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Ch 39: Wave Functions and Uncertainty
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 39: Wave Functions and UncertaintyProblema 38b
Capitolo 39, Problema 38b

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\(\text{ mm}\) \(\end{cases}\) mm where L = 2.0 mm. Determine the normalization constant c.

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To normalize the wave function, we use the condition that the total probability of finding the particle over all space is equal to 1. Mathematically, this is expressed as: ∫|ψ(x)|² dx = 1, where the integral is taken over all space.
Substitute the given wave function into the normalization condition. Since the wave function is piecewise, split the integral into two parts: ∫|ψ(x)|² dx = ∫[c²e²ˣ/ᴸ] dx (for x ≤ 0) + ∫[c²e⁻²ˣ/ᴸ] dx (for x ≥ 0).
Evaluate each integral separately. For the first integral (x ≤ 0), calculate ∫[c²e²ˣ/ᴸ] dx from -∞ to 0. For the second integral (x ≥ 0), calculate ∫[c²e⁻²ˣ/ᴸ] dx from 0 to ∞. Use the standard integral formula for exponential functions: ∫eᵃˣ dx = (1/a)eᵃˣ + C, where a ≠ 0.
Combine the results of the two integrals and set the total equal to 1. This will give you an equation involving the normalization constant c. Solve for c by isolating it on one side of the equation.
Substitute the given value of L = 2.0 mm into the equation to express c in terms of numerical values. This will provide the normalized constant c, ensuring the wave function satisfies the normalization condition.

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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 system and is used to calculate probabilities of finding a particle in a given position. The wave function must be normalized, meaning the total probability of finding the particle in all space must equal one.
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Intro to Wave Functions

Normalization

Normalization is the process of adjusting the wave function so that the integral of the absolute square of the wave function over all space equals one. This ensures that the total probability of finding the particle in any location is 100%. For a piecewise wave function, normalization involves integrating each segment and setting the sum equal to one to solve for the normalization constant.
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Integration

Integration is a fundamental mathematical operation used to calculate areas under curves, which in quantum mechanics is essential for determining probabilities. In the context of normalization, it involves integrating the square of the wave function over the entire space to find the normalization constant. The integration limits will depend on the defined regions of the wave function.
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