In quantum mechanics it is easy to expand the number of dimensions where a particle can exist. 2-Dimensional box Consider a particle that can move in 2 independent directions, \(x,y\). When each coordinate can be treated independently, the solution to the 2-D Schcrödinger equation yields a wavefunction \(\psi(x,y)\) which depends on both coordinates and its characterized by 2 independent quantum numbers \(n_x\) and \(n_y\). \[\Psi_{n_x,n_y}(x,y)= \psi_{n_x}(x) \cdot \psi_{n_y}(y) = \sqrt{\dfrac{2}{L_x}\cdot \dfrac{2}{L_y} } \cdot \sin(\dfrac{n_x\pi}{L_x}x) \cdot \sin(\dfrac{n_y\pi}{L_y}y)\] The energy of the states depends on those two quantum numbers \[E_{n_x,n_y} = \dfrac{h^2}{8mL^2_x}n^2_x+\dfrac{h^2}{8mL^2_y}n^2_y= \dfrac{h^2}{8m}\cdot (\dfrac{n^2_x}{L^2_x}+\dfrac{n^2_y}{L^2_y})\] where \(m\) is the mass of the particle, \(L_x\) and \( L_y\) are the length of the box in the \(x\) and \(y\) directions, respectively. Each state of the system is given by a specific combination of quantum numbers \(n_x\) and \(n_y\) which are integer numbers that vary independently. Looking at a square box (where \(L_x =L_y =L\) we can evaluate the probability density and the energy of each state, with \[|\Psi_{n_x,n_y}(x,y)|^2 = |\dfrac{2}{L} \cdot \sin(\dfrac{n_x\pi}{L}x) \cdot \sin(\dfrac{n_y\pi}{L}y)|^2 \hspace{2cm} E_{n_x,n_y}=\dfrac{h^2}{8mL^2}\cdot (n^2_x+n^2_y)\]. The presence of two quantum numbers provides the possibility of degeneracy , which occurs when different states, each one with its own probability density, have the same total energy. In this case, the states described by \(\Psi_{a,b}(x,y)\) and \(\Psi_{b,a}(x,y)\) will have different probability densities, but the same energy. 0 1 2 \(n_x,n_y\) \(\Psi_{n_x,n_y}(x,y)|^2 \) \(E_{n_x,n_y}\) (1,1) NaN \(E_{1,1}= 0.752\ eV\) (1,2) NaN \(\textcolor{darkblue}{E_{1,2}= 1.880\ eV}\) (2,1) NaN \(\textcolor{darkblue}{E_{2,1}= 1.880\ eV}\) (2,2) NaN \(E_{2,2}= 3.008\ eV\) (1,3) NaN \(\textcolor{darkgreen}{E_{1,3}= 3.761\ eV}\) (3,1) NaN \(\textcolor{darkgreen}{E_{3,1}= 3.761\ eV}\) (2,3) NaN \(\textcolor{darkmagenta}{E_{2,3}= 4.889\ eV}\) (3,2) NaN \(\textcolor{darkmagenta}{E_{2,3}= 4.889\ eV}\) 3-Dimensional box Moving into 3 dimensions is done in a similar fashion, where the wavefucntion will depend on 3 coordinates (\(x,y,z\)); three independent quantum numbers will identify each state (\(n_x,n_y,n_z)\) and the energies will dependend on those 3 independent quantum numbers (each one with values \(1, 2, ...,\infty\). The additional dimension increases the possibility of more degenerate states. For example, for a cubic box (all side lenghts being equal), we obtain \(E_{1,3,6}= E_{1,6,3}=E_{3,6,1}=E_{3,1,6}=E_{6,1,3}=E_{6,3,1}\), although each of those states will have a different probability density in space.