Chapter 4, Problem 42P

(a)

Interpretation Introduction

Interpretation:

The wave function for a highly excited state ${\stackrel{\sim }{\psi }}_{100,100}(\stackrel{\sim }{x},\stackrel{\sim }{y})$ for a particle in square box should be written and nodal lines number should be determined along with the number of local probability maxima for this state.

Concept introduction:

The function which describes the position of an electron and quantum state of an isolated quantum system is known as wave function.

The general formula of the wave function for a particle in a square box is:

  ${\psi }_{n_x}{}_{n_y}(x,y)=\frac{2}{L}\sin \left(\frac{n_x\pi x}{L}\right)\sin \left(\frac{n_y\pi y}{L}\right)$

Where, ${\psi }_{n_x}{}_{n_y}(x,y)$ is wave function and $n_x,n_y$ have values 1, 2, 3 etc.

L = length of the box.

The nodes that lie along straight lines are known as nodal lines. The points in space around the nucleus where the probability of finding electron is zero is known as nodes.

(b)

Interpretation Introduction

Interpretation:

The motion of particle should be described in this state.

Concept introduction:

The function which describes the position of an electron and quantum state of an isolated quantum system is known as wave function.

The general formula of the wave function for a particle in a square box is:

  ${\psi }_{n_x}{}_{n_y}(x,y)=\frac{2}{L}\sin \left(\frac{n_x\pi x}{L}\right)\sin \left(\frac{n_y\pi y}{L}\right)$

Where, ${\psi }_{n_x}{}_{n_y}(x,y)$ is wave function and $n_x,n_y$ have values 1, 2, 3 etc.

L = length of the box.