Key Points from this Chapter
The Schrödinger equation is an eigenvalue equation which, if it can be solved, reveals the possible wave functions, ψ(x) of a particle and their energies, E. The equation, in 1-dimension, is:
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where V(x) is the potential energy and m is the mass of the particle.
The equation is constructed by requiring consistency with the de Broglie equation, the wave equation and the principle of conservation of energy.
The d2ψ/dx2 term relates to the curvature of the wave function: the greater the curvature, the higher the kinetic energy.
The infinite possible solutions for a free particle allow values of the energy: there is no quantisation.
Quantisation arises when there are boundary conditions, placing limitations on the possible wave functions.
The particle in a 1-dimensional box is a valuable model system. The wavefunction is constrained to go to zero at the walls: this causes quantisation
Each solution is distinguished by a different quantum number (n). Increasing n is associated with more nodes (points where the wave function goes to zero) and higher kinetic energy.
In the lowest energy state (n=1) the particle still has some energy. This is the zero point energy. This satisfies the uncertainty principle.
ψ2 is the probability density function, determining where the particle is likely to be found, if a measurement is made. Therefore ψ2 must be normalised, so that the total probability = 1.
The particle in a box serves as a reasonable model for delocalised electrons in a conjugated polyene hydrocarbon.
High particle masses and large boxes make the wavelength of the wavefunction so short that the probability density approaches a continuum. Thus quantum behaviour approaches classical behaviour under these circumstances: the correspondence principle.
Atomic orbitals are analogous eigenfunctions of the Schrödinger equation, solved in three dimensions, in the presence of the electrostatic potential field around the nucleus.
Eigenfunctions of the Schrödinger equation can add together to form superposition states but these are not themselves eigenfunctions, so if a measurement is made, the wavefunction collapses, so that a specific eigenvalue is measured.
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