For a particle in a finite potential well of width L and depth U0, what is the ratio of the probability Prob(in δx at x=L+η) to the probability Prob(in δx at x=L)?

Verified step by step guidance

1. Identify the regions of interest: Region I (x < 0), Region II (0 ≤ x ≤ L), and Region III (x > L). The potential energy is zero in Region I and III, and -U0 in Region II.
2. Understand that the wave function ψ(x) behaves differently in these regions. In Region II, the wave function can be represented by a sinusoidal function due to the finite potential, while in Region I and III, it decays exponentially due to the classically forbidden nature of these regions.
3. Recognize that the probability density is given by |ψ(x)|^2. This represents the probability of finding the particle per unit length at position x.
4. Analyze the behavior of the wave function at x = L and x = L + η. At x = L, the wave function transitions from a sinusoidal form to an exponentially decaying form. At x = L + η (where η > 0), the wave function is in the exponentially decaying region.
5. To find the ratio of the probabilities, calculate the ratio |ψ(L + η)|^2 / |ψ(L)|^2. This involves evaluating the exponential decay of the wave function beyond x = L and comparing it to the value at x = L.

Key Concepts

Finite Potential Well

A finite potential well is a region where a particle experiences a potential energy lower than its surroundings, allowing for bound states. The well has a specific width (L) and depth (U0), which influence the energy levels and wave functions of particles within it. Understanding the characteristics of the potential well is crucial for analyzing the behavior of particles, particularly in quantum mechanics.

Quantum Probability Density

In quantum mechanics, the probability density describes the likelihood of finding a particle in a specific region of space. It is derived from the square of the wave function's amplitude. For a particle in a potential well, the probability density varies with position, and calculating the ratio of probabilities at different locations requires understanding how the wave function behaves at the boundaries of the well.

Boundary Conditions

Boundary conditions are essential in quantum mechanics as they determine the allowed wave functions for a particle in a potential well. At the edges of the well, the wave function must satisfy specific criteria, such as continuity and differentiability. These conditions affect the probability distribution and are critical for calculating probabilities at specific points, such as at x=L and x=L+η.