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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 16b
Chapter 39, Problem 16b

FIGURE EX39.16 shows the wave function of an electron. Draw a graph of |ψ(x)|2.

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Step 1: Understand the problem. The wave function ψ(x) represents the quantum state of an electron, and |ψ(x)|^2 is the probability density function. This means that |ψ(x)|^2 gives the probability of finding the electron at a specific position x. To draw the graph of |ψ(x)|^2, we need to square the magnitude of ψ(x) at each point along the x-axis.
Step 2: Analyze the given wave function ψ(x). Look at the graph provided in FIGURE EX39.16 and identify the key features of ψ(x), such as its amplitude, nodes (points where ψ(x) = 0), and regions of positive and negative values. These features will influence the shape of |ψ(x)|^2.
Step 3: Square the wave function. For each value of x, calculate |ψ(x)|^2 = ψ(x) × ψ*(x), where ψ*(x) is the complex conjugate of ψ(x). If ψ(x) is purely real, this simplifies to |ψ(x)|^2 = ψ(x)^2. Note that squaring the wave function removes any negative values, so the graph of |ψ(x)|^2 will only have positive values.
Step 4: Plot the graph. Using the squared values from Step 3, plot |ψ(x)|^2 as a function of x. Pay attention to the nodes (where ψ(x) = 0), as |ψ(x)|^2 will also be zero at these points. The peaks in the graph of |ψ(x)|^2 will correspond to the regions where ψ(x) has the highest amplitude.
Step 5: Label the graph. Clearly label the axes of the graph, with the x-axis representing position (x) and the y-axis representing |ψ(x)|^2 (probability density). Include any important features, such as nodes or regions of high probability density, to make the graph informative and accurate.

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Key Concepts

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

Wave Function (ψ)

The wave function, denoted as ψ, is a fundamental concept in quantum mechanics that describes the quantum state of a particle, such as an electron. It contains all the information about the system and is a complex-valued function of position and time. The square of the absolute value of the wave function, |ψ(x)|^2, represents the probability density of finding the particle at a given position x.
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Probability Density

Probability density is a measure that describes the likelihood of finding a particle in a specific region of space. In quantum mechanics, it is calculated as the square of the wave function's absolute value, |ψ(x)|^2. This concept is crucial for interpreting the wave function, as it provides a statistical interpretation of the particle's position, allowing us to visualize where the electron is most likely to be found.
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Graphing Functions

Graphing functions involves plotting the values of a function on a coordinate system to visualize its behavior. In the context of |ψ(x)|^2, this means creating a graph where the x-axis represents position and the y-axis represents the probability density. Understanding how to graph functions is essential for analyzing quantum states and interpreting the spatial distribution of particles, such as electrons.
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Related Practice
Textbook Question

FIGURE EX39.14 is a graph of |ψ(x)|2 for an electron. What is the probability that the electron is located between x = 1.0 nm and x = 2.0 nm?

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Textbook Question

What minimum bandwidth is needed to transmit a pulse that consists of 100 cycles of a 1.0 MHz oscillation?

Textbook Question

FIGURE EX39.13 shows the probability density for an electron that has passed through an experimental apparatus. What is the probability that the electron will land in a 0.010-mm-wide strip at x = 0.000 mm?

Textbook Question

A 1.5-μm-wavelength laser pulse is transmitted through a 2.0-GHz-bandwidth optical fiber. How many oscillations are in the shortest-duration laser pulse that can travel through the fiber?

Textbook Question

FIGURE EX39.14 is a graph of |ψ(x)|2 for an electron. What is the value of a?

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Textbook Question

What is the minimum uncertainty in position, in nm, of an electron whose velocity is known to be between 3×105 m/s and 4 ×105 m/s? Give your answer to one significant figure.