Special functions

Some useful functions.

std::complex<double> mrcpp::free_particle_analytical_solution(double x, double x0, double t, double sigma)

Free-particle time evolution on real line.

Analytical solution of a one dimensional free-particle movement

\[ \psi(x, t) = \sqrt{ \frac{ \sigma }{ 4it + \sigma } } e^{ - \frac { (x - x_0)^2 }{ 4it + \sigma } } \]

where \( t, \sigma > 0 \).

Parameters:

• x – [in] space coordinate in \( \mathbb R \).

• x0 – [in] \( x_0 \) center of gaussian function at zero time moment.

• t – [in] time moment.

• sigma – [in] \( \sigma \) width of the initial gaussian wave.

Returns:

The complex-valued wave function \( \psi(x, t) \) at the specified space coordinate and time.

double mrcpp::smooth_compact_function(double x, double a, double b)

A smooth compactly supported non-negative function.

Smooth function on the real line \( \mathbb R \) defined by the formula

\[ g_{a,b} (x) = \exp \left( - \frac{b - a}{(x - a)(b - x)} \right) , \quad a < x < b \]

and \( g_{a,b} (x) = 0 \) elsewhere.

Parameters:

• x – [in] space coordinate in \( \mathbb R \).

• a – [in] the left support boundary.

• b – [in] the right support boundary.

Returns:

The non-negative value \( g_{a,b} (x) \) at the specified space coordinate \( x \in \mathbb R \).