A particle is described by the wave function where L = 2.0 mm. Sketch graphs of both the wave function and the probability density as functions of x.

Verified step by step guidance

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.

Key Concepts

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.