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. Calculate the probability of finding the particle within 1.0 mm of the origin.

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 38c

Chapter 39, Problem 38c

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.

Verified step by step guidance

Step 1: Understand the problem. The wave function ψ(x) describes the probability amplitude of the particle's position. To find the probability of the particle being within 1.0 mm of the origin, we need to integrate the square of the wave function |ψ(x)|² over the interval from -1.0 mm to 1.0 mm.

Step 2: Write the expression for the probability. The probability P is given by the integral:

P_{(- 1.0, 1.0)} = \int_{- 1.0}^{1.0} | ψ (x) | ²^{d} x

Step 3: Substitute the given wave function into the integral. The wave function is piecewise defined, so split the integral into two parts: one for x ≤ 0 and one for x ≥ 0. For x ≤ 0, ψ(x) = ceˣ/ᴸ, and for x ≥ 0, ψ(x) = ce−ˣ/ᴸ.

Step 4: Perform the integration for each piece. For x ≤ 0, integrate |ψ(x)|² = c²e²ˣ/ᴸ over the interval [-1.0 mm, 0]. For x ≥ 0, integrate |ψ(x)|² = c²e−²ˣ/ᴸ over the interval [0, 1.0 mm]. Combine the results of both integrals to find the total probability.

Step 5: Normalize the wave function. The constant c is determined by the normalization condition, which requires the total probability over all space to equal 1. Use this condition to solve for c², then substitute it into the probability calculation to ensure the result is properly normalized.

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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 in space.

Probability Density

Probability density is a measure that describes 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. To find the probability of locating the particle within a specific interval, one must integrate the probability density over that interval.

Normalization of the Wave Function

Normalization ensures that the total probability of finding the particle in all space equals one. This is achieved by adjusting the constant 'c' in the wave function so that the integral of |ψ(x)|² over all space equals one. Proper normalization is crucial for accurate probability calculations in quantum mechanics.

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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

Consider the electron wave function $ψ (x) = {\begin{matrix} c \sqrt{1 - x^{2}} & ∣ x ∣ \leq 1 cm \\ 0 & ∣ x ∣ \geq 1 cm \end{matrix}$ where x is in cm. Draw a graph of |ψ(x)|² over the interval −2 cm ≤ x ≤ 2 cm. Provide numerical scales.

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

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A pulse of light is created by the superposition of many waves that span the frequency range f₀ − (1/2) Δf ≤ f ≤ f₀ + (1/2) Δf, where f₀ = c/λ is called the center frequency of the pulse. Laser technology can generate a pulse of light that has a wavelength of 600 nm and lasts a mere 6.0 fs (1 fs = 1 femtosecond =10⁻¹⁵ s). What is the spatial length of the laser pulse as it travels through space?

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

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Textbook Question

Consider the electron wave function $ψ (x) = {\begin{matrix} c \sqrt{1 - x^{2}} & ∣ x ∣ \leq 1 cm \\ 0 & ∣ x ∣ \geq 1 cm \end{matrix}$ where x is in cm. If 10⁴ electrons are detected, how many will be in the interval 0.00 cm ≤ x ≤ 0.50 cm?

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