All Problems Wave Properties of particles. Schrodinger Equation.
Problem 6.81
A particle of mass \(m\) is located in a two-dimensional square potential well with absolutely impenetrable walls. Find: (a) the particle's permitted energy values if the sides of the well are \(l_{1}\) and \(l_{2}\) (b) the energy values of the particle at the first four levels if the well has the shape of a square with side \(l\).
6.81. (a) In this case the Schrödinger equation takes the form \[ \frac{\partial^{2} \psi}{\partial x^{2}}+\frac{\partial^{2} \psi}{\partial y^{2}}+k^{2} \psi=0, k^{2}=2 m E / \hbar^{2} \] Let us take the origin of coordinates at one of the corners of the well. On the sides of the well the function \(\psi(x, y)\) must turn into zero (according to the condition), and therefore it is convenient to seek this function inside the well in the form \(\psi(x, y)=a \sin k_{1} x \times\) \(\times \sin k_{2} y,\) since on the two sides \((x=0\) and \(y=0) \psi=0\) automatically. The possible values of \(k_{1}\) and \(k_{2}\) are found from the condition of \(\psi\) turning into zero on the opposite sides of the well: \[ \begin{array}{l} \psi\left(l_{1}, y\right)=0, \quad k_{1}=\pm\left(\pi / l_{1}\right) n_{1}, \quad n_{1}=1,2,3, \ldots, \\ \psi\left(x, l_{2}\right)=0, \quad k_{2}=\pm\left(\pi / l_{2}\right) n_{2}, \quad n_{2}=1,2,3, \ldots \end{array} \] The substitution of the wave function into the Schrödinger equation leads to the relation \(k_{1}^{2}+k_{2}^{2}=k^{2},\) whence \[ E_{n_{1} n_{2}}=\left(n_{1}^{2} / l_{1}^{2}+n_{2}^{2} / l_{2}^{2}\right) \pi^{2} \hbar^{2} / 2 m \] (b) \(9.87,24.7,39.5,\) and 49.4 units of \(\hbar^{2} / m l^{2}\).