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

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 14a

Chapter 39, Problem 14a

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

Verified step by step guidance

Understand the problem: The graph of |ψ(x)|² represents the probability density function of the electron's position. The value of 'a' is likely a parameter related to the spatial extent or characteristic length scale of the wavefunction. This could be inferred from the graph's features, such as its width or periodicity.

Analyze the graph: Examine the graph of |ψ(x)|² to identify key features such as the distance between peaks (if periodic), the width of the central peak, or any other characteristic length scale. These features are often related to 'a' in quantum mechanics problems.

Relate the graph to the wavefunction: Recall that |ψ(x)|² is the square of the wavefunction ψ(x). If the wavefunction is given by a specific form (e.g., a Gaussian or sinusoidal function), 'a' might represent a parameter such as the standard deviation of a Gaussian or the wavelength of a sinusoidal function.

Use the mathematical relationship: If the wavefunction is known or can be inferred, write down its mathematical form. For example, if ψ(x) = A * exp(-x² / (2a²)), then 'a' is the parameter that determines the width of the Gaussian. Alternatively, if ψ(x) = A * sin(πx / a), then 'a' is related to the wavelength or periodicity.

Determine 'a' from the graph: Measure the relevant feature from the graph (e.g., the full width at half maximum for a Gaussian or the distance between peaks for a sinusoidal function). Use this measurement in the mathematical relationship to solve for 'a'.

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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. 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.

Probability Density

Probability density is a measure used in quantum mechanics to determine the likelihood of finding a particle in a specific region of space. It is derived from the wave function, where |ψ(x)|^2 gives the probability per unit length for a one-dimensional system. Understanding probability density is crucial for interpreting the results of quantum experiments and predicting particle behavior.

Normalization Condition

The normalization condition is a requirement in quantum mechanics that ensures the total probability of finding a particle in all space equals one. Mathematically, this is expressed as the integral of |ψ(x)|^2 over all space being equal to one. This condition is essential for the physical interpretation of the wave function and ensures that the probabilities derived from it are meaningful.

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Related Practice

Textbook Question

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

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

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

When 5×10¹² photons pass through an experimental apparatus, 2.0×10⁹ land in a 0.10-mm-wide strip. What is the probability density at this point?

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 EX39.12 shows the probability density for an electron that has passed through an experimental apparatus. If 1.0×10⁶ electrons are used, what is the expected number that will land in a 0.010-mm-wide strip at 2.000 mm?