Question

Write the wave function for the highly excited state ${\overline{\mathbf{\psi }}}_{\mathbf{100}\mathbf{,}\mathbf{100}}\mathbf{\left(}\overline{\mathbf{x}\mathbf{,}}\overline{\mathbf{y}}\mathbf{\right)}$for a particle in a square box.

(a) Determine the number of nodal lines and the number of local probability maxima for this state.

(b) Describe the motion of the particle in this state.

Short answer

a.

If the quantum number for wave function is n then the number of nodal lines is n=1. Therefore, the nodal planes for the wave function${\overline{\psi }}_{100,100}\left(\overline{x},\overline{y}\right)$ are 99 along the x-axis and 99 along the y-axis. The number of probability maxima is 100.

b.

Since the quantum number in the x and y directions are the same, i.e.,$n_x=100$ and $n_y=100$, the motion of the particle is symmetrical.

Step-by-step solution

Schrodinger Equation

The linear partial differential equation operates the wave function of a quantum mechanical system. The detection of the place of the electron at any moment was known by the Schrodinger equation.

Explanation

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

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

Here, ${\psi }_{n_x{n}_y}\left(x,y\right)$is the wave function, $n_x$and $n_y$have the values 1,2,3 and L is the length of the box.

${\psi }_{100,100}\left(x,y\right)=\frac{2}{L}\sin \left(\frac{100\pi x}{L}\right)\sin \left(\frac{100\pi y}{L}\right)$

For a particle in a square box, the wave function${\overline{\psi }}_{12}\left(\overline{x},\overline{y}\right)$is defined as follows:

${\overline{\psi }}_{12}\left(\overline{x},\overline{y}\right)=\frac{{\psi }_{12}\left(\overline{x},\overline{y}\right)}{{\psi }_{max}}$

Here, $\overline{x}=\frac{x}{L}$and

On substituting in the given equation, we get

$\begin{array}{rcl}{\psi }_{100,100}\left(\overline{x},\overline{y}\right)& =& \frac{2}{L}\sin \left(\frac{100\pi \overline{x}}{L}\right)\sin \left(\frac{100\pi \overline{y}}{L}\right)\\ {\psi }_{100,100}\left(\overline{x},\overline{y}\right)& =& \frac{2}{L}\sin \left(\frac{100\pi \left(x/L\right)}{L}\right)\sin \left(\frac{100\pi \left(y/L\right)}{L}\right)\\ {\psi }_{100,100}\left(\overline{x},\overline{y}\right)& =& \frac{2}{L}\sin \left(100\pi x\right)\sin \left(100\pi \right)\\ {\psi }_{100,100}\left(\overline{x},\overline{y}\right)& =& \frac{\left\{\frac{2}{L}\sin \left(100\pi x\right)\sin \left(100\pi y\right)\right\}}{{\psi }_{\max }}\\ & & \end{array}$

a.

If the quantum number for wave function is n then the number of nodal lines is n=1. Therefore, the nodal planes for the wave function${\overline{\psi }}_{100,100}\left(\overline{x},\overline{y}\right)$are 99 along x-axis and 99 along y-axis. The number of probability maxima is 100.

b.

Since the quantum number in the x and y directions are the same, i.e.,$n_x=100$and $n_y=100$, the motion of the particle is symmetrical.