Hamiltonian (quantum mechanics)

The Hamiltonian determined in quantum mechanics the time evolution and the possible energy measurements, it is therefore the energy operator. It provides, for example, the energy levels of the electron in the hydrogen atom. It is named after William Rowan Hamilton. On him the Hamiltonian formulation of classical mechanics goes back, in which the Hamiltonian function determines the time evolution and energy.

Time and energy development

In quantum mechanics, each state of the considered physical system is represented by an associated vector in Hilbert space. Its time evolution is determined by the Schrödinger equation by the Hamiltonian:

With

• The imaginary unit
• The reduced Planck 's constant

The Hamiltonian is obtained in many cases by so-called canonical quantization of the Hamiltonian of the corresponding classical system ( with the generalized coordinate x and the canonical momentum p ). For this purpose the algebraic expression is read for the Hamiltonian as a function of operators ( position operator and momentum operator ) satisfying the canonical commutation relations. However, this is not unique, since the function has the value, the operator function but the value addition is real, but is Hermitian. In addition, there are quantum mechanical quantities such as the spin that do not occur in classical physics. How do they affect the time evolution does not follow from analogies with classical physics, but must be inferred from the physical findings.

The eigenvalue equation

Determines the eigenvectors of the Hamiltonian. They are stationary in time- independent Hamiltonian, ie time-independent in every observable property. The eigenvalues ​​are the corresponding energies.

Since the Hamiltonian is Hermitian ( precisely essentially self-adjoint ), the spectral theorem implies that the energies are real and that the eigenvectors form an orthonormal basis of the Hilbert space. Depending on the system, the energy spectrum may be discrete or continuous. Some systems, such as the hydrogen atom or a particle in the potential well, have a down limited, discrete spectrum and over a continuum of possible energies.

The Hamiltonian generates unitary time evolution. If so effected for all times and between and the Hamiltonian commutes with

The unitary map each initial condition on the associated state at time

If the Hamiltonian does not depend on the time, this simplifies to

Operators that commute with, for time -independent Hamiltonian are conserved quantities of the system. In particular, the energy is conserved.

For energy, a energy-time uncertainty principle applies, only one has a different approach than for example in the place -momentum uncertainty principle in quantum mechanics in their derivation.

Examples

Quantum mechanical particle in the potential

From the Hamiltonian

For a nonrelativistic, classical particle of mass moving in the potential, a Hamiltonian can be read. For this purpose, the expressions for the momentum and the potential of the corresponding operators are replaced:

In the position representation of the momentum operator acts as a derivation and the operator as multiplication by the function The application of this Hamiltonian of a point particle of mass in the potential on the spatial wave function of the particle is then translated by

Here, the Laplace operator.

The Schrödinger equation is thus

This Schrödinger equation of a point mass in the potential is the basis for the explanation of the tunnel effect. It provides for inserting the Coulomb potential ( a potential for interaction between an electron and a proton), the spectral lines of the hydrogen atom. By insertion of appropriate potentials and the other spectral light atoms can be calculated.

One-dimensional harmonic oscillator

Analogously, one obtains for the quantum harmonic oscillator, which can only move along a line, the Hamiltonian

The energies can be determined algebraically. obtained

It is at the same energy as that of a ground state energy, the one -way quantum of energy has been added.

Spin in a magnetic field

For spin of an electron bound to an atom and is located in an unpaired state ( alone in the electron cloud) in the magnetic field, is one of the Hamiltonian

It is

• The gyromagnetic ratio of the electron
• The spin operator.

Since the spin in the direction of the magnetic field only the eigenvalues ​​or may assume (spin polarization) are the potential energies. In the inhomogeneous magnetic field of the Stern-Gerlach experiment therefore cleaves a particle beam of silver atoms into two partial beams.

Loaded, spinless particle in an electromagnetic field

The Hamiltonian of a particle with charge in an external electromagnetic field is obtained by minimal substitution

Referred to here

• The vector potential
• The scalar potential.

In multiplying the bracket is to be noted that the operators only when Coulomb gauge exchange.