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)?

Ch 40: One-Dimensional Quantum Mechanics

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 40: One-Dimensional Quantum Mechanics

문제 36

40장, 문제 36

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)?

검증된 단계별 안내

Step 1: Recognize that the problem involves a quantum mechanical particle in a finite potential well. The probability density is related to the square of the wavefunction, |ψ(x)|², at the respective positions.

Step 2: For a finite potential well, the wavefunction inside the well (x ≤ L) is typically sinusoidal, while outside the well (x > L) it decays exponentially. The general form of the wavefunction outside the well is ψ(x) = A * e^(-κx), where κ = sqrt(2m(U₀ - E))/ħ, m is the particle's mass, U₀ is the well depth, E is the particle's energy, and ħ is the reduced Planck constant.

Step 3: To find the ratio of probabilities, calculate the square of the wavefunction at the two positions: Prob(in δx at x=L+η) ∝ |ψ(L+η)|² and Prob(in δx at x=L) ∝ |ψ(L)|². Using the exponential decay form of the wavefunction, |ψ(L+η)|² = |A|² * e^(-2κ(L+η)) and |ψ(L)|² = |A|² * e^(-2κL).

Step 4: Divide the probability at x=L+η by the probability at x=L to find the ratio: Prob(in δx at x=L+η) / Prob(in δx at x=L) = e^(-2κ(L+η)) / e^(-2κL). Simplify the expression to get e^(-2κη).

Step 5: Conclude that the ratio of probabilities depends only on the exponential decay factor e^(-2κη), where κ is determined by the particle's mass, the well depth U₀, and the energy E. This shows how the probability decreases as the particle moves further outside the well.

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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 quantum particles, particularly in terms of their probability distributions.

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, which represents the particle's state. The probability of locating the particle in a small interval (δx) can be calculated by integrating the probability density over that interval, making it essential for comparing probabilities at different positions within the potential well.

Boundary Conditions

Boundary conditions are constraints that the wave function must satisfy at the edges of the potential well. For a finite potential well, these conditions determine how the wave function behaves at the boundaries (x = 0 and x = L). They are critical for solving the Schrödinger equation and finding the allowed energy levels and corresponding wave functions, which directly affect the probability calculations at specific points within the well.