Consider the electron wave function ψ(x)={c1−x2∣x∣≤1 cm0∣x∣≥1 cm\psi (x)=\begin{cases} c\sqrt{$$1-x^{2}$$} & \left|x\right|\leq 1\text{ cm} \\ 0 & \left|x\right|\geq 1\text{ cm} \end{cases} where x is in cm. If 104 electrons are detected, how many will be in the interval 0.00 cm ≤ x ≤ 0.50 cm?

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 39d

Chapter 39, Problem 39d

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?

Verified step by step guidance

Step 1: Understand the problem. The wave function ψ(x) describes the probability amplitude of finding an electron at position x. To determine the number of electrons in the interval 0.00 cm ≤ x ≤ 0.50 cm, we need to calculate the probability of finding an electron in this interval and multiply it by the total number of electrons detected (10^4).

Step 2: Recall that the probability density function is proportional to the square of the wave function, |ψ(x)|². For |x| ≤ 1 cm, ψ(x) = √c(1 − x²). Therefore, the probability density function is |ψ(x)|² = c(1 − x²)².

Step 3: To find the probability of detecting an electron in the interval 0.00 cm ≤ x ≤ 0.50 cm, integrate the probability density function |ψ(x)|² over this interval. The integral is: ∫[0.00 to 0.50] c(1 − x²)² dx.

Step 4: Normalize the wave function to determine the constant c. The total probability of finding an electron in the range |x| ≤ 1 cm must equal 1. Set up the normalization condition: ∫[−1 to 1] c(1 − x²)² dx = 1. Solve this integral to find the value of c.

Step 5: Once c is determined, evaluate the integral ∫[0.00 to 0.50] c(1 − x²)² dx to find the probability of detecting an electron in the interval 0.00 cm ≤ x ≤ 0.50 cm. Multiply this probability by 10^4 to calculate the number of electrons detected in this interval.

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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), describes the quantum state of a particle, such as an electron, in terms of its position. It contains all the information about the system and is used to calculate probabilities of finding the particle in a specific region of space. The square of the wave function's absolute value, |ψ(x)|², gives the probability density, which indicates the likelihood of locating the particle at a given position.

Probability Density

Probability density is a measure derived from the wave function that indicates the likelihood of finding a particle in a specific interval of space. For a one-dimensional wave function, the probability density is given by |ψ(x)|². To find the total probability of locating the particle within a certain range, one must integrate the probability density over that interval.

Normalization

Normalization is a crucial concept in quantum mechanics that ensures the total probability of finding a particle in all possible positions equals one. This is achieved by adjusting the wave function so that the integral of the probability density over the entire space equals one. In this context, it is important to confirm that the wave function is properly normalized before calculating the number of electrons in a specific interval.

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