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.

Ch 39: Wave Functions and Uncertainty

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 39: Wave Functions and Uncertainty

Problem 36c

Chapter 39, Problem 36c

Consider the electron wave function $ψ (x) = {\begin{matrix} c \sqrt{1 - x^{2}} & ∣ x ∣ \leq 1 cm \\ 0 & ∣ x ∣ \geq 1 cm \end{matrix}$ where x is in cm. Draw a graph of |ψ(x)|² over the interval −2 cm ≤ x ≤ 2 cm. Provide numerical scales.

Verified step by step guidance

Understand the problem: 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. The task is to graph |ψ(x)|², which represents the probability density, over the interval −2 cm ≤ x ≤ 2 cm.

Step 1: Recall that |ψ(x)|² is the square of the magnitude of the wave function. For |x| ≤ 1 cm, |ψ(x)|² = (√c(1 − x²))² = c(1 − x²)². For |x| > 1 cm, |ψ(x)|² = 0 because ψ(x) = 0 in this region.

Step 2: Identify the domain of the function. The non-zero part of |ψ(x)|² exists only for |x| ≤ 1 cm, i.e., −1 cm ≤ x ≤ 1 cm. Outside this range (|x| > 1 cm), |ψ(x)|² = 0.

Step 3: Plot the function |ψ(x)|² = c(1 − x²)² for −1 cm ≤ x ≤ 1 cm. This is a symmetric function about x = 0 because it depends on x². The value of |ψ(x)|² is maximum at x = 0 (where |ψ(x)|² = c) and decreases to 0 at x = ±1 cm.

Step 4: Extend the graph to the interval −2 cm ≤ x ≤ 2 cm. For −2 cm ≤ x < −1 cm and 1 cm < x ≤ 2 cm, |ψ(x)|² = 0. Label the x-axis with a scale from −2 cm to 2 cm and the y-axis with a scale that includes the maximum value of |ψ(x)|², which is c.

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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 a complex-valued function. The square of the absolute value of the wave function, |ψ(x)|^2, represents the probability density of finding the particle at position x.

Probability Density

Probability density is a measure that describes the likelihood of finding a particle in a specific region of space. For a wave function ψ(x), the probability density is given by |ψ(x)|^2. In this context, it indicates how the electron's presence is distributed across the defined interval, which is crucial for understanding its behavior in quantum mechanics.

Graphing Functions

Graphing functions involves plotting the values of a function on a coordinate system to visualize its behavior. In this case, we will graph |ψ(x)|^2 over the interval from -2 cm to 2 cm, which helps illustrate the regions where the electron is most likely to be found. Proper numerical scales on the axes are essential for accurately interpreting the graph.

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Related Practice

Textbook Question

A particle is described by the wave function $ψ (x) = {\begin{matrix} c e^{x / L} & x \leq 0 mm \\ c e^{- x / L} & x \geq 0 mm \end{matrix}$ mm where L = 2.0 mm. Determine the normalization constant c.

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 $ψ (x) = {\begin{matrix} c e^{x / L} & x \leq 0 mm \\ c e^{- x / L} & x \geq 0 mm \end{matrix}$ where L = 2.0 mm. Sketch graphs of both the wave function and the probability density as functions of x.

Textbook Question

Consider the electron wave function $ψ (x) = {\begin{matrix} c \sqrt{1 - x^{2}} & ∣ x ∣ \leq 1 cm \\ 0 & ∣ x ∣ \geq 1 cm \end{matrix}$ 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.

Textbook Question

Consider the electron wave function $ψ (x) = {\begin{matrix} c \sqrt{1 - x^{2}} & ∣ x ∣ \leq 1 cm \\ 0 & ∣ x ∣ \geq 1 cm \end{matrix}$ where x is in cm. Determine the normalization constant c.

Textbook Question

A particle is described by the wave function $ψ (x) = {\begin{matrix} c e^{x / L} & x \leq 0 mm \\ c e^{- x / L} & x \geq 0 mm \end{matrix}$ where L = 2.0 mm. Calculate the probability of finding the particle within 1.0 mm of the origin.

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