3.The Schrodinger equation for a particle (of mass m) in a box, with potential
V = 0 at 0<x<l and V = ∞ at x≦0 and x≧l, is
h^2 (d^2)Ψ
-── ──── = EΨ
2m d (x^2)
(a)Derive the quantized energies and corresponding wavefunctions of the
particle. (10%)
(b)Calculate the average position (<x>) and average linear momentum (<p>) for
the ground state (n=1). (5%)
As a rough approximation, conjugated polyenes, such as butadiene, vitamin A,
and carotene, maybe considered as linear one-dimensional boxes. Vitamin A
absorbs electromagnetic radiation most strongly at 332nm, where vitamin A is
excited from n = 1 to n = 2 quantum states.
(c)What wavelengths of electronic radiation would absorbed by butadiene and
carotene? (Shorter or longer than 332nm?) Explain the reasons as to why you
think so. (5%)
(d)Estimate the length of the vitamin A "box". (5%)
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