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Physical chemistry 222

A Schrödinger equation can be written to describe any particular physical system. For
a particle of mass m moving in one dimension only (along the x-axis) the equation is:

2
2
2
where −ћ /2m d /dx and V(x) are the operators for (one-dimensional) kinetic and
potential energy, respectively, that together constitute the Hamiltonian operator. The
symbol ħ is short-hand notation for h/2π (h is Planck’s constant).
The Schrödinger equation can be shown to be consistent with experimental
observation by considering the equation for a freely-moving particle that possesses
kinetic energy only:

Rearranging gives:

A solution to this equation is:

which may be verified by differentiating the function twice:

The wavelength, λ, of a sine wave of form sin(kx) is:

so the wavelength of the wavefunction associated with the freely-moving particle is:

Substituting using the relationship between kinetic energy E and momentum p: