In quantum mechanics, the wave function of a free particle is described by the time-independent Schrödinger equation, which is a differential equation that governs the behavior of quantum systems. The wave function represents the probability distribution of finding the particle at different positions in space.
When considering a free particle, it means that the particle is not subjected to any potential energy, such as an external force or a potential well. In this case, the Schrödinger equation simplifies to the time-independent Schrödinger equation for a free particle, which is given by:
(-ħ²/2m)∇²ψ + Eψ = 0
Here, ħ is the reduced Planck's constant, m is the mass of the particle, ∇² is the Laplacian operator, E is the energy of the particle, and ψ is the wave function.
For a free particle, the energy E can take any value, including negative values. When E ≤ 0, it means that the energy of the particle is non-positive. In this case, the time-independent Schrödinger equation becomes:
(-ħ²/2m)∇²ψ - |E|ψ = 0
Notice that the term involving the energy becomes negative due to the absolute value. This equation is still a valid form of the Schrödinger equation for a free particle with non-positive energy.
Solutions to this equation yield wave functions that describe the behavior of the particle. However, when E ≤ 0, the solution to the equation becomes problematic. This is because the Laplacian operator is a positive operator, and when combined with the negative term (-|E|), it leads to unbound and unphysical solutions.
In the context of a free particle, physically meaningful solutions require positive energies (E > 0). When E ≤ 0, there is no physically realizable solution that represents a free particle. In practice, for a free particle, the energy is typically taken to be positive, corresponding to bound states with a well-defined energy.
It's worth noting that for particles with negative energy (E ≤ 0), there are alternative approaches, such as the Dirac equation, that are used to describe their behavior. These equations incorporate the principles of relativistic quantum mechanics and are used for particles like positrons, which have negative energy solutions within those frameworks. However, for the non-relativistic case of a free particle, the energy is typically restricted to positive values.