FIGURE EX39.13 shows the probability density for an electron that has passed through an experimental apparatus. What is the probability that the electron will land in a 0.010-mm-wide strip at (a) x=0.000 mm,

Verified step by step guidance

1. Identify the probability density function (PDF) given in the problem or figure. This function, often denoted as \( \psi(x)^2 \), represents the probability density of finding the electron at a particular position x.
2. Determine the width of the strip where you want to find the probability. In this case, the width \( \Delta x \) is given as 0.010 mm.
3. Set up the integral of the probability density function over the desired interval. Since you are looking for the probability at x = 0.000 mm, you will integrate from \( x = -0.005 \) mm to \( x = 0.005 \) mm, which centers the interval at 0.000 mm and covers the 0.010 mm width.
4. Evaluate the integral \( \int_{-0.005}^{0.005} \psi(x)^2 \, dx \). This integral calculates the total probability of finding the electron within the specified strip around x = 0.000 mm.
5. Interpret the result of the integral. The value obtained from this integral represents the probability that the electron will land within the 0.010-mm-wide strip at x = 0.000 mm.

Key Concepts

Probability Density

Probability density is a statistical measure that describes the likelihood of finding a particle, such as an electron, in a specific region of space. It is represented as a function, where the value at any point indicates the probability per unit length. In quantum mechanics, the square of the wave function's amplitude gives the probability density, allowing us to determine where an electron is likely to be found.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of particles at the atomic and subatomic levels. It introduces concepts such as wave-particle duality, where particles exhibit both wave-like and particle-like properties. Understanding quantum mechanics is essential for interpreting phenomena like electron behavior in experimental setups, as it governs the probabilities associated with their positions and momenta.

Measurement in Quantum Systems

In quantum systems, measurement plays a crucial role in determining the state of a particle. When measuring the position of an electron, the act of measurement collapses its wave function, resulting in a specific outcome. The probability of finding the electron in a given region, such as a 0.010-mm-wide strip, is calculated using the probability density function, which reflects the inherent uncertainties of quantum mechanics.