Die mechanics

The matrix mechanics is by the German physicist Werner Heisenberg , Max Born and Pascual Jordan developed formulation of quantum mechanics . It forms the counterpart to the equivalent wave mechanics coined by Erwin Schrödinger . In a certain way, matrix mechanics (see also Heisenberg picture ) offers a more natural and fundamental description of a quantum mechanical system than the wave-mechanical Schrödinger picture , especially for relativistic theories, since it brings with it Lorentz invariance . It also shows a strong formal resemblance to classical mechanics because Heisenberg's equations of motion are similar to classical Hamilton's equations of motion .

history

In 1925 Heisenberg wrote a treatise on the quantum theoretical reinterpretation of kinematic and mechanical relationships in order to clarify inconsistencies in quantum theory on the way to a non-classical atomic theory, and thus created a basis for strictly valid quantum mechanics. The starting thesis was that research in microphysics does not have to be done for the orbits or times of rotation of the electrons in the atom , but for measurable differences in the radiation frequencies and spectral line intensities in order to develop “a quantum theoretical mechanics analogous to classical mechanics, in which only relationships between observable quantities occur (Q4-66) ".

The matrix mechanics was then worked out jointly by Max Born, Werner Heisenberg and Pascual Jordan in a publication for the Zeitschrift für Physik in 1926, the so-called "three-man work" . In this way of looking at quantum mechanics, the state vector of a system does not change over time. Instead, the dynamics of the system are only described by the time dependence of the operators ("matrices").

Regarding the physical predictions, the Schrödinger and Heisenberg mechanics are equivalent. This equivalence was first demonstrated by Schrödinger , then also by Pauli , Eckart , Dirac , Jordan and von Neumann in different ways.

General matrix representation of quantum mechanics

In the following, the vector or matrix representation is to be derived from an abstract Hilbert space vector and an operator on this Hilbert space .

First choose a basis (complete orthonormal system) in the Hilbert space describing the system , whereby the dimension of the Hilbert space is countable. ${\displaystyle \left\{|n\rangle \right\}}$

For the following scalar product, insert a 1 twice using the completeness of the basis and : ${\displaystyle \langle \phi |{\hat {A}}|\psi \rangle }$ ${\displaystyle 1=\sum _{m}|m\rangle \langle m|}$ ${\displaystyle 1=\sum _{n}|n\rangle \langle n|}$

${\displaystyle \langle \phi |{\hat {A}}|\psi \rangle =\sum _{m,n}\underbrace {\langle \phi |m\rangle } _{\phi _{m}}\underbrace {\langle m|{\hat {A}}|n\rangle } _{A_{mn}}\underbrace {\langle n|\psi \rangle } _{\psi _{n}}}$

The projections onto the base vectors give the coordinate representation with vectors and matrices with regard to : ${\displaystyle \left\{|n\rangle \right\}}$

• Bra row vector: (can also be written as a complex-conjugated, transposed column vector) ${\displaystyle \phi _{m}=\langle \phi |m\rangle =\langle m|\phi \rangle ^{*}}$

${\displaystyle \left(\langle \phi |\right)_{\left\{|n\rangle \right\}}=\left(\langle \phi |1\rangle ,\,\langle \phi |2\rangle ,\,\langle \phi |3\rangle ,\,\ldots \right)=\left({\begin{array}{c}\langle 1|\phi \rangle ^{*}\\\langle 2|\phi \rangle ^{*}\\\vdots \end{array}}\right)^{t}}$

• Operator matrix: ${\displaystyle A_{mn}=\langle m|{\hat {A}}|n\rangle }$

${\displaystyle \left({\hat {A}}\right)_{\left\{|n\rangle \right\}}=\left({\begin{array}{ccc}\langle 1|{\hat {A}}|1\rangle &\langle 1|{\hat {A}}|2\rangle &\cdots \\\langle 2|{\hat {A}}|1\rangle &\langle 2|{\hat {A}}|2\rangle \\\vdots &&\ddots \end{array}}\right)}$

• Ket column vector: ${\displaystyle \psi _{n}=\langle n|\psi \rangle }$

${\displaystyle \left(|\psi \rangle \right)_{\left\{|n\rangle \right\}}=\left({\begin{array}{c}\langle 1|\psi \rangle \\\langle 2|\psi \rangle \\\vdots \end{array}}\right)}$

In the matrix representation, an adjunction corresponds to a complex conjugation and an additional transposition: ${\displaystyle A_{mn}^{\dagger }=A_{nm}^{*}}$

${\displaystyle \left({\hat {A}}^{\dagger }\right)_{\left\{|n\rangle \right\}}=\left({\hat {A}}^{*t}\right)_{\left\{|n\rangle \right\}}}$

If the basis vectors are eigenvectors of an operator , then the matrix representation of the operator with regard to this basis is diagonal: ${\displaystyle |n\rangle }$ ${\displaystyle {\hat {O}}}$ ${\displaystyle {\hat {O}}|n\rangle =o_{n}|n\rangle }$

${\displaystyle O_{mn}=\langle m|{\hat {O}}|n\rangle =o_{n}\langle m|n\rangle =o_{n}\delta _{mn}}$

Matrix representation of the Heisenberg picture

Heisenberg's equation of motion

In the Heisenberg picture the states are independent of time and the operators are time-dependent. The time dependence of an operator is given by the Heisenberg equation of motion :

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\hat {A}}^{\rm {H}}={\frac {\mathrm {i} }{\hbar }}\left[{\hat {H}},{\hat {A}}^{H}\right]+{\hat {U}}(t)\left({\frac {\partial }{\partial t}}{\hat {A}}^{S}\right){\hat {U}}^{\dagger }(t)}$

Here is the operator for the unitary transformation from the Schrödinger image to the Heisenberg image and the commutator . In the matrix representation with respect to any base, this means that the vectors are time-independent and the matrices are time-dependent. The summation convention is used from now on . ${\displaystyle {\hat {U}}(t)=\exp \left({\frac {\mathrm {i} }{\hbar }}{\hat {H}}t\right)}$ ${\displaystyle [\cdot ,\cdot ]}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}A_{\rm {nm}}^{H}={\frac {\mathrm {i} }{\hbar }}\left(H_{nk}^{H}A_{km}^{H}-A_{nk}^{H}H_{km}^{H}\right)+U_{nk}\left({\frac {\partial }{\partial t}}A_{\rm {kl}}^{S}\right)U_{lm}^{\dagger }}$

With regard to the eigenbase of energy, the representation is simplified because the Hamilton operator is diagonal (let the Hamilton operator be explicitly independent of time ): ${\displaystyle {\tfrac {\partial }{\partial t}}H=0}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}A_{\rm {nm}}^{H}={\frac {\mathrm {i} }{\hbar }}\left(E_{n}\delta _{nk}A_{km}^{H}-A_{nk}^{H}\delta _{km}E_{m}\right)+U_{nk}\left({\frac {\partial }{\partial t}}A_{\rm {kl}}^{S}\right)U_{lm}^{\dagger }={\frac {\mathrm {i} }{\hbar }}\left(E_{n}-E_{m}\right)A_{nm}^{H}+U_{nk}\left({\frac {\partial }{\partial t}}A_{\rm {kl}}^{S}\right)U_{lm}^{\dagger }}$

Solution of the equation for special cases

If is not explicitly time-dependent ( ), the development over time is given by ${\displaystyle {\hat {A}}}$ ${\displaystyle {\tfrac {\partial }{\partial t}}A^{H}=0}$

${\displaystyle {\hat {A}}^{H}(t)={\hat {U}}(t)^{\dagger }{\hat {A}}^{H}(0){\hat {U}}(t).}$

Here is the time evolution operator and the adjoint time evolution operator. ${\displaystyle {\hat {U}}(t)}$ ${\displaystyle {\hat {U}}(t)^{\dagger }}$

If, in addition, the Hamilton operator is not explicitly time-dependent ( ), the time evolution operator takes the simple form : ${\displaystyle {\tfrac {\partial }{\partial t}}H=0}$ ${\displaystyle U(t)=\exp \left(-{\tfrac {\mathrm {i} }{\hbar }}H\,t\right)}$

${\displaystyle {\hat {A}}^{H}(t)=\exp \left({\tfrac {\mathrm {i} }{\hbar }}H\,t\right){\hat {A}}^{H}(0)\,\exp \left(-{\tfrac {\mathrm {i} }{\hbar }}H\,t\right)}$

In matrix representation with respect to any basis (the exponential function of matrices, like the exponential function of operators, can be evaluated using a series representation):

${\displaystyle A_{\rm {nm}}^{H}(t)=\exp \left({\tfrac {\mathrm {i} }{\hbar }}H\,t\right)_{nk}\,A_{kl}^{H}(0)\,\exp \left(-{\tfrac {\mathrm {i} }{\hbar }}H\,t\right)_{lm}}$

With regard to the inherent energy basis, the time development becomes easier again:

${\displaystyle A_{\rm {nm}}^{H}(t)=\exp \left({\tfrac {\mathrm {i} }{\hbar }}E_{n}\,t\right)\delta _{nk}\,A_{kl}^{H}(0)\,\delta _{lm}\exp \left(-{\tfrac {\mathrm {i} }{\hbar }}E_{m}\,t\right)=\exp \left({\tfrac {\mathrm {i} }{\hbar }}\left(E_{n}-E_{m}\right)t\right)A_{nm}^{H}(0)}$

By plugging in, one checks that this equation solves Heisenberg's equation of motion . ${\displaystyle {\tfrac {\mathrm {d} }{\mathrm {d} t}}A_{\rm {nm}}^{H}={\tfrac {\mathrm {i} }{\hbar }}\left(E_{n}-E_{m}\right)A_{nm}^{H}}$

Individual evidence

1. ↑ John Gribbin: In search of Schrödinger's cat , quantum physics and reality - translated from the English by Friedrich Griese; Scientific advice for the German edition: Helmut Rechenberg. - 2nd edition, 6.-8. Tausend - Piper, Munich 1987 - p. 129

Web links

• Reconstruction of the mathematical derivation
• Max Born's biography on Wissenschaft-online.de