A polyene molecule is a linear hydrocarbon chain of alternating single and double bonded C-atoms. Polyene molecules are important in biological processes and below Figure shows such a short polyene molecule.

Because of the alternation of the single and double bonds, the n-electrons are mobile and may move along the length of the polyene. Thus we may treat the polyene molecule as a one-dimensional box of length L which confines the n-electrons. This interesting and important quantum mechanical concept will be discussed in your Chemistry degree. Below Figure shows some of the different wavefunctions for the allowed solutions for the mobile n-electrons as the quantum number n increases so does the energy of the n-electron in that level.

The total probability of finding a π-electron at a certain position x, measured from one end of the one-dimensional box of length L is given below.

Where the sine is in radians not degrees hence the need for π; x is the distance along the molecule from one end; n is a constant called a quantum number which may take any of the values n = 1, 2, 3, and B is a constant to be evaluated. Use the standard integral below to integrate the probability equation from x = 0 to x = L and thus find B in terms of L the length of the molecule (C is the constant of integration).

**ʃ sin ^{2} (ax) dx = x/2 – [(sin 2ax) / 4a] + C**

Now

Using the standard integral,

**ʃ sin ^{2} (ax) dx = x/2 – [(sin 2ax) / 4a] + C**

Take the constant B2 outside the integral as B is not a function of x, let a = nria in the standard integral gives

Substituting the limits x = L and x = 0 and the constant of integration cancels out.

The term 2nπ is in radians not degrees and as n is an integer, sin(2nπ) = 0, prove this by using your calculator to find sin(π) in radians. The second term of the upper limit is zero and both **terms of the lower limit are also zero.**

**B ^{2} (L/2) = 1**

**B ^{2} = 2/L **

**B = √ (2/L)**

Although not directly part of the question, we now have the full equation for the wavefunctions of a π-electron in a given quantum level n of a polyene molecule as shown in above Figure.

**Ψ _{n} = √ (2/L) sin (nπx / L)**