Von Neumann equation

The Von Neumann equation (after John von Neumann ) represents the quantum mechanical analogue to the Liouville equation of classical statistical mechanics . It describes the development of the density operator over time in the Schrödinger picture : ${\displaystyle {\hat {\rho }}}$

${\displaystyle {\frac {\partial {\hat {\rho }}}{\partial t}}=-{\frac {i}{\hbar }}\left[{\hat {H}},{\hat {\rho }}\right]}$

It is

• ${\displaystyle {\hat {H}}}$the Hamilton operator of the system
• ${\displaystyle [{\hat {H}},{\hat {\rho }}]={\hat {H}}{\hat {\rho }}-{\hat {\rho }}{\hat {H}}}$a commutator .

The density operator is . In this case, referred to the probability , in a mixture of the pure state to be measured, if the states of orthogonal are. The trace of a density operator equals 1 because . ${\displaystyle {\hat {\rho }}=\sum \nolimits _{k}\,p_{k}|\psi _{k}\rangle \langle \psi _{k}|}$ ${\displaystyle p_{k}}$ ${\displaystyle |\psi _{k}\rangle }$ ${\displaystyle |\psi \rangle }$ ${\displaystyle \operatorname {Tr} ({\hat {\rho }})}$ ${\displaystyle \operatorname {Tr} ({\hat {\rho }})=\sum \nolimits _{k}p_{k}=1}$

discussion

The general solution of the Von Neumann equation is, using the time evolution operator and its adjoint operator : ${\displaystyle {\hat {U}}(t)}$ ${\displaystyle {\hat {U}}^{\dagger }(t)}$

${\displaystyle {\hat {\rho }}(t)={\hat {U}}(t){\hat {\rho }}(0){\hat {U}}^{\dagger }(t)}$

The density operator is stationary when it is exchanged with the Hamilton operator . ${\displaystyle {\tfrac {\partial }{\partial t}}{\hat {\rho }}=0}$ ${\displaystyle \left[{\hat {H}},{\hat {\rho }}\right]=0}$

With the help of the Von Neumann equation one can show that the trace of the quadratic density operator is constant over time:

${\displaystyle \mathrm {Tr} ({\hat {\rho }}^{2}(t))=\mathrm {Tr} ({\hat {\rho }}(t){\hat {\rho }}(t))=\mathrm {Tr} ({\hat {U}}(t){\hat {\rho }}(0)\underbrace {{\hat {U}}^{\dagger }(t){\hat {U}}(t)} _{=1}{\hat {\rho }}(0){\hat {U}}^{\dagger }(t))=\mathrm {Tr} (\underbrace {{\hat {U}}^{\dagger }(t){\hat {U}}(t)} _{=1}{\hat {\rho }}(0){\hat {\rho }}(0))=\mathrm {Tr} ({\hat {\rho }}^{2}(0))}$

${\displaystyle \Rightarrow \ {\frac {\partial }{\partial t}}\operatorname {Tr} \left({\hat {\rho }}^{2}\right)=0}$

The cyclical invariance of the track was used in the penultimate step. Because of equality if and only when describing a pure state, it follows that pure states remain pure and mixed mixed. ${\displaystyle \operatorname {Tr} \left({\hat {\rho }}^{2}\right)\leq 1}$ ${\displaystyle \rho }$

Expected values ​​of operators are expressed by. The time dependence of the expected values ${\displaystyle \langle {\hat {A}}\rangle =\mathrm {Tr} ({\hat {\rho }}{\hat {A}})}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\langle {\hat {A}}\rangle ={\frac {\mathrm {d} }{\mathrm {d} t}}\mathrm {Tr} ({\hat {\rho }}{\hat {A}})=\mathrm {Tr} \left({\frac {\partial {\hat {\rho }}}{\partial t}}{\hat {A}}+{\hat {\rho }}{\frac {\partial {\hat {A}}}{\partial t}}\right)=\mathrm {Tr} \left(-{\frac {i}{\hbar }}\left[{\hat {H}},{\hat {\rho }}\right]{\hat {A}}+{\hat {\rho }}{\frac {\partial {\hat {A}}}{\partial t}}\right)}$

is the same in the stationary case:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\langle {\hat {A}}\rangle =\mathrm {Tr} \left({\hat {\rho }}{\frac {\partial {\hat {A}}}{\partial t}}\right)=\left\langle {\frac {\partial {\hat {A}}}{\partial t}}\right\rangle }$

The expected value of a measurement of time-independent observables is time-independent in the stationary case . ${\displaystyle {\tfrac {\partial }{\partial t}}{\hat {A}}=0}$ ${\displaystyle {\tfrac {\mathrm {d} }{\mathrm {d} t}}\langle {\hat {A}}\rangle =0}$

Derivation

The Von Neumann equation can be derived from the Schrödinger equation.

The partial derivative of the statistical operator is formed, taking into account the product rule:

${\displaystyle {\frac {\partial }{\partial t}}{\hat {\rho }}=\sum _{k}p_{k}\left({\frac {\partial }{\partial t}}|\psi _{k}\rangle \right)\langle \psi _{k}|+\sum _{k}p_{k}|\psi _{k}\rangle \left({\frac {\partial }{\partial t}}\langle \psi _{k}|\right)}$

The Schrödinger equation for Hilbert space vectors (Ket) reads

${\displaystyle i\hbar {\frac {\partial }{\partial t}}|\psi \rangle ={\hat {H}}|\psi \rangle \quad \Rightarrow \quad {\frac {\partial }{\partial t}}|\psi \rangle =-{\frac {i}{\hbar }}{\hat {H}}|\psi \rangle }$

and for dual Hilbert space vectors (Bra)

${\displaystyle -i\hbar {\frac {\partial }{\partial t}}\langle \psi |=\langle \psi |{\hat {H}}\quad \Rightarrow \quad {\frac {\partial }{\partial t}}\langle \psi |={\frac {i}{\hbar }}\langle \psi |{\hat {H}}}$

This is used above:

${\displaystyle {\frac {\partial }{\partial t}}{\hat {\rho }}=\sum _{k}p_{k}\left(-{\frac {i}{\hbar }}{\hat {H}}|\psi _{k}\rangle \right)\langle \psi _{k}|+\sum _{k}p_{k}|\psi _{k}\rangle \left({\frac {i}{\hbar }}\langle \psi _{k}|{\hat {H}}\right)}$

Simplifying delivers the Von Neumann equation:

${\displaystyle {\frac {\partial }{\partial t}}{\hat {\rho }}=-{\frac {i}{\hbar }}\left({\hat {H}}\sum _{k}p_{k}|\psi _{k}\rangle \langle \psi _{k}|-\sum _{k}p_{k}|\psi _{k}\rangle \langle \psi _{k}|{\hat {H}}\right)=-{\frac {i}{\hbar }}\left({\hat {H}}{\hat {\rho }}-{\hat {\rho }}{\hat {H}}\right)=-{\frac {i}{\hbar }}\left[{\hat {H}},{\hat {\rho }}\right]}$

This result obtained in the Schrödinger picture for the density operator of a closed quantum system must not be compared with Heisenberg's equation of motion for an operator that is not explicitly time-dependent

${\displaystyle {\frac {\mathrm {d} {\hat {A}}_{\rm {H}}}{\mathrm {d} t}}={\frac {\mathrm {i} }{\hbar }}\left[{\hat {H}}_{\rm {H}},{\hat {A}}_{\rm {H}}\right]}$

which describes the time evolution of observables and only formally agrees with the Von Neumann equation except for one sign.

The formal similarity of the equations is explained by the fact that the observables in the Heisenberg picture form the C * -algebra of the bounded linear operators, whereas the space of the density operators (as trace class operators ) corresponds to the predual of this C * -algebra. In the case of a concrete Hilbert space representation , the duality of vector space and the corresponding dual space description in the one-parameter unitary group dynamics always implies a different sign of the time parameter, which, due to the time derivative on the respective left side of the Heisenberg or Von-Neumann equation, has a different sign.

This difference becomes particularly clear if, analogously to the derivation method above, the Heisenberg equation is obtained from the Schrödinger equation, which is always possible for quantum systems with finite-dimensional Hilbert space.

literature

• Franz Schwabl : Quantum Mechanics (QM I) . 5th expanded edition. Springer, Berlin et al. 1998, ISBN 3-540-63779-6 (Springer textbook).
• Ola Bratteli : Operator Algebras and Quantum Statistical Mechanics 1 . 2nd Edition. Springer, Berlin et al. 1987, ISBN 3-540-17093-6 (English, Springer textbook).