x=0.50cmand x=0.999cmis 0.00,0.0005and 0.00010respectively.

07

Part (d) Step 1: Given Information 

Total number of electrons=104

08

Part (d) Step 2: solution

Formula Used:

The probability function is given by:

P(x)=|ψ(x)|2

Calculation:

Probability to find an electron in the interval -0.30cm<x<0.30cmis:

-0.30-xdx+00.3xdx=0-0.3xdx+00.3xdx=x220-0.3+x2200.3=(-0.3)22+(0.3)22=0.09

Number of electrons that would land in the interval -0.30cm<x<0.30cmare:


role="math" localid="1650894290307" N=NoP(x)N=104×0.09=900electrons

Conclusion:900 electrons are expected to land in the interval -0.30cm<x<0.30cm.

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