Dyson series

The Dyson series is a by Freeman Dyson named series which, in the time-dependent perturbation in the quantum mechanics and in the quantum electrodynamics occurs. In quantum electrodynamics the series is divergent, but leads to results that agree well with the experiment if it is broken off after a finite number of terms.

Wave function representation

In the interaction picture of quantum mechanics is split the Hamiltonian in an undisturbed part and an interaction term to: . The interaction picture (index for interaction ) is related to the Schrödinger picture (without index) as follows: ${\displaystyle H}$ ${\displaystyle H_{0}}$ ${\displaystyle V}$ ${\displaystyle H=H_{0}+V}$ ${\displaystyle {\text{I}}}$

• Conditions: ${\displaystyle \psi _{\text{I}}(t):=e^{\mathrm {i} H_{0}t/\hbar }\;\psi (t)}$
• Disturbance operator: ${\displaystyle V_{\text{I}}(t):=e^{\mathrm {i} H_{0}t/\hbar }\;V(t)\;e^{-\mathrm {i} H_{0}t/\hbar }}$

If you now derive the definition equation for after time and insert the Schrödinger equation , the result for the time evolution equation is: ${\displaystyle \psi _{\text{I}}(t)}$ ${\displaystyle t}$

${\displaystyle \mathrm {i} \hbar {\frac {\partial }{\partial t}}\psi _{\text{I}}(t)=V_{\text{I}}(t)\psi _{\text{I}}(t)\;\;.}$

A time integration gives the integral equation:

${\displaystyle \psi _{\text{I}}(t)=\psi _{\text{I}}(t_{0})+{\frac {1}{\mathrm {i} \hbar }}\int _{t_{0}}^{t}\mathrm {d} t_{1}\,V_{\text{I}}(t_{1})\,\psi _{\text{I}}(t_{1})\;\;.}$

This can be read as the recursion equation for . Iterative substitution of yields: ${\displaystyle \psi _{\text{I}}(t)}$ ${\displaystyle \psi _{\text{I}}(t_{(n)})}$

${\displaystyle \psi _{\text{I}}(t)=\psi _{\text{I}}(t_{0})+{\frac {1}{\mathrm {i} \hbar }}\int _{t_{0}}^{t}\mathrm {d} t_{1}\,V_{\text{I}}(t_{1})\,\psi _{\text{I}}(t_{0})+{\frac {1}{(\mathrm {i} \hbar )^{2}}}\int _{t_{0}}^{t}\mathrm {d} t_{1}\int _{t_{0}}^{t_{1}}\mathrm {d} t_{2}\,V_{\text{I}}(t_{1})\,V_{\text{I}}(t_{2})\,\psi _{\text{I}}(t_{0})+\ldots \;\;.}$

In general, commute and not . That is why the time order in the integration area is important. Due to the symmetry of the integrand, it is still possible to run all integrals from to instead . Then, however, the integrand has to be time-ordered afterwards and the nth summand corrected by the factor ( stands for the time-ordering operator): ${\displaystyle V_{\text{I}}(t_{1})}$ ${\displaystyle V_{\text{I}}(t_{2})}$ ${\displaystyle [V_{\text{I}}(t_{1}),V_{\text{I}}(t_{2})]\neq 0}$ ${\displaystyle t_{0}\leq t_{1}\leq t_{2}\leq t}$ ${\displaystyle t_{0}}$ ${\displaystyle t}$ ${\displaystyle 1/n!}$ ${\displaystyle {\mathcal {T}}}$

${\displaystyle \psi _{\text{I}}(t)=\psi _{\text{I}}(t_{0})+{\frac {1}{\mathrm {i} \hbar }}\int _{t_{0}}^{t}\mathrm {d} t_{1}\,V_{\text{I}}(t_{1})\,\psi _{\text{I}}(t_{0})+{\frac {1}{(\mathrm {i} \hbar )^{2}}}{\frac {1}{2!}}\int _{t_{0}}^{t}\mathrm {d} t_{1}\int _{t_{0}}^{t}\mathrm {d} t_{2}\,{\mathcal {T}}\,V_{\text{I}}(t_{1})\,V_{\text{I}}(t_{2})\,\psi _{\text{I}}(t_{0})+\ldots \;\;.}$

This is the Dyson series for wave functions :

${\displaystyle \psi _{\text{I}}(t)=\sum _{n=0}^{\infty }{\frac {1}{(\mathrm {i} \hbar )^{n}}}{\frac {1}{n!}}\int _{t_{0}}^{t}\mathrm {d} t_{1}\int _{t_{0}}^{t}\mathrm {d} t_{2}\cdots \int _{t_{0}}^{t}\mathrm {d} t_{n}\,{\mathcal {T}}\,\left(\prod _{k=1}^{n}V_{\text{I}}(t_{k})\right)\psi _{\text{I}}(t_{0})\;\;.}$

The overlap elements are also useful for perturbation theory:

${\displaystyle \langle \psi (t)\mid \psi (t_{0})\rangle =\sum _{n=0}^{\infty }{\frac {1}{(\mathrm {i} \hbar )^{n}}}\underbrace {\int \mathrm {d} t_{1}\cdots \mathrm {d} t_{n}} _{t\geq t_{1}\geq \cdots \geq t_{n}\geq t_{0}}\,\langle \psi (t)|e^{-\mathrm {i} H_{0}(t-t_{1})}Ve^{-\mathrm {i} H_{0}(t_{1}-t_{2})}\cdots Ve^{-\mathrm {i} H_{0}(t_{n}-t_{0})}|\psi (t_{0})\rangle \;.}$

Due to the above definition of , the scalar products in the interaction image and the Schrödinger image are identical. ${\displaystyle \psi _{\text{I}}(t)}$

Operator representation

It is also possible to develop the Dyson series directly with the time expansion operator without using wave functions explicitly. is also called propagator - or in this context also Dyson operator. By definition, the time evolution operator applied to a wave function at time yields the wave function at time : ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle U(t,t_{0})}$ ${\displaystyle t_{0}}$ ${\displaystyle t}$

${\displaystyle \psi _{\text{I}}(t)=U_{\text{I}}(t,t_{0})\psi _{\text{I}}(t_{0})}$

If you put this into the above time evolution equation, you get, assuming that this applies to every wave function : ${\displaystyle \psi }$

${\displaystyle \mathrm {i} \hbar {\frac {\partial }{\partial t}}U_{\text{I}}(t,t_{0})=V_{\text{I}}(t)U_{\text{I}}(t,t_{0})\;\;.}$

A recursion equation is then obtained through formal integration analogous to the above:

${\displaystyle U_{\text{I}}(t,t_{0})=1+{\frac {1}{\mathrm {i} \hbar }}\int _{t_{0}}^{t}\mathrm {d} t_{1}\,V_{\text{I}}(t_{1})\,U_{\text{I}}(t_{1},t_{0})\;\;.}$

With an analogous derivation (see article on time-dependent perturbation theory ), this leads to the Dyson series for the time evolution operator :

${\displaystyle U_{\text{I}}(t,t_{0})={\mathcal {T}}\,e^{{\frac {1}{\mathrm {i} \hbar }}\int _{t_{0}}^{t}V_{\text{I}}(t_{1})\,\mathrm {d} t_{1}}.}$

In the Schrödinger picture the following applies accordingly:

${\displaystyle U(t,t_{0})={\mathcal {T}}\,e^{{\frac {1}{\mathrm {i} \hbar }}\int _{t_{0}}^{t}H(t_{1})\,\mathrm {d} t_{1}}.}$

literature

• Franz Schwabl, Quantum Mechanics (QM I). An introduction , 7th edition, Springer Verlag, Munich 2007, ISBN 978-3-540-73674-5 . Chapter 16.3.1 Fault development

Individual evidence

1. ^ FJ Dyson: The Radiation Theories of Tomonaga, Schwinger, and Feynman . In: Phys. Rev. . 75, No. 3, 1949, pp. 486-502. doi : 10.1103 / PhysRev.75.486 .
2. ^ FJ Dyson: The S Matrix in Quantum Electrodynamics . In: Phys. Rev. . 75, No. 11, 1949, pp. 1736-1755. doi : 10.1103 / PhysRev.75.1736 .
3. ^ FJ Dyson: Divergence of Perturbation Theory in Quantum Electrodynamics . In: Phys. Rev. . 85, No. 4, 1952, pp. 631-632. doi : 10.1103 / PhysRev.85.631 .