(a) Write down and sketch the probability distribution |ψ(x)|^2 for the second excited state (n=

step 1 In this question we have to find the probability distribution and also find the most probable po

(a) Write down and sketch the probability distribution...

(a) Write down and sketch the probability distribution $|\psi(x)|^{2}$ for the second excited state $(n=3)$ of a particle in a rigid box of length $a$. (b) What are the most probable positions, $x_{\mathrm{mp}} ?$ (c) What are the probabilities of finding the particle in the intervals $[0.50 a, 0.51 a]$ and $[0.75 a, 0.76 a] ?$

Key Concepts

Quantum Particle in a Box

This concept refers to a fundamental quantum mechanics problem where a particle is confined in a region with impenetrable walls (i.e., a rigid box) and is free to move within that region. The boundary conditions (the wavefunction being zero at the walls) lead to a quantization of energy levels and specific forms for the allowed wavefunctions, making it an ideal system for illustrating quantum confinement and quantization.

Wavefunction and Probability Density

The wavefunction, denoted by ?(x), contains the complete information about the state of the quantum system. The probability density, given by |?(x)|², represents the likelihood of finding the particle at a given position. In stationary states, this probability density is time-independent and its normalization ensures that the total probability over the entire region is one.

Eigenstates and Quantization

Eigenstates are the solutions to the Schrödinger equation for a given potential, like the rigid box, and are associated with discrete energy levels. The quantum number (n) labels these states, and each quantum number corresponds to a unique wavefunction and energy level. This quantization arises from the boundary conditions and yields a set of orthogonal wavefunctions.

Nodes, Antinodes, and Most Probable Positions

Nodes are points where the wavefunction (and thus the probability density) is zero, while antinodes are regions of maximum probability density. The most probable positions for the particle correspond to these antinodes, where |?(x)|² reaches its maximum value. The pattern of nodes and antinodes reflects the standing wave nature of the quantum state in the box.

Integration of Probability Densities

To find the probability of locating the particle within a specific interval, one must integrate the probability density over that range. This process highlights the probabilistic interpretation of quantum mechanics, linking the mathematical form of the wavefunction with measurable outcomes in experiments.

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