Write the wave function for a particle in a three dimensional box.
Understand the Problem
The question is asking for the wave function of a particle confined in a three-dimensional box, a fundamental concept in quantum mechanics. The wave function describes the quantum state of the particle and is typically represented by a mathematical formula that depends on the box dimensions.
Answer
The wave function for a particle in a three-dimensional box is: \( \psi_{n_x,n_y,n_z}(x,y,z) = \left(\frac{8}{L_xL_yL_z}\right)^{1/2}\sin \left(\frac{n_x \pi x}{L_x}\right) \sin \left(\frac{n_y \pi y}{L_y}\right) \sin \left(\frac{n_z \pi z}{L_z}\right)\).
The wave function for a particle in a three-dimensional box is given by:
$$egin{equation*} \psi_{n_x,n_y,n_z}(x,y,z) = \left(\frac{8}{L_x L_y L_z}\right)^{1/2}\sin \left(\frac{n_x \pi x}{L_x}\right) \sin \left(\frac{n_y \pi y}{L_y}\right) \sin \left(\frac{n_z \pi z}{L_z}\right) \end{equation*}$$
Here, ( L_x, L_y, ) and ( L_z ) are the dimensions of the box along the x, y, and z axes respectively, and ( n_x, n_y, ) and ( n_z ) are the quantum numbers associated with each dimension.
Answer for screen readers
The wave function for a particle in a three-dimensional box is given by:
$$egin{equation*} \psi_{n_x,n_y,n_z}(x,y,z) = \left(\frac{8}{L_x L_y L_z}\right)^{1/2}\sin \left(\frac{n_x \pi x}{L_x}\right) \sin \left(\frac{n_y \pi y}{L_y}\right) \sin \left(\frac{n_z \pi z}{L_z}\right) \end{equation*}$$
Here, ( L_x, L_y, ) and ( L_z ) are the dimensions of the box along the x, y, and z axes respectively, and ( n_x, n_y, ) and ( n_z ) are the quantum numbers associated with each dimension.
More Information
The wave function describes the probability amplitude of finding a particle at a particular point inside the box. The quantum numbers must be positive integers, and the dimensions of the box influence the frequency and amplitude of the wave function.
Tips
Common mistakes include not normalizing the wave function properly and assuming incorrect quantum numbers.
Sources
- 3.9: A Particle in a Three-Dimensional Box - Chemistry LibreTexts - chem.libretexts.org
- 3.4: A Particle in a Three-Dimensional Box - Chemistry LibreTexts - chem.libretexts.org
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