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. Interpret your answer to part b by shading the region representing this probability on the appropriate graph in part 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 38d

Chapter 39, Problem 38d

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. Interpret your answer to part b by shading the region representing this probability on the appropriate graph in part a.

Verified step by step guidance

Step 1: Begin by understanding the wave function ψ(x) provided in the problem. It is a piecewise function defined differently for x ≤ 0 and x ≥ 0. The constants c and L are given, where L = 2.0 mm.

Step 2: Recall that the probability density function is proportional to the square of the wave function, |ψ(x)|². To interpret the probability, calculate |ψ(x)|² for both regions (x ≤ 0 and x ≥ 0). For x ≤ 0, |ψ(x)|² = |c²e²ˣ/ᴸ|, and for x ≥ 0, |ψ(x)|² = |c²e⁻²ˣ/ᴸ|.

Step 3: Normalize the wave function to ensure the total probability over all space equals 1. This involves integrating |ψ(x)|² over all x (from -∞ to ∞) and solving for the constant c. The integral will be split into two parts: one for x ≤ 0 and one for x ≥ 0.

Step 4: Once the wave function is normalized, calculate the probability for specific regions of interest by integrating |ψ(x)|² over those regions. For example, if you want the probability for x ≤ 0, integrate |ψ(x)|² from -∞ to 0. Similarly, for x ≥ 0, integrate from 0 to ∞.

Step 5: Shade the region on the graph corresponding to the calculated probabilities. The graph should represent the probability density function |ψ(x)|², and the shaded regions should visually indicate the areas where the particle is most likely to be found based on the calculated probabilities.

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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), is a mathematical description of the quantum state of a particle. 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, which is crucial for interpreting quantum behavior.

Probability Density

Probability density is derived from the wave function and represents the likelihood of finding a particle in a given region of space. For a one-dimensional wave function, the probability density is calculated as |ψ(x)|². This concept is essential for visualizing the distribution of a particle's position and understanding how quantum mechanics differs from classical mechanics.

Graphical Representation

Graphical representation in quantum mechanics often involves plotting the wave function and its associated probability density. In this context, shading the region under the probability density curve on a graph helps visualize the likelihood of finding the particle in specific areas. This visual aid is important for interpreting results and communicating findings in quantum physics.

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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.

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Textbook Question

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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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