For a model system of three non-interacting electrons confined in a two dimensional square box of length L, the ground state energy in units of \(\left( {\frac{h^2}{{8mL^2}}}\right)\) is

Concept:

• For a particle in a 2D box, the wavefunction is given by

\(\Psi {n_x},{n_y} = N\sin \left( {{{{n_x}\pi x} \over L}} \right)\sin \left( {{{{n_y}\pi y} \over L}} \right)\), where N is the normalization constant.

• The energy of a particle in a 2D box of side length l is given by,

\(E = {{({n_x}^2 + {n_y}^2){h^2}} \over {8m{l^2}}}\), where n_x and n_y is the quantum number.

Explanation:

• The three non-interacting electrons can be arranged in a 2D square box as follows:

(1, 2) (2, 1) 1 - \(\rm E_1=1\times5\frac{h^2}{8ml^2}\)

(1, 1) \(\rm E_2=2\times\frac{2h^2}{8ml^2}\)

\(\rm E_T=\frac{9h^2}{8ml^2}=E_1+E_2\)

• In the ground state (g.s), two electrons can be placed in a singly degenerated energy state (1,1). The energy is given by,

\({E_1} = 2 \times {{(1 + 1){h^2}} \over {8m{l^2}}}\)

\( = {{4{h^2}} \over {8m{l^2}}}\)

• ​In the first exited state, one electron can be placed in a doubly degenerated energy level (2,1) and (1,2). The energy is given by,

​\({E_2} = {{({2^2} + 1){h^2}} \over {8m{l^2}}}\)

\( = {{5{h^2}} \over {8m{l^2}}}\)

The energy of the system is thus,

\( = {{4{h^2}} \over {8m{l^2}}} + {{5{h^2}} \over {8m{l^2}}}\)

\( = {{9{h^2}} \over {8m{l^2}}}\)

Conclusion:

Hence, the ground state energy in units of \(\left( {\frac{h^2}{{8mL^2}}}\right)\) is 9