Kavli Affiliate: Felix Fischer
| First 5 Authors: Felix Fischer, Daniel Burgarth, Davide Lonigro, ,
| Summary:
When numerically simulating the unitary time evolution of an
infinite-dimensional quantum system, one is usually led to treat the
Hamiltonian $H$ as an ”infinite-dimensional matrix” by expressing it in some
orthonormal basis of the Hilbert space, and then truncate it to some finite
dimensions. However, the solutions of the Schr"odinger equations generated by
the truncated Hamiltonians need not converge, in general, to the solution of
the Schr"odinger equation corresponding to the actual Hamiltonian.
In this paper we demonstrate that, under mild assumptions, they converge to
the solution of the Schr"odinger equation generated by a specific Hamiltonian
which crucially depends on the particular choice of basis: the Friedrichs
extension of the restriction of $H$ to the space of finite linear combinations
of elements of the basis. Importantly, this is generally different from $H$
itself; in all such cases, numerical simulations will unavoidably reproduce the
wrong dynamics in the limit, and yet there is no numerical test that can reveal
this failure, unless one has the analytical solution to compare with.
As a practical demonstration of such results, we consider the quantum
particle in the box, and we show that, for a wide class of bases (which include
associated Legendre polynomials as a concrete example) the dynamics generated
by the truncated Hamiltonians will always converge to the one corresponding to
the particle with Dirichlet boundary conditions, regardless the initial choice
of boundary conditions. Other such examples are discussed.
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