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Mathematics for Physics

120 Hamiltonian

An operator that is used to represent the total energy of a quantum system.

\(H=P^{2}/2m+V(X)\). For one particle the Hamiltonian is kinetic energy plus potential energy, written with the operators \(P\) and \(X\). This principle is used to build \(\hat{H}\) from a classical energy function.

The Hamiltonian operator is

\[ H = \dfrac{P^{2}}{2m} + V(X) \]

where

  • \(H\) is the Hamiltonian operator.
  • \(P\) is the momentum operator.
  • \(m\) is the mass.
  • \(V(X)\) is the potential energy as a function of the position operator.
  • \(X\) is the position operator.

Hamiltonian time evolution. The Hamiltonian generates time evolution through the Schrödinger equation. This principle is used to compute the state at a later time.

The Schrödinger equation is

\[ i\hbar\dfrac{\partial\Psi}{\partial t} = \hat{H}\Psi \]

where

  • \(\hat{H}\) is the Hamiltonian.
  • \(\Psi\) is the state.
  • \(t\) is time.
  • \(\hbar\) is the reduced Planck constant.

Energy eigenvalues. The eigenvalues of \(\hat{H}\) are the allowed energies. This principle is used to find stationary states and discrete spectra.

The energy eigenvalue equation is

\[ \hat{H}\psi = E\psi \]

where

  • \(\psi\) is an energy eigenfunction.
  • \(E\) is the energy eigenvalue.

Note: Also denoted \(H\). Also denoted \(\hat{H}\).

120.1 References

  1. Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — \(H=\dfrac{P^{2}}{2m}+V(X)\).
  2. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — energy observable; Schrödinger evolution.
  3. Das, T. K. Quantum Mechanics: Axiomatic Approach and Understanding Through Mathematics. Springer, 2023. — \(i\hbar\dfrac{\partial\Psi}{\partial t}=\hat{H}\Psi\).