MATSLISE: Schrödinger |
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At the beginning of the twentieth century, experimental evidence suggested that atomic particles were also wave-like in nature. For example, electrons were found to give diffraction patterns when passed through a double slit in a similar way to light waves. Therefore, it was reasonable to assume that a wave equation could explain the behaviour of atomic particles.
Schrödinger was the first person to write down such a wave equation. Much discussion then centred on what the equation meant. The eigenvalues of the wave equation were shown to be equal to the energy levels of the quantum mechanical system, and the best test of the equation was when it was used to solve for the energy levels of the Hydrogen atom.
It was initially much less obvious what the wavefunction of the equation was. After much debate, the wavefunction is now accepted to be a probability distribution. The Schrödinger equation is used to find the allowed energy levels of quantum mechanical systems (such as atoms, or transistors). The associated wavefunction gives the probability of finding the particle at a certain position.
Solving the Schrödinger equation with MATSLISE: problem specification
MATSLISE solves the boundary value problem for the Schrödinger equation y"=(V(x)-E)y
where in quantum mechanics V(x)
(potential function) describes a potential field and E
is an energy level (eigenvalue). y"
is the second order derivative of the wavefunction y
.
Solving this equation means calculating the eigenvalues E
and the associated eigenfunctions y
.
Before the eigenvalues and eigenfunctions can be computed, the user must first specify the problem to solve in the correct way. That is: the user must choose the appropriate inputwindow (Schrödinger, Sturm-Liouville or Distorted Coulomb) and fill in the inputfields properly. Figure 1 shows the inputwindow for the Schrödinger problem (here the Harmonic Oscillator). This window can be used for regular Schrödinger problems or Schrödinger problems with an infinite integration interval for which interval truncation is effective.
In order to specify the Schrödinger problem to solve, the user should provide the following input:
V(x)
: The (continuous) potential function (e.g. V(x)=x^2
)
a
: The lower bound of the integration interval
b
: The upper bound of the integration interval
A0*y(a) + B0*y'(a) = 0,
A1*y(b) + B1*y'(b) = 0
,|A0|+|B0| > 0
and |A1|+|B1| > 0
.
tol
: a positive constant representing the accuracy requested by the user in the results produced by MATSLISE.The user is allowed to enter -inf
or inf
in the integration-interval-fields
a
and b
. When an inf
value is entered,
MATSLISE automatically determines good truncated endpoints for every eigenvalue.
There are some additional inputfields which allow to use parameters in the problem-specification, see Using parameters in the inputfields.
![]() | MATSLISE: a solver for Schrodinger and Sturm-Liouville equations | Using parameters in the inputfields |
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