Recall that $\left(\mid \psi {\mid }^2\right)dx$ is the probability of finding the par­ticle that has normalized wave function $\psi \left(x\right)$ in the interval $x$ to $x+dx$. Consider a particle in a box with rigid walls at $x=0$ and $x=L$. Let the particle be in the ground level and use ${\psi }_n$ as given in Eq. ($40.35$) ${\psi }_n\left(x\right)=\sqrt{\frac{2}{L}}sin\left[\right(\frac{n\pi x}{L}\left)\right]$ where $n=1,2,3,\dots$.
(a) For which values of $x$, if any, in the range from $0$ to $L$ is the probability of finding the particle zero?
(b) For which values of $x$ is the probability highest?
(c) In parts (a) and (b) are your answers consistent with Fig. $40.12$? Explain.

Verified step by step guidance

1. Step 1: Understand the problem and the given wave function. The wave function ψ_n(x) = √(2/L) sin⁡[(nπx/L)] describes the quantum state of a particle in a box with rigid walls at x = 0 and x = L. The probability density is given by |ψ(x)|², and the probability of finding the particle in a small interval dx is |ψ(x)|² dx.
2. Step 2: Analyze part (a). To find where the probability of finding the particle is zero, we need to determine where the wave function ψ_n(x) is zero. Since ψ_n(x) = √(2/L) sin⁡[(nπx/L)], the sine function is zero when its argument is an integer multiple of π. Solve sin⁡[(nπx/L)] = 0 for x in the range 0 ≤ x ≤ L.
3. Step 3: Analyze part (b). To find where the probability is highest, we need to determine where |ψ_n(x)|² is maximized. Since |ψ_n(x)|² is proportional to sin²[(nπx/L)], the sine squared function is maximized when sin[(nπx/L)] = ±1. Solve for x in the range 0 ≤ x ≤ L where this condition is met.
4. Step 4: Address part (c). Compare the results from parts (a) and (b) with the graphical representation of the wave function in Fig. 40.12. Check if the nodes (where ψ_n(x) = 0) and antinodes (where |ψ_n(x)|² is maximum) match the expected behavior of the ground state wave function (n = 1).
5. Step 5: Summarize the findings. For part (a), the probability is zero at the nodes of the wave function, which occur at specific x values. For part (b), the probability is highest at the antinode, which is the midpoint of the box for the ground state (n = 1). Confirm that these results align with the physical interpretation and Fig. 40.12.

Key Concepts

Wave Function and Probability Density

The wave function, denoted as ψ(x), describes the quantum state of a particle. The square of its absolute value, |ψ(x)|², represents the probability density of finding the particle at a specific position x. In a one-dimensional box, this probability density is crucial for determining where the particle is likely to be located within the confines of the box.

Boundary Conditions in Quantum Mechanics

In quantum mechanics, boundary conditions define the behavior of a wave function at the edges of a system. For a particle in a box with rigid walls, the wave function must be zero at the boundaries (x = 0 and x = L). This condition leads to quantized energy levels and specific forms of the wave function, which are essential for solving problems related to confined particles.

Nodes and Antinodes in Standing Waves

In the context of a particle in a box, the wave function can be visualized as a standing wave. Nodes are points where the wave function is zero, indicating zero probability of finding the particle, while antinodes are points of maximum amplitude, corresponding to the highest probability. Understanding the distribution of nodes and antinodes helps in identifying where the particle is most likely to be found within the box.