Investigating Localized Quantum Forces Using the Virial Theorem

In classical mechanics, the force acting on a particle at a specific position governs its motion according to Newton's second law. However, in quantum bound states, the force is defined through the gradient of the potential energy, quantum-mechanically averaged over a range of positions, making the notion of force at a specific location less meaningful. We propose that the notion of a localized quantum mechanical force can be explored using the virial theorem. To investigate this, we used a model of a particle in a finite square well and found that the force at the boundary can be decomposed into two components, a characteristic not addressed in earlier studies. These force components depend on the height and width of the well, allowing us to investigate their impact on the net force. Our analysis is further extended to include the infinite square well and the quantum harmonic oscillator.