Matrix mechanics
The matrix mechanics is a technology developed by the German physicist Werner Heisenberg, Max Born and Pascual Jordan formulation of quantum mechanics. It forms the counterpart to the embossed by Erwin Schrödinger wave mechanics.
1925 Heisenberg developed a treatise On the quantum- theoretical reinterpretation of kinematic and mechanical relations to clarify inconsistencies of quantum theory on the way to a non-classical atomic theory, and thus created a basis of a strictly valid quantum mechanics. Initial hypothesis was that in the microphysics should not be researched in the atom according to paths or cycle times of the electrons, but according to measurable differences in the frequencies of radiation and spectral line intensities to own it "one of the classical mechanics form analogous quantum- theoretical mechanics, in which occur only relations between observable quantities (Q4 -66) ".
Matrix mechanics was Elaborated then developed by Max Born, Werner Heisenberg and Pascual Jordan in a paper for the Journal of Physics in 1926, the so-called "three men's work ". In this view, quantum mechanics, the state vector of a system does not change with time. Instead, the dynamics of the system is limited only by the time dependence of the operators ( " matrices " ) described (see Heisenberg picture).
In a way, the matrix mechanics provides a more natural and more fundamental description of a quantum system as the wave-mechanical Schrödinger picture, especially for relativistic theories, as it brings the Lorentz invariance with it. It also has a strong formal similarity to classical mechanics, because the Heisenberg equations of motion are similar to the classical Hamiltonian equations of motion.
The physical predictions concerning the Schrödinger and Heisenberg's mechanics equivalent. This equivalence was first proved by Schrödinger, then Pauli, Eckhart, Dirac, Jordan, and by von Neumann in different ways.
General matrix representation of quantum mechanics
The following is to be derived from an abstract Hilbert space vector and an operator on this Hilbert space whose vector or matrix representation.
First you choose in describing the system Hilbert space has a base ( complete orthonormal normal system ), the dimension of the Hilbert space is countable.
In the following scalar product you push twice, type 1 by exploiting the completeness of the basis and:
Due to the projections onto the basis vectors, the coordinate representation is obtained with vectors and matrices related to:
- Bra- row vector: ( can also be complex -conjugated, transposed column vector write )
- Operator matrix:
- Ket - column vector:
One adjunction corresponds to the matrix representation of a complex conjugation and an additional transposition:
Are the basis vectors of the eigenvectors of an operator, that is, it is the matrix representation of the operator with respect to the base diagonal:
Matrix representation of the Heisenberg picture
Heisenberg equation of motion
In the Heisenberg picture the states of time-independent and the operators are time dependent. The time-dependence of an operator is given by the Heisenberg 's equation of motion:
In the matrix representation with respect to an arbitrary basis, this means that the time-independent vectors and matrices are time dependent. As of now, the summation convention is used.
With regard to the energy eigenbasis simplifies the presentation because the Hamiltonian is diagonal ( the Hamiltonian is explicitly time -independent):
Solution of the equation for special cases
If it is not explicitly time dependent ( ), the time evolution is given by
This is the time evolution operator and the adjoint time evolution operator.
If in addition the Hamiltonian is not explicitly time dependent ( ), the time evolution operator takes the simple form:
In a matrix representation with respect to any base ( the exponential of the exponential function is the same as matrices by means of operators evaluate number representation):
Regarding energy eigenbasis the time evolution is simple again:
By inserting it verifies that solves this equation is the Heisenberg's equation of motion.