Simple models are very useful. Consider the twin finite wells shown in the figure, at First with a tiny separation. Then with increasingly distant separations, In all case, the four lowest allowed wave functions are planned on axes proportional to their energies. We see that they pass through the classically forbidden region between the wells, and we also see a trend. When the wells are very close, the four functions and energies are what we might expect of a single finite well, but as they move apart, pairs of functions converge to intermediate energies.

(a) The energies of the second and fourth states decrease. Based on changing wavelength alone, argue that is reasonable.

(b) The energies of the first and third states increase. Why? (Hint: Study bow the behaviour required in the classically forbidden region affects these two relative to the others.)

(c) The distant wells case might represent two distant atoms. If each atom had one electron, what advantage is there in bringing the atoms closer to form a molecule? (Note: Two electrons can have the same wave function.)

The separation of a twin well, initially having a tiny separation, increases. The four wave functions pass through the classical forbidden region between the walls. When the wells are very close, the four wave functions and energies are very similar to a single finite well, but as they move apart, pairs of functions converge to immediate energies.

The second and fourth have nodes in the middle, and simply assume a longer wavelength as the wells separate. A larger wavelength implies a smaller momentum and thus kinetic energy.

The separation of a twin well, initially having a tiny separation, increases. The four wave functions pass through the classical forbidden region between the walls. When the wells are very close, the four wave functions and energies are very similar to a single finite well, but as they move apart, pairs of functions converge to immediate energies.

The first and third state energies are required to die off in the classically forbidden region, where they would naturally have an antinode. Thus, they assume effectively shorter wavelengths than in one single well. A shorter wavelength implies a larger value of kinetic energy.

The separation of a twin well, initially having a tiny separation, increases. The four wave functions pass through the classical forbidden region between the walls. When the wells are very close, the four wave functions and energies are very similar to a single finite well, but as they move apart, pairs of functions converge to immediate energies.

When an antinode is approached from middle it will generate a value of wavelength. So the energy corresponding to the antinode point is low to form a ground state.