Question

4) pi electrons in a conjugated molecule can be modelled simply as particles confined to the single particle states allowed in a rectangular box of length L (which is slightly longer than the conjugated chain) and width W and height H. The Pauli Exclusion Principle states that no more than 2 electrons (with opposite spins) can occupy one quantum state. For butatriene (a molecular wire for nanotechnology), CH2=C=C=CH2, you may take the box length as L = 7.0 Angstroms, and W = H = L/2 with potential V = 0 inside the box and infinite potential outside. Use this model, for wavefunction $\psi(x, y, z)$, to estimate the wavelength of light absorbed when a pi electron is excited from the highest occupied to the lowest unoccupied box level. Set the length to run in the x direction and width and height in y and z respectively. The Hamiltonian is: $\hat{H} = -\frac{\hbar^2}{2m} \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right) + V$ a) Write down the 3-D time independent Schrödinger equation for this problem (1 mark) b) Show that the wavefunction $\psi(x, y, z)$ given below, where A, B, C and D are constants, is a solution of the three dimensional time independent Schrödinger equation. $\psi(x, y, z) = A\sin(Bx)\sin(Cy)\sin(Dz)$ (4 marks) c) Use the boundary conditions to find the allowed values of B, C and D by introducing quantum numbers p, q and r. (4 marks)

          4) pi electrons in a conjugated molecule can be modelled simply as particles confined to the single
particle states allowed in a rectangular box of length L (which is slightly longer than the conjugated
chain) and width W and height H. The Pauli Exclusion Principle states that no more than 2 electrons
(with opposite spins) can occupy one quantum state. For butatriene (a molecular wire for
nanotechnology), CH2=C=C=CH2, you may take the box length as L = 7.0 Angstroms, and W = H =
L/2 with potential V = 0 inside the box and infinite potential outside. Use this model, for
wavefunction $\psi(x, y, z)$, to estimate the wavelength of light absorbed when a pi electron is excited
from the highest occupied to the lowest unoccupied box level. Set the length to run in the x direction
and width and height in y and z respectively.
The Hamiltonian is:
$\hat{H} = -\frac{\hbar^2}{2m} \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right) + V$
a) Write down the 3-D time independent Schrödinger equation for this problem
(1 mark)
b) Show that the wavefunction $\psi(x, y, z)$ given below, where A, B, C and D are constants, is a
solution of the three dimensional time independent Schrödinger equation.
$\psi(x, y, z) = A\sin(Bx)\sin(Cy)\sin(Dz)$
(4 marks)
c) Use the boundary conditions to find the allowed values of B, C and D by introducing quantum
numbers p, q and r.
(4 marks)

4) pi electrons in a conjugated molecule can be modelled simply as particles confined to the single
particle states allowed in a rectangular box of length L (which is slightly longer than the conjugated
chain) and width W and height H. The Pauli Exclusion Principle states that no more than 2 electrons
(with opposite spins) can occupy one quantum state. For butatriene (a molecular wire for
nanotechnology), CH2=C=C=CH2, you may take the box length as L = 7.0 Angstroms, and W = H =
L/2 with potential V = 0 inside the box and infinite potential outside. Use this model, for
wavefunction ψ(x, y, z), to estimate the wavelength of light absorbed when a pi electron is excited
from the highest occupied to the lowest unoccupied box level. Set the length to run in the x direction
and width and height in y and z respectively.
The Hamiltonian is:
Ĥ = -(ħ^2)/(2m)((∂^2)/(∂ x^2) + (∂^2)/(∂ y^2) + (∂^2)/(∂ z^2)) + V
a) Write down the 3-D time independent Schrödinger equation for this problem
(1 mark)
b) Show that the wavefunction ψ(x, y, z) given below, where A, B, C and D are constants, is a
solution of the three dimensional time independent Schrödinger equation.
ψ(x, y, z) = Asin(Bx)sin(Cy)sin(Dz)
(4 marks)
c) Use the boundary conditions to find the allowed values of B, C and D by introducing quantum
numbers p, q and r.
(4 marks)

Added by Tina B.

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