Chapter 39: Q.38 (page 1139)

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

Short Answer

Expert verified

the normalization constant c is 0.707 mm^-1/2

prob= $= 2 c^{2} (- \frac{L}{2}) {[e^{- \frac{2 x}{L}}]}_{0}^{1.0 m m}$

Thus the probability of finding the particle within 1.0mm for the origin is 63.2%

Step by step solution

To find the probability of a particle

The expression for the probability of a particle at position x is given from probability density is as follows:

$p (x) = {|ψ (x)|}^{2}$

Here ${|ψ (x)|}^{2}$ is the probability density

A particle is described by the wave function as follows: $ψ (x) = \{\overset{C e^{\frac{x}{L}} X \leq 0 n m}{C e^{\frac{- x}{L}} X \geq 0 n m}\}$

Here L is 2.0nm

The probability density of the particle at $x \leq 0 n m$ is as follows:

${|ψ (x)|}^{2} = c^{2} e^{\frac{2 π}{L}}$

The probability density of the particle at $x \geq 0 n m$ is as follows:

${|ψ (x)|}^{2} = c^{2} e^{\frac{2 π}{L}}$

Table for the values of ψ and ψ2 is as follows:

(b)Determine the normalization constant c∫-∞+∞ψx2dx=12∫-∞+∞ψx2dx=12c3-L2e-2xl=1 Lc21-0=1 c=1Lsubstitute 2.0mm for Lc=12.0mm =0.707 mm-12Thus the normalization constant c is 0.707 mm-1/2

These are the constant

(c) Calculate the probability of finding the particle within 1.0mmof the origin.

$P r o b (- 1.0 m m \leq x \leq + 1.0 m m) = \int_{- 1.0 m m}^{+ 1.0 m m} {|ψ (x)|}^{2} d x = 2 \int_{- 1.0 m m}^{+ 1.0 m m} {|ψ (x)|}^{2} d x = 2 c^{2} (- \frac{L}{2}) {[e^{- \frac{2 x}{L}}]}_{0}^{1.0 m m}$

Simplify the above equation furtherProb-1.0mm≤x≤+1.0mm=Lc2e0-e2.0mm/Lsubstitute 2.0mm for L and 0.707 mm-1/2 for cprob-1.0mm≤x≤+1.0mm=2.0mm0.707mm-1/22e0-e-2.0mm 2.0mm =63.2%

Thus the probability of finding the particle within 1.0mm for the origin is 63.2%

(d)The following is the graph showing the shaded region representing the finding of the particle within 1.0 mm of the origin.

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Short Answer

Step by step solution

To find the probability of a particle

Table for the values of ψ and ψ2 is as follows:

(b)Determine the normalization constant c∫-∞+∞ψx2dx=12∫-∞+∞ψx2dx=12c3-L2e-2xl=1 Lc21-0=1 c=1Lsubstitute 2.0mm for Lc=12.0mm =0.707 mm-12Thus the normalization constant c is 0.707 mm-1/2

(c) Calculate the probability of finding the particle within 1.0mmof the origin.

Simplify the above equation furtherProb-1.0mm≤x≤+1.0mm=Lc2e0-e2.0mm/Lsubstitute 2.0mm for L and 0.707 mm-1/2 for cprob-1.0mm≤x≤+1.0mm=2.0mm0.707mm-1/22e0-e-2.0mm 2.0mm =63.2%

(d)The following is the graph showing the shaded region representing the finding of the particle within 1.0 mm of the origin.

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