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In the QM Hamiltonian, I keep seeing h-bar/2m instead of p/2m for the kinetic energy term. H-bar is not equivalent to momentum. What am I missing here?
The Hamiltonian Kinetic Energy Operator is a mathematical operator used in quantum mechanics to describe the kinetic energy of a particle in a particular quantum state. It is part of the larger Hamiltonian operator, which includes both the kinetic and potential energy of a system.
The classical kinetic energy formula, 1/2 mv^2, only applies to macroscopic objects and is based on Newtonian mechanics. The Hamiltonian Kinetic Energy Operator, on the other hand, is a quantum mechanical operator that takes into account the wave-like nature of particles and is used to describe the energy of particles on a microscopic scale.
The Hamiltonian Kinetic Energy Operator is represented by the symbol T̂ and is defined as T̂ = -ħ^2/2m * ∂^2/∂x^2, where ħ is the reduced Planck's constant, m is the mass of the particle, and ∂^2/∂x^2 is the second derivative of the wavefunction with respect to position.
The Hamiltonian Kinetic Energy Operator is used in the Schrödinger equation, which is a fundamental equation in quantum mechanics that describes the time evolution of a quantum state. It is also used in the calculation of energy levels and transition probabilities in quantum systems.
Yes, the Hamiltonian Kinetic Energy Operator can be applied to all types of particles, including electrons, protons, and atoms. It is a fundamental operator in quantum mechanics and plays a crucial role in understanding the behavior of particles on a microscopic scale.