Time evolution operator

The time evolution operator is a quantum mechanical operator with which the temporal evolution of a physical system can be calculated. The quantum mechanical operator is closely related to the propagator in quantum field or many-body theory . Usually it is written as and describes the evolution of the system from time to time . ${\displaystyle U}$ ${\displaystyle U(t,t_{0})}$ ${\displaystyle t_{0}}$ ${\displaystyle t}$

construction

The time evolution operator can be constructed using the formal exponentiation of the Schrödinger equation or directly using its properties.

About the exponentiation of the Schrödinger equation

The time evolution operator is defined by the time evolution of any state at a point in time up to the point in time : ${\displaystyle U(t,t_{0})}$ ${\displaystyle |\psi \rangle }$ ${\displaystyle t_{0}}$ ${\displaystyle t}$

${\displaystyle |\psi (t)\rangle =U(t,t_{0})\,|\psi (t_{0})\rangle \quad \forall |\psi \rangle }$

Insertion into the Schrödinger equation yields a set of ordinary first-order differential equations :

${\displaystyle \mathrm {i} \hbar {\tfrac {\partial }{\partial t}}U(t,t_{0})=H(t)\,U(t,t_{0})}$

These equations are equivalent to the Schrödinger equation insofar as they describe the extension of the time evolution operator by an infinitesimal time step : ${\displaystyle \delta t}$

${\displaystyle U(t+\delta t,t_{0})=(1-{\tfrac {\mathrm {i} }{\hbar }}H(t)\,\delta t)\,U(t,t_{0})+O(\delta t^{2})}$

with the Hamilton operator , which represents the generator of the time developments. ${\displaystyle H}$

From these equations some properties of can be read off: ${\displaystyle U(t,t_{0})}$

1. Continuity : ${\displaystyle U(t_{0},t_{0})=1}$
2. Unitarity : ${\displaystyle U^{\dagger }(t,t_{0})\,U(t,t_{0})=1}$
3. Propagator property: ${\displaystyle U(t,t_{0})=U(t,t^{\prime })\,U(t^{\prime },t_{0})\quad \forall t^{\prime }}$

Continuity and propagator properties follow directly from the definition of . The unitarity follows from the self adjointness of and ensures the maintenance of the norm and thus the overall probability: ${\displaystyle U}$ ${\displaystyle H}$

${\displaystyle \langle \psi (t)|\psi (t)\rangle =\langle \psi (t_{0})|U^{\dagger }(t,t_{0})\,U(t,t_{0})|\psi (t_{0})\rangle =\langle \psi (t_{0})|\psi (t_{0})\rangle }$

Via the properties of the operator

Conversely, the development of time can also be used as a starting point: The system is defined by a time development operator that uses the above. Criteria 1 to 3 must meet. Then it can be shown that this is generated by a self-adjoint operator , which bridges the gap between the Hamilton operator and the Schrödinger equation. ${\displaystyle U(t,t_{0})}$ ${\displaystyle H(t)}$

Explicit form

From the Schrödinger equation we only get the infinitesimal form of the time evolution operator (see above):

${\displaystyle U(t+\delta t,t)=1-{\tfrac {\mathrm {i} }{\hbar }}H(t)\,\delta t+O(\delta t^{2})}$

Roughly speaking, finite time developments are obtained either by linking an infinite number of infinitesimal time steps or by a series development .

Time-independent systems

If the Hamilton operator is not explicitly time-dependent, an analytical solution for the time evolution operator can be found. For the stringing together of small time developments , a matrix exponential is obtained in the limit value towards 0 : ${\displaystyle H}$ ${\displaystyle \delta t}$ ${\displaystyle \delta t}$

${\displaystyle U(t,t_{0})=\lim _{\delta t\to 0}\left(1-{\tfrac {\mathrm {i} }{\hbar }}H\delta t\right)^{\frac {t-t_{0}}{\delta t}}=\exp \left(-{\tfrac {\mathrm {i} }{\hbar }}H\cdot (t-t_{0})\right)}$

The same result also follows directly from the Schrödinger equation for if the operator is expanded into a power series in and a coefficient comparison is carried out. We then get the power series expansion of the exponential function: ${\displaystyle U}$ ${\displaystyle t-t_{0}}$

${\displaystyle U(t,t_{0})=\sum _{n=0}^{\infty }{\frac {1}{n!}}\left(-{\tfrac {\mathrm {i} }{\hbar }}H\cdot (t-t_{0})\right)^{n}=\exp \left(-{\tfrac {\mathrm {i} }{\hbar }}H\cdot (t-t_{0})\right)}$

${\displaystyle U(t,t_{0})}$is only dependent on the time difference for time-independent systems , which expresses the independence of the system from the choice of the time origin (temporal translation invariance ). ${\displaystyle \Delta t=t-t_{0}}$

For practical calculations one usually uses the spectral representation of the time evolution operator, in which the "impractical" operator in the exponent becomes a phase factor:

${\displaystyle U(t,t_{0})=\sum _{n}\exp \left(-{\tfrac {\mathrm {i} }{\hbar }}E_{n}(t-t_{0})\right)|n\rangle \langle n|}$,

where is the eigenbase of energy ( ). This form fits together with the solution of the time-separated Schrödinger equation. ${\displaystyle {|n\rangle }}$ ${\displaystyle H|n\rangle =E_{n}|n\rangle }$

Explicitly time-dependent systems

If it is time-dependent, then in general only numerical solutions for are possible, unless the Hamilton operator commutes, evaluated at different times, with itself. Then, as in the one-dimensional case, the differential equation can be solved by exponentiation and one obtains: ${\displaystyle H=H(t)}$ ${\displaystyle U}$

${\displaystyle U(t,t_{0})=\exp \left(-{\tfrac {\mathrm {i} }{\hbar }}\int _{t_{0}}^{t}H(\tau )d\tau \right)\quad \Leftrightarrow \quad [H(t),H(t^{\prime })]=0}$

Usually this is not the case and one has to generalize one of the two techniques above for time-independent systems in order to arrive at a solution.

If one considers the time development again as a series of small (in this case not equivalent) time steps , the limit value towards 0 formally defines a product integral according to Vito Volterra : ${\displaystyle \delta t}$ ${\displaystyle \delta t}$

${\displaystyle U(t,t_{0})=\lim _{\delta t\to 0}\prod _{t_{i}}\left(1-{\tfrac {\mathrm {i} }{\hbar }}H(t_{i})\delta t\right)=:\prod _{\tau =t_{0}}^{t}\left(1-{\tfrac {\mathrm {i} }{\hbar }}H(\tau )d\tau \right)}$

This picture of the lined up small time steps ("time slicing") is an essential ingredient for the definition of the path integral . For practical calculations, time-dependent perturbation theory is usually used , in which a series expansion is used in the Schrödinger equation in integral form. The disturbance series then results in the so-called Dyson series : ${\displaystyle U}$

${\displaystyle U(t,t_{0})=\sum _{n=0}^{\infty }\left(-{\frac {\mathrm {i} }{\hbar }}\right)^{n}\int _{t_{0}}^{t}dt_{1}H(t_{1})\int _{t_{0}}^{t_{1}}dt_{2}H(t_{2})\cdots \int _{t_{0}}^{t_{n-1}}dt_{n}\;H(t_{n})}$

${\displaystyle =\sum _{n=0}^{\infty }{\frac {1}{n!}}\left(-{\frac {\mathrm {i} }{\hbar }}\right)^{n}\int _{t_{0}}^{t}dt_{1}\cdots \int _{t_{0}}^{t}dt_{n}\;T\left[H(t_{1})\cdots H(t_{n})\right]}$

${\displaystyle =:T\left[\exp \left(-{\frac {\mathrm {i} }{\hbar }}\int _{t_{0}}^{t}H(\tau )d\tau \right)\right].}$

with the timing operator . In the one-dimensional case, this operator is trivial and the above equation is obtained again. While for many systems one can restrict oneself to a few terms of the perturbation series, there are some systems, such as non- Fermi liquids , in which the series does not converge. ${\displaystyle T}$

literature

• JJ Sakurai: Modern Quantum Mechanics . Prentice Hall, 1993. ISBN 978-0201539295

Remarks

1. ↑ The exponential function for operators is often also defined via its power series
2. ↑ In this form, the formula is only valid for countable bases. However, it can also be generalized to continuous spectra (see also spectral theorem ).