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Using a variational method to obtain the ground state of the quantum Hamiltonian: symbolic computation approach. (English) Zbl 1421.81043

Summary: The importance of learning symbolic computation in research in theoretical sciences cannot be overemphasised. While Fortran and/or C programming laboratories have become an essential part of MSc curricula now, symbolic computing is almost never taught. We demonstrate how a freeware, SAGE, can be employed for the variational solution of simple (or complex) Hamiltonians encountered in quantum mechanics in one dimension. One can easily change the trial wavefunction and the Hamiltonian and obtain estimates of ground state energy. This should lead to a qualitative understanding of the physics of the problem. A brief extension to the first excited state for potentials with parity is discussed. Finally, we give a brief overview of the range of problems which can be introduced in a physics laboratory by using SAGE.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35A15 Variational methods applied to PDEs
68W30 Symbolic computation and algebraic computation
97M50 Physics, astronomy, technology, engineering (aspects of mathematics education)

Software:

SAGE Interacts
Full Text: DOI

References:

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