Question

(a) Explain (using words and, where required, mathematical symbols, but no diagrams) how the application to the Schrödinger Equation to model systems such as the particle in a box or particle on a ring can help us to understand each of the following features of real molecule systems: quantisation of energy and angular momentum, nodes, and degeneracy of atomic orbitals.

Answer 1

Schrodinger equation is used to find out the states of wavefunction and their corresponding energies.

Schrodinger equation gives the wavefunctions for a particle in box, particle on ring, particle on sphere, hydrogen atom, rigid rotor, harmonic oscillator etc. Schrodinger equation makes the quantization which distributes the energies in discrete manner.

a) nodes: nodes can be found out by setting the wavefunction to zero where the wavefunctions are derived from schrodinger equation.

b) angular momentum: (along z axis) it is obtained when the momentum operator ( -ihd/dx) is applied over a wavefunction.

c) degeneracy can be measured by checking where the energies are equal. Like for example in particle in 2D box

States (1,2) and (2,1) are having same energy and thus degenerate.

The 1,3-butadiene resembles particle in 1D box, electron un benzene ring resembles particle on ring.