Kinetic Energy Term in the Hamiltonian

Explain why the kinetic energy term in the Hamiltonian is written as $\frac{p^2}{2m}$, knowing that $p$ is the momentum operator and $m$ is the mass of the system.

In quantum mechanics, the Hamiltonian operator is synonymous with the total energy of the system, comprising both kinetic and potential energies. The kinetic term in the Hamiltonian, written as , stems from classical mechanics where kinetic energy is half the mass times velocity squared. However, in quantum mechanics, physical properties like position and momentum are represented by operators. Here, momentum $p$ is an operator, which is a central concept in quantum mechanics. Understanding operators and their algebra is crucial, as it underpins many quantum phenomena and calculations. The representation of kinetic energy emphasizes the wave-particle duality in quantum systems, showing how classical descriptions must be amended using operators to describe the behavior of particles accurately. The formulation also allows seamless transfer into analyzing wavefunctions through Schrodinger's framework, highlighting the transition from classical to quantum thought. Thus, the operator form of kinetic energy is pivotal in comprehending quantum motion and related phenomena.