78
2 The Quantum Approach to the Two-Body Problem
IF(f(x_istep)
END REPEAT ISTEP
The accuracy of the location of the zero points and minima and maxima obviously
depends on the number of the grid points considered within the interval under consideration. The same method can be used by halving the interval (or dividing it into
smaller subintervals) in several subsequent iterations in order to refine the solutions
found until you reach the desired accuracy (this method is rather slow but can be
easily parallelized by subdividing the intervals).
Another well-known method is the Newton–Raphson one that employs the formula
x i+1 = x i −
f (x i )
f (x i )
(2.150)
using the values obtained by the previous iteration for the function and its derivatives
to generate the next value. Applying the same algorithm to the derivative gives the
minimum and maximum points. One should be careful with the Newton–Raphson
method which can often diverge or never converge.
2.5.3 The Time-Dependent Method
As discussed at the beginning of the chapter the integration of the time-dependent
Schrödinger equation (see Eq. 2.16) is usually performed by factorizing the timedependent part of the wave function ((r, t) = (r)χ(t)) and integrating separately the stationary Schrödinger equation (see Eq. 2.103). Alternatively, the timedependent formalism offers the possibility of integrating Eq. 2.16 by integrating in
time the equation:
({r}, t) = ˆ
U (t, t 0 ))({r}, t 0 )
(2.151)
where ˆ
U (t, t 0 ) is the time evolution operator.
This means that one can define the initial shape of the wave function (or wave
packet) at time t = t 0 and apply the Hamiltonian operator until the wavepacket has
propagated into the asymptotic region. This approach allows you to use the time variable t as a continuity variable though at the price of keeping an additional variable in
the formalism. This makes the approach very simple (because it is possible to evaluate
the elements of the matrix S (the probability is the squared modulus of the corresponding S element) by reiterating the application of the operator evolution e
−i ˆ
Hτ /
to the wave function of the system starting from the reagents in a predetermined initial state. The procedure is repeated until the wave function is distributed across the
accessible configuration space. At each time interval, the shape of the wave function
in the region of interest for the evaluation of the final arrangement is analyzed. Using
2 The Quantum Approach to the Two-Body Problem
IF(f(x_istep)
END REPEAT ISTEP
The accuracy of the location of the zero points and minima and maxima obviously
depends on the number of the grid points considered within the interval under consideration. The same method can be used by halving the interval (or dividing it into
smaller subintervals) in several subsequent iterations in order to refine the solutions
found until you reach the desired accuracy (this method is rather slow but can be
easily parallelized by subdividing the intervals).
Another well-known method is the Newton–Raphson one that employs the formula
x i+1 = x i −
f (x i )
f (x i )
(2.150)
using the values obtained by the previous iteration for the function and its derivatives
to generate the next value. Applying the same algorithm to the derivative gives the
minimum and maximum points. One should be careful with the Newton–Raphson
method which can often diverge or never converge.
2.5.3 The Time-Dependent Method
As discussed at the beginning of the chapter the integration of the time-dependent
Schrödinger equation (see Eq. 2.16) is usually performed by factorizing the timedependent part of the wave function ((r, t) = (r)χ(t)) and integrating separately the stationary Schrödinger equation (see Eq. 2.103). Alternatively, the timedependent formalism offers the possibility of integrating Eq. 2.16 by integrating in
time the equation:
({r}, t) = ˆ
U (t, t 0 ))({r}, t 0 )
(2.151)
where ˆ
U (t, t 0 ) is the time evolution operator.
This means that one can define the initial shape of the wave function (or wave
packet) at time t = t 0 and apply the Hamiltonian operator until the wavepacket has
propagated into the asymptotic region. This approach allows you to use the time variable t as a continuity variable though at the price of keeping an additional variable in
the formalism. This makes the approach very simple (because it is possible to evaluate
the elements of the matrix S (the probability is the squared modulus of the corresponding S element) by reiterating the application of the operator evolution e
−i ˆ
Hτ /
to the wave function of the system starting from the reagents in a predetermined initial state. The procedure is repeated until the wave function is distributed across the
accessible configuration space. At each time interval, the shape of the wave function
in the region of interest for the evaluation of the final arrangement is analyzed. Using