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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 36b
Chapter 39, Problem 36b

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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Understand the given wave function: The wave function ψ(x) is defined piecewise. For |x| ≤ 1 cm, ψ(x) = √c (1 − x²), and for |x| > 1 cm, ψ(x) = 0. This means the wave function is nonzero only in the interval −1 cm ≤ x ≤ 1 cm and is zero elsewhere.
Identify the domain for the graph: The problem asks for the graph of ψ(x) over the interval −2 cm ≤ x ≤ 2 cm. Since ψ(x) = 0 for |x| > 1 cm, the graph will show ψ(x) = 0 for x < −1 cm and x > 1 cm.
Plot the nonzero part of ψ(x): For −1 cm ≤ x ≤ 1 cm, calculate ψ(x) = √c (1 − x²). This is a parabolic function that opens downward because of the negative x² term. The maximum value of ψ(x) occurs at x = 0, where ψ(0) = √c. At x = ±1 cm, ψ(x) = 0.
Add numerical scales to the axes: The x-axis should range from −2 cm to 2 cm, and the y-axis should range from 0 to the maximum value of ψ(x), which is √c. Label the key points, such as ψ(0) = √c and ψ(±1) = 0.
Combine the pieces: Draw the graph by plotting the parabolic curve for −1 cm ≤ x ≤ 1 cm and horizontal lines at ψ(x) = 0 for x < −1 cm and x > 1 cm. Ensure the graph transitions smoothly at x = ±1 cm, where ψ(x) becomes zero.

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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), describes the quantum state of a particle, such as an electron, in a given position x. It contains all the information about the system and is used to calculate probabilities of finding the particle in various states. The square of the wave function's absolute value, |ψ(x)|², gives the probability density of the particle's position.
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Normalization

Normalization is a crucial concept in quantum mechanics that ensures the total probability of finding a particle in all possible positions equals one. For a wave function to be physically meaningful, it must be normalized over its entire domain. This involves integrating the square of the wave function over the specified range and adjusting the constant factor (c in this case) accordingly.
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Graphing Functions

Graphing functions involves plotting the values of a function on a coordinate system, where the x-axis represents the independent variable (position x) and the y-axis represents the dependent variable (the wave function ψ(x)). In this case, the graph will illustrate the behavior of the wave function within the specified interval, highlighting regions where the wave function is non-zero and its shape, which is essential for visualizing quantum states.
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Related Practice
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\(\text{ mm}\) \(\end{cases}\) mm where L = 2.0 mm. Determine the normalization constant c.

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.

Textbook Question

FIGURE P39.31 shows the wave function of a particle confined between x = 0 nm and x = 1.0 nm. The wave function is zero outside this region. Calculate the probability of finding the particle in the interval 0 nm ≤ x ≤ 0.25 nm.

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.

Textbook Question

FIGURE P39.31 shows the wave function of a particle confined between x = 0 nm and x = 1.0 nm. The wave function is zero outside this region. Draw a graph of the probability density P(x)=|ψ(x)|2

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. Determine the normalization constant c.