Hamiltonian

Examples of Hamiltonian in the following topics:

• Atomic Structure

  • where $ H$ is the Hamiltonian operator.
  • If the Hamiltonian is independent of time we can solve this equation by
  • We can imagine the operator $H$ as a matrix that multiplies the state vector $\psi$, so this equation is an eigenvalue equation with $E$ as the eigenvalue and $\psi$ as an eigenvector (or eigenfunction) of the matrix (or operator) $H$.The Hamiltonian classically is the sum of the kinetic energy and the potential energy of the particles.

• The Perturbation to the Hamiltonian

  • The Hamiltonian of an electron in a external electromagnetic field is given by
  • Let's suppose that $H_0 \psi_f= E_f \psi_f$ and $H_0 \psi_i = E_i \psi_i $i.e. they are eigenstates of the unperturbed Hamiltonian) we have
  • when and only when ${\bf d}$ operates on two eigenstates of the unperturbed Hamiltonian.

• Quantum Numbers

  • Formally, the dynamics of any quantum system are described by a quantum Hamiltonian (H) applied to the wave equation.
  • There is one quantum number of the system corresponding to the energy—the eigenvalue of the Hamiltonian.
  • There is also one quantum number for each operator (O) that commutes with the Hamiltonian (i.e. satisfies the relation HO = OH).
  • For particles in a time-independent potential, per the Schrödinger equation, it also labels the nth eigenvalue of Hamiltonian (H) (i.e. the energy E with the contribution due to angular momentum, the term involving J2, left out).

• A single electron in a central field

  • This means quantum-mechanically that the Hamiltonian commutes with the angular momentum operator, and that the wavefunctions that satisfy the Hamiltonian also are eigenfunction of the angular momentum operator (${\bf L}={\bf r}\times {\bf p}$).We have

• The H_2^+ Molecular Ion

  • One can prove that the ground state eigenvalue $E$ of the Hamiltonian $H$
  • We can substitute these trial wavefunctions into the Hamiltonian in the second equation in this section to find an upper limit on the value of $E_j({\bf R})$.

• Description of the Hydrogen Atom

  • The eigenstates of the Hamiltonian (that is, the energy eigenstates) can be chosen as simultaneous eigenstates of the angular momentum operator.

• The Transfer of Power between the Federalists and the Republicans

  • For instance, Madison largely wrote the Constitution of 1789 and published prolifically on supporting ratification (the Federalist Papers), but began to vehemently oppose the program of the Hamiltonians and their new Federalist Party from 1789-1800.

• Promoting Economic Development

  • With the support of Washington, the entire Hamiltonian economic program received the necessary support in Congress to be implemented.

• Wave Equation for the Hydrogen Atom

  • The eigenstates of the Hamiltonian (that is, the energy eigenstates) can be chosen as simultaneous eigenstates of the angular momentum operator.

• Domestic Turmoil During the Adams Presidency

  • The "Hamiltonians who surround him," Democratic-Republican Vice President Thomas Jefferson soon remarked, "are only a little less hostile to him than to me."