Question

What is the probability of finding a particle between x = 0 and x = 0.25 nm in a box of length 1.0 nm in (a) its lowest energy state (n = 1) and (b) when n = 100. Relate your answer to the correspondence principle.

This is an illustration of the correspondence principle, which states that classical mechanics emerges from quantum mechanics as high quantum numbers are reached.

What does this all mean?

• Only certain (discrete) energies are allowed (the energy is quantized). This is actually a result of the boundary conditions (i.e. we are restricting the electron to a region of space).

• The energy levels are dependent on the box length. That is, the longer the box (or longer the wire), the closer the energy levels. This is the forerunner to delocalization in conjugated molecules. That is, the longer the conjugation, the closer the energy levels. This enables the colour of a dye to be ’tuned’ by altering the conjugated chain length.

Answer 1

Syr y de 'Protability (p) for 1-0 box pa 11 2 sin nga I. 0:25 ? sin²nua da l x P = 2 ex2 Cos 2nna l majdn a D Sinanna l l 2 - (simamo 2nn o for n=1 P = 1 l {a-sin arx l e 20 here 2-0.25 nm, & l = lnm. 1 20:25 - Bin(28 920-25) ] 1 1416-9x109 0.09 5 for n=100 Pa L 0.25 I {sin (1. Mony 3:14 3 025)) { try Sin (2x) 60 x 3.14X0:25) 1 P = 0.26

Only certain (discrete) energies are allowed (the energy is quantized). This is actually a result of the boundary conditions (i.e. we are restricting the electron to a region of space).