CALC Consider a particle in a rigid box of length L. For each of the states n = 1,n = 2, and n = 3: Where, in terms of L, are the positions at which the particle is most likely to be found?

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Understand the problem: The particle in a rigid box (also called an infinite potential well) is described by quantum mechanics. The wavefunction for the particle in the nth energy state is given by ψₙ(x) = √(2/L) * sin(nπx/L), where L is the length of the box, and x is the position within the box (0 ≤ x ≤ L). The probability of finding the particle at a position x is proportional to |ψₙ(x)|².

To find the positions where the particle is most likely to be found, we need to identify the maxima of the probability density function |ψₙ(x)|². This involves squaring the wavefunction: |ψₙ(x)|² = (2/L) * sin²(nπx/L).

The maxima of sin²(nπx/L) occur where sin(nπx/L) = ±1. This happens when nπx/L = (2k + 1)π/2, where k is an integer. Solve for x to find the positions: x = [(2k + 1)L]/(2n), where k = 0, 1, 2, ..., and x must lie within the box (0 ≤ x ≤ L).

For n = 1, substitute n = 1 into the formula x = [(2k + 1)L]/(2n). This gives x = L/2, meaning the particle is most likely to be found at the center of the box.

For n = 2 and n = 3, repeat the process by substituting n = 2 and n = 3 into the formula x = [(2k + 1)L]/(2n). This will yield multiple positions within the box where the particle is most likely to be found. These positions correspond to the maxima of the probability density function for each state.

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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 and quantization of energy levels, which are essential for understanding how particles behave in confined spaces, like a rigid box.

Wave Function and Probability Density

In quantum mechanics, the wave function describes the quantum state of a particle, and its square gives the probability density of finding the particle in a particular position. For a particle in a box, the wave function takes specific forms for different energy states, indicating where the particle is most likely to be found.

Standing Waves

In a rigid box, the allowed states of a particle correspond to standing wave patterns. These patterns arise from the boundary conditions of the box, leading to specific nodes and antinodes where the probability of finding the particle is maximized or minimized, respectively, for each quantum state.