The purpose of this exercise is to study the basic properties of localized states of quantum systems in a confining potential.
The restriction of free motion by potential energy is called a
quantum well.
The simplest type of quantum well is an infinite potential well from which the particle cannot escape at all (the particle is completely localized inside this well). Another simplest type of a quantum well is a finite-square well, in which the particle is also limited by a box, but has finite-height potential walls. In this case, unlike the infinite potential well, there is a possibility that the particle can be found outside the box. It is worth noticing that, in the classical interpretation, if the total energy of a particle is less than the barrier of the potential energy of the walls, it cannot be found outside. In the quantum mechanics, there is a non-zero probability that the particle can be found outside the box, even if the particle’s energy is less than the barrier of the potential energy of the walls.
In this section you can set the basic quantum well profile. Please note that the units used here are "a.u." for "atomic units", where 1 a.u.\(= m_e = q_e = 2 E_I = a_0\) (\(m_e = 9.109 \times 10^{-31}\) kg, \(q_e = 1.6 \times 10^{-19}\) C, \(E_I = 13.6\) eV and \(a_0=0.53\) Å).
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system in position or momentum space. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements performed on the system can be derived from it. The most common symbols used to denote the wave function, are the Greek letters \(\psi \) or \(\Psi \) (lowercase and uppercase psi, respectively).
The comparison between numerical and analytical solutions is presented in this section.
In quantum mechanics, the expected value is the probabilistic expected value of the result (measurement) of an experiment. It can be considered as an average of all possible measurement results, weighted by their probability.
Consider an operator \( \hat{A} \). The expectation value is then \( \langle A \rangle = \langle \psi |\hat{A}| \psi \rangle \) in Dirac notation with \( | \psi \rangle \) a normalized state vector.
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