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Time-Dependent Expectation Values from Integral Equations for Quantum Flux and Probability Densities.
P Schürger1, K Renziehausen2,3, T Schaupp1
1Institut für Physikalische und Theoretische Chemie, Universität Würzburg, Emil-Fischer-Strasse 42, 97074 Würzburg, Germany.
The Journal of Physical Chemistry. A
|November 22, 2022
Summary
Calculating quantum expectation values can be done using probability density or flux density integrals. Both methods yield identical results with exact wave functions, but approximations can cause discrepancies, impacting Born-Oppenheimer approximation validity.
Area of Science:
- Quantum mechanics
- Computational chemistry
- Theoretical physics
Background:
- Calculating time-dependent quantum expectation values is crucial for understanding molecular dynamics.
- Different computational approaches exist, utilizing probability density or flux density.
- The accuracy of these calculations can be affected by approximations in the wave function.
Purpose of the Study:
- To compare two distinct methods for calculating time-dependent quantum expectation values.
- To investigate the impact of wave function approximations on these calculations.
- To assess the validity of the Born-Oppenheimer approximation in this context.
Main Methods:
- Calculated expectation values using integrals over probability density functions.
- Calculated expectation values using integrals over probability flux density functions.
- Employed the adiabatic expansion of the total wave function for decomposition.
Main Results:
- Identical results were obtained when using exact wave functions for both integration methods.
- Discrepancies arose when approximate wave functions were used.
- The Born-Oppenheimer approximation's validity was discussed in relation to both calculation methods.
Conclusions:
- The choice of integrand (probability density vs. flux density) does not affect the accuracy of expectation value calculations with exact wave functions.
- Wave function approximations necessitate careful consideration of the chosen calculation method.
- The study highlights the importance of accurate wave functions for reliable quantum mechanical calculations, particularly concerning the Born-Oppenheimer approximation.
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