In this chapter we apply the principles of Quantum Mechanics to the simplest possible physical system, a free particle in one dimension. This particle could be an electron or, if we only consider translational motion, an atom or a molecule. Free means that no forces are acting on the particle. Since a force is produced by a change in the potential energy, the potential energy must be constant if there is no force. This constant can be taken to be zero because energy is relative not absolute. By saying energy is relative, we mean we are concerned with adding and removing energy from systems not with the absolute value of the energy content. The discussion of the free particle in this chapter further illustrates the fundamental ideas of Quantum Mechanics and introduces solutions to new problems. Specifically the energy, momentum and probability density for a free particle are discussed, and a connection is made between the wave property of matter and the uncertainty principle.
- 5.1: The Free Particle
- In classical physics, a free particle is one that is present in a "field-free" space. In quantum mechanics, it means a region of uniform potential, usually set to zero in the region of interest since potential can be arbitrarily set to zero at any point (or surface in three dimensions) in space. Here, we obtain the Schrödinger equation for the free particle in one dimension.
- 5.2: The Uncertainty Principle
- The uncertainty principle is a consequence of the wave property of matter. A wave has some finite extent in space and generally is not localized at a point. Consequently there usually is significant uncertainty in the position of a quantum particle in space. Activity 1 at the end of this chapter illustrates that a reduction in the spatial extent of a wavefunction to reduce the uncertainty in the position of a particle increases the uncertainty in the momentum of the particle.
- 5.3: Linear Combinations of Eigenfunctions
- Many problems encountered by quantum chemists and computational chemists lead to wavefunctions that are not eigenfunctions of the Hamiltonian operator. The wavefunction in this state belongs to a class of functions known as superposition functions, which are linear combinations of eigenfunctions. A linear combination of functions is a sum of functions, each multiplied by a weighting coefficient, which is a constant.
- 5.E: Translational States (Exercises)
- Exercises for the "Quantum States of Atoms and Molecules" TextMap by Zielinksi et al.