Zero-Point Energy

The zero-point energy of a quantum mechanical system is the energy of the system in its ground state (when the system has the lowest possible energy). Take for example, a harmonically oscillating particle in a potential well. While classically, it may be possible to suppose that the particle could have zero energy if its kinetic energy and the energy potential at its position are both zero, quantum mechanically speaking, such a system is impossible. The Heisenberg uncertainty relation forbids such a scenario where both the particle's position and momentum are known. Here, the Heisenberg uncertainty relation and the Schrodinger equation are used to derive the zero-point energy of a quantum harmonic oscillator.

Rectangular Potential Barrier

Rectangular potential barriers, also called square potential barriers, are formed by energy potentials which create wall-like barricades for particles. Essentially, a potential barrier is a potential step except the energy potential returns to zero at some finite positive $x$-position, $a$, and remains zero beyond that point. Here, we'll derive the wave function of a particle facing a rectangular potential barrier, then find the transmission and reflection coefficients of the particle upon encountering the barrier.

Potential Step

Potential steps are created by energy potentials which form step-like barricades for particles. Before the potential step, the energy potential is uniformly zero, but at the step, the energy potential rises instantaneously to a finite value and remains constant at that value for all positions beyond the step. Here, we'll derive the wave function of a particle facing a potential step, then find the transmission and reflection coefficients of the particle upon encountering the step.

Probability Current

The probability current, also known as the probability flux, of a wave function at a certain point describes the rate of flow at which probability passes through that point, analogous to the way electrical current describes the rate of flow at which electrical charge passes through a point in a medium. Probability currents are used, for example, when calculating reflection and transmission coefficients for particles encountering potential steps or potential barriers.

Infinite Square & Box Potential Wells

In quantum mechanics, energy wells are are formed by energy potentials that hinder a particle's movement within a certain area. An example of a particle stuck in a potential well is an electron caged in a negatively charged box. The only way for the electron to escape is if it has enough kinetic energy to trade off for potential energy as it approaches the negatively charged electric field. Infinite square potential wells are one-dimensional energy wells which restrict particles inside within its infinitely high potential walls. Inside the well, the energy potential is uniformly zero, but to leave the square well, an infinite amount of energy is required. Infinite box wells are simply three-dimensional rectangular cages formed by square wells in each of the three dimensions.

Hamiltonian & Schrodinger Equation

The Hamiltonian is an operator which gives the total energy of a system by adding together the system's kinetic energy and potential energy. Schrodinger's time-independent equation is a simple mathematical equivocation of this relation between Hamiltonians and total energy. Schrodinger's time-independent equation, more difficultly derived, shows how the the total energy of a system can also be found using operations which rely on the time evolution of wave functions.