In quantum mechanics, the Hamiltonian operator represents the total energy of a system. It is the sum of kinetic energy operator T hat and potential energy operator V hat. For a particle of mass m, the Hamiltonian is given by minus h bar squared over two m times the Laplacian plus the potential V.
The expectation value represents the average value of an observable when measured many times on identical quantum systems. It is calculated using the integral of psi star times the operator times psi over all space. This gives us the statistical average of what we would measure.
The expectation value of the Hamiltonian gives us the average total energy of the quantum system. It equals the integral of psi star times H hat times psi over all space. This can be split into kinetic and potential energy contributions, providing the complete energy picture of the quantum state.
To calculate the Hamiltonian expectation value, we follow four key steps. First, identify the normalized wavefunction psi. Second, apply the Hamiltonian operator to the wavefunction. Third, calculate the integral of psi star times H psi over all space. Finally, this gives us the average energy of the quantum system.
To summarize what we have learned: The Hamiltonian operator represents the total energy of quantum systems. Expectation values provide the average measurements of observables. The energy expectation value combines kinetic and potential energy contributions through quantum integration over all space. This mathematical framework forms the foundation for energy calculations in quantum mechanics.