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

문제 38c

39장, 문제 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.

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.