Particle in a Box Wave Function Analysis

This solution analyzes the correctness of the given statements regarding the wave function ($\Psi_n$) of a particle confined within a one-dimensional box.

Analyzing Particle in a Box Statements

• Statement A: $\Psi_n = \text{A} \sin \frac{\sqrt{2m\text{E}_n}}{\hbar} x$

  The energy levels for a particle in a 1D box are $E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$. Substituting this into the argument $\frac{\sqrt{2mE_n}}{\hbar} x$ correctly yields $\frac{n\pi x}{L}$. Thus, this statement represents the functional form of the wave function.

  Correct
• Statement B: Normalized wave function $\Psi_n = \sqrt{\frac{2}{L}} \sin \frac{n\pi x}{L}$

  This is the standard, correctly normalized wave function for a particle in a 1D box of length L (from $x=0$ to $x=L$).

  Correct
• Statement C: $\Psi_n$ cannot be negative

  The wave function $\Psi_n$ can be negative in certain regions. For example, the $\Psi_2$ state is negative for $L/2 < x < L$. Only the probability density $|\Psi_n|^2$ must be non-negative.

  Incorrect
• Statement D: $\Psi_n = \text{A} \sin \frac{n\pi x}{L}$

  This represents the general form of the wave function for a particle in a 1D box, where 'A' is the normalization constant.

  Correct
• Statement E: $|\Psi_n|^2$ can be less than zero

  The term $|\Psi_n|^2$ represents the probability density. Probability density is always non-negative ($|\Psi_n|^2 \ge 0$). It cannot be less than zero.

  Incorrect

Correct Particle in a Box Statements Identified

Based on the analysis, statements A, B, and D are correct.

Selecting Correct Answer Combination

The combination of correct statements is A, B, and D only.