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. Sketch graphs of both the wave function and the probability density as functions of x.

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

문제 38a

39장, 문제 38a

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.

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Understand the given wave function: The wave function ψ(x) is piecewise-defined, with an exponential form that depends on whether x is less than or greater than 0. The constant c is a normalization constant, and L is given as 2.0 mm. The wave function is ψ(x) = c * e^(x/L) for x ≤ 0 and ψ(x) = c * e^(-x/L) for x ≥ 0.

Normalize the wave function: To ensure the total probability is 1, integrate the square of the wave function over all space and solve for c. The normalization condition is ∫|ψ(x)|² dx = 1, which becomes ∫[c² * e^(2x/L)] dx from -∞ to 0 + ∫[c² * e^(-2x/L)] dx from 0 to ∞ = 1.

Calculate the probability density: The probability density is given by |ψ(x)|². For x ≤ 0, |ψ(x)|² = c² * e^(2x/L), and for x ≥ 0, |ψ(x)|² = c² * e^(-2x/L). These expressions describe how the probability of finding the particle varies with position x.

Sketch the wave function ψ(x): Plot ψ(x) as a function of x. For x ≤ 0, the wave function increases exponentially as x approaches 0, and for x ≥ 0, it decreases exponentially as x moves away from 0. The graph should be continuous at x = 0.

Sketch the probability density |ψ(x)|²: Plot |ψ(x)|² as a function of x. For x ≤ 0, the probability density decreases exponentially as x becomes more negative, and for x ≥ 0, it decreases exponentially as x becomes more positive. The graph is symmetric about x = 0 and peaks at x = 0.

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주요 개념

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Wave Function

The wave function, denoted as ψ(x), is a fundamental concept in quantum mechanics that describes 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 x.

Probability Density

Probability density is derived from the wave function and represents the likelihood of locating a particle within a given region of space. For a one-dimensional wave function, the probability density is calculated as |ψ(x)|². This concept is crucial for interpreting quantum mechanics, as it allows us to predict where a particle is likely to be found upon measurement.

Exponential Decay

Exponential decay is a mathematical function that describes how a quantity decreases over time or space at a rate proportional to its current value. In the context of the given wave function, the terms eˣ/ᴸ and e−ˣ/ᴸ illustrate how the wave function behaves differently in the regions x ≤ 0 mm and x ≥ 0 mm, leading to distinct probability densities that reflect the particle's confinement and behavior.