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

Knight Calc 5th Edition

Ch 39: Wave Functions and Uncertainty

Problem 31d

Chapter 39, Problem 31d

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.

Verified step by step guidance

Step 1: Understand the problem. The wave function ψ(x) represents the probability amplitude of a particle confined between x = 0 nm and x = 1.0 nm. The probability of finding the particle in a specific interval is calculated using the integral of the square of the wave function over that interval.

Step 2: Analyze the graph. The wave function ψ(x) is a triangular function that starts at ψ(0) = 0, increases linearly to a maximum value of c at x = 0.5 nm, and then decreases linearly back to ψ(1) = 0. The function is zero outside the region 0 ≤ x ≤ 1 nm.

Step 3: Write the expression for the probability. The probability of finding the particle in the interval 0 nm ≤ x ≤ 0.25 nm is given by: P = ∫[0 to 0.25] |ψ(x)|² dx. Since ψ(x) is linear in this region, its equation can be determined using the slope-intercept form of a line.

Step 4: Determine the equation of ψ(x) for 0 ≤ x ≤ 0.5 nm. The slope of the line is m = c / 0.5 = 2c. Thus, ψ(x) = 2cx for 0 ≤ x ≤ 0.5 nm. Substitute this into the probability expression: P = ∫[0 to 0.25] (2cx)² dx.

Step 5: Simplify the integral. Expand (2cx)² to get 4c²x². The integral becomes P = ∫[0 to 0.25] 4c²x² dx. Solve this integral by applying the power rule for integration: ∫x² dx = (x³ / 3). After integrating, evaluate the result at the limits x = 0 and x = 0.25.

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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 in a given region. It contains all the information about the particle's position and momentum. The square of the wave function's absolute value, |ψ(x)|², gives the probability density of finding the particle at a specific position within the defined boundaries.

Probability Density

Probability density is a measure derived from the wave function that indicates the likelihood of finding a particle in a particular region of space. For a one-dimensional case, it is calculated as |ψ(x)|². To find the probability of locating the particle within a specific interval, one must integrate the probability density over that interval.

Normalization of the Wave Function

Normalization ensures that the total probability of finding the particle within the entire space is equal to one. This is achieved by integrating the probability density over the entire range of the wave function and setting the result equal to one. A properly normalized wave function is essential for accurate probability calculations in quantum mechanics.

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Related Practice

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)|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. Determine the value of the constant c, as defined in the figure.

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

An experiment finds electrons to be uniformly distributed over the interval 0 cm ≤ x ≤ 2 cm, with no electrons falling outside this interval. What is the probability density at x = 0.80 cm?

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