I have implemented a version of Visscher's method for numerically solving the TDSE (A fast explicit algorithm for the time-dependent Schrdinger equation) (also described in Are there simple ways to numerically solve the time-dependent Schödinger equation? ).$$R(t+\frac{1}{2} \Delta t)=R(t-\frac{1}{2} \Delta t)+\Delta t HI(t)$$$$I(t+\frac{1}{2} \Delta t)=I(t-\frac{1}{2} \Delta t)-\Delta t HR(t)$$I was wondering how to implement transparent boundary conditions for such a scheme?  If you leave the boundaries uncomputed and set to zero, it produces perfect reflection.  I have also tried pretending boundary elements are surrounded by zero on all sides other than the non-boundary side - to mimic a wavefront propagating into nothing - this also reflects.
Transparent boundary conditions for finite element simulation of TDSE
finite element;boundary conditions;quantum mechanics
Kirill's answer is a good method but it is hampered that the fact that you need to tune it to a specific range in energy, so it is not appropriate if you have a broad-band wavepacket you need to absorb.An alternative approach is exterior complex scaling, which is reviewed well inInfinite-range exterior complex scaling as a perfect absorber in time-dependent problems. A. Scrinzi. Phys. Rev. A 81 no. 5, 053845 (2010), arXiv:1002.2520.The idea is to solve the Schrdinger equation not for $\psi(x)$ but for its analytical continuation, branching off from the real line into the complex plane, at an angle, after all the interesting stuff is done. Thus you solve for $\xi\equiv x\in\mathbb R$ if $x<R$, but instead of a real $x>R$ you change to $x=e^{i\theta}(\xi-R)+R$. The advantage of this is that plane waves of the form $\psi(x)=e^{ikx}$ will change into$$\psi(\xi)=\exp(ik(e^{i\theta}(\xi-R)+R))=e^{ikR}e^{ik\cos(\theta)(\xi-R)}\exp(-k\sin(\theta)(\xi-R)),$$with an exponentially decreasing amplitude regardless of the value of $k$. For more details, see Scrinzi's paper (which deals with a specific discretization scheme for this method) and references therein.Another method that can work is a complex absorbing potential, of the form $-i\eta x^n$. This is described in e.g.Investigation on the reflection and transmission properties of complex absorbing potentials. U.V. Riss and H.D. Meyer. J. Chem. Phys. 105 no. 4, 1409 (1996)and as I see it the main ingredient is having a slightly non-hermitian hamiltonian such that $e^{i Ht}$ will tend to decrease the norm of the wavefunction - and choosing the anti-hermitian part $H-H^\dagger$ cleverly enough that only the parts of the wavefunction on the outside are killed off, without introducing discontinuities which would themselves cause reflections. However, this is not usually perfect and can require a fair bit of fiddling and tuning to get it to work well.A third method, known as the use of an absorbing mask, is relatively crude: take a 'buffer' region just before the boundary and, at every timestep, multiply your wavefunction by a factor which goes to zero at the boundary. There's a bunch of such masks out there and - at a very rough glance - there don't seem to be any specific choices that stick out. You do want your mask to go down smoothly to zero, or otherwise you're essentially imposing a Dirichlet boundary condition and that will generate reflections. Beyond that, you need to work your specific problem into something that will work for you - which is to say, the method is case-by-case and fairly fiddly to get the reflection coefficient to sufficiently low values.