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Ch 39: Wave Functions and Uncertainty
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 39: Wave Functions and UncertaintyProblem 38b
Chapter 39, Problem 38b

A particle is described by the wave function ψ(x)={cex/Lx≤0 mmce−x/Lx≥0 mm\(\psi\) (x)=\(\begin{cases}\) ce^{x/L} & x\(\leq\) 0\(\text{ mm}\) \\ ce^{-x/L} & x\(\geq\) 0\(\text{ mm}\) \(\end{cases}\) mm where L = 2.0 mm. Determine the normalization constant c.

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To normalize the wave function, we use the condition that the total probability of finding the particle over all space is equal to 1. Mathematically, this is expressed as: ∫|ψ(x)|² dx = 1, where the integral is taken over all space.
Substitute the given wave function into the normalization condition. Since the wave function is piecewise, split the integral into two parts: ∫|ψ(x)|² dx = ∫[c²e²ˣ/ᴸ] dx (for x ≤ 0) + ∫[c²e⁻²ˣ/ᴸ] dx (for x ≥ 0).
Evaluate each integral separately. For the first integral (x ≤ 0), calculate ∫[c²e²ˣ/ᴸ] dx from -∞ to 0. For the second integral (x ≥ 0), calculate ∫[c²e⁻²ˣ/ᴸ] dx from 0 to ∞. Use the standard integral formula for exponential functions: ∫eᵃˣ dx = (1/a)eᵃˣ + C, where a ≠ 0.
Combine the results of the two integrals and set the total equal to 1. This will give you an equation involving the normalization constant c. Solve for c by isolating it on one side of the equation.
Substitute the given value of L = 2.0 mm into the equation to express c in terms of numerical values. This will provide the normalized constant c, ensuring the wave function satisfies the normalization condition.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Wave Function

The wave function, denoted as ψ(x), is a mathematical description of the quantum state of a particle. It contains all the information about the system and is used to calculate probabilities of finding a particle in a given position. The wave function must be normalized, meaning the total probability of finding the particle in all space must equal one.
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Normalization

Normalization is the process of adjusting the wave function so that the integral of the absolute square of the wave function over all space equals one. This ensures that the total probability of finding the particle in any location is 100%. For a piecewise wave function, normalization involves integrating each segment and setting the sum equal to one to solve for the normalization constant.
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Integration

Integration is a fundamental mathematical operation used to calculate areas under curves, which in quantum mechanics is essential for determining probabilities. In the context of normalization, it involves integrating the square of the wave function over the entire space to find the normalization constant. The integration limits will depend on the defined regions of the wave function.
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Related Practice
Textbook Question

Consider the electron wave function ψ(x)={c1−x2∣x∣≤1 cm0∣x∣≥1 cm\(\psi\) (x)=\(\begin{cases}\) c\(\sqrt{1-x^{2}\)} & \(\left\)|x\(\right\)|\(\leq\) 1\(\text{ cm}\) \\ 0 & \(\left\)|x\(\right\)|\(\geq\) 1\(\text{ cm}\) \(\end{cases}\) where x is in cm. Draw a graph of |ψ(x)|2 over the interval −2 cm ≤ x ≤ 2 cm. Provide numerical scales.

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Textbook Question

A particle is described by the wave function ψ(x)={cex/Lx≤0 mmce−x/Lx≥0 mm\(\psi\) (x)=\(\begin{cases}\) ce^{x/L} & x\(\leq\) 0\(\text{ mm}\) \\ ce^{-x/L} & x\(\geq\) 0\(\text{ mm}\) \(\end{cases}\) where L = 2.0 mm. Sketch graphs of both the wave function and the probability density as functions of x.

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Textbook Question

Consider the electron wave function ψ(x)={c1−x2∣x∣≤1 cm0∣x∣≥1 cm\(\psi\) (x)=\(\begin{cases}\) c\(\sqrt{1-x^{2}\)} & \(\left\)|x\(\right\)|\(\leq\) 1\(\text{ cm}\) \\ 0 & \(\left\)|x\(\right\)|\(\geq\) 1\(\text{ cm}\) \(\end{cases}\) where x is in cm. Draw a graph of ψ(x) over the interval −2 cm ≤ x ≤ 2 cm. Provide numerical scales on both axes.

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Textbook Question

A particle is described by the wave function ψ(x)={cex/Lx≤0 mmce−x/Lx≥0 mm\(\psi\) (x)=\(\begin{cases}\) ce^{x/L} & x\(\leq\) 0\(\text{ mm}\) \\ ce^{-x/L} & x\(\geq\) 0\(\text{ mm}\) \(\end{cases}\) where L = 2.0 mm. Interpret your answer to part b by shading the region representing this probability on the appropriate graph in part a.

1579
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Textbook Question

Consider the electron wave function ψ(x)={c1−x2∣x∣≤1 cm0∣x∣≥1 cm\(\psi\) (x)=\(\begin{cases}\) c\(\sqrt{1-x^{2}\)} & \(\left\)|x\(\right\)|\(\leq\) 1\(\text{ cm}\) \\ 0 & \(\left\)|x\(\right\)|\(\geq\) 1\(\text{ cm}\) \(\end{cases}\) where x is in cm. If 104 electrons are detected, how many will be in the interval 0.00 cm ≤ x ≤ 0.50 cm?

1588
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Textbook Question

A particle is described by the wave function ψ(x)={cex/Lx≤0 mmce−x/Lx≥0 mm\(\psi\) (x)=\(\begin{cases}\) ce^{x/L} & x\(\leq\) 0\(\text{ mm}\) \\ ce^{-x/L} & x\(\geq\) 0\(\text{ mm}\) \(\end{cases}\) where L = 2.0 mm. Calculate the probability of finding the particle within 1.0 mm of the origin.

1579
views