A 16-nm-long box has a thin partition that divides the box into a 4-nm-long section and a 12-nm-long section. An electron confined in the shorter section is in the n = 2 state. The partition is briefly withdrawn, then reinserted, leaving the electron in the longer section of the box. What is the electron’s quantum state after the partition is back in place?

Verified step by step guidance

1. Understand the problem: The electron is initially confined in the shorter section of the box (4 nm) in the n=2 quantum state. When the partition is removed, the box becomes a single 16-nm-long box. The electron's wavefunction spreads across the entire box. When the partition is reinserted, the electron ends up in the longer section (12 nm). We need to determine the quantum state of the electron in the longer section.
2. Step 1: Recall the energy quantization for a particle in a box. The energy levels are given by the formula: $E$_n = (n²h²)/(8mL²), where n is the quantum number, h is Planck's constant, m is the mass of the particle, and L is the length of the box. The wavefunctions are sinusoidal and depend on the quantum number n and the box length.
3. Step 2: When the partition is removed, the electron's wavefunction in the 4-nm box (n=2 state) is expressed in terms of the wavefunctions of the 16-nm box. This involves expanding the initial wavefunction as a linear combination of the eigenfunctions of the 16-nm box. The coefficients of this expansion determine the probability of the electron transitioning to each quantum state in the 16-nm box.
4. Step 3: When the partition is reinserted, the electron's wavefunction in the 16-nm box is projected onto the eigenfunctions of the 12-nm box (the longer section). This projection determines the probability of the electron being in each quantum state of the 12-nm box. The quantum state with the highest probability is the most likely state of the electron after the partition is reinserted.
5. Step 4: To calculate the probabilities, use the overlap integral between the wavefunctions of the 16-nm box and the 12-nm box. The wavefunctions for a particle in a box are given by: $\backslash (\backslash text\{\psi \}\backslash )$_n(x) = \(\text{√}\)(2/L) \(\text{sin}\)(nπx/L), where L is the box length. Perform the integration to find the coefficients for each quantum state in the 12-nm box.

Key Concepts

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, quantization of energy levels, and the uncertainty principle, which are essential for understanding how particles like electrons behave in confined spaces, such as in a quantum box.

Particle in a Box Model

The particle in a box model is a fundamental concept in quantum mechanics that describes a particle free to move in a small space with infinitely high potential walls. This model helps to determine the allowed energy levels and wave functions of the particle, which are quantized. The length of the box directly influences the energy states available to the particle.

Quantum State and Superposition

A quantum state describes the state of a quantum system, encapsulating all information about the system's properties. When the partition is removed, the electron can occupy a superposition of states in the longer section. Upon reinserting the partition, the electron's state will collapse to one of the allowed energy states of the new section, which can be determined using the particle in a box model.