In quantum mechanics, the state of a physical system is represented by a wave function.
In Born’s interpretation, the square of the particle’s wave function represents the probability density of finding the particle around a specific location in space.
Wave functions must first be normalized before using them to make predictions.
The expectation value is the average value of a quantity that requires a wave function and an integration.

The Heisenberg uncertainty principle states that it is impossible to simultaneously measure the x-components of position and of momentum of a particle with an arbitrarily high precision. The product of experimental uncertainties is always larger than or equal to $ℏ / 2 .$
The limitations of this principle have nothing to do with the quality of the experimental apparatus but originate in the wave-like nature of matter.
The energy-time uncertainty principle expresses the experimental observation that a quantum state that exists only for a short time cannot have a definite energy.

The Schrӧdinger equation is the fundamental equation of wave quantum mechanics. It allows us to make predictions about wave functions.
When a particle moves in a time-independent potential, a solution of the time-dependent Schrӧdinger equation is a product of a time-independent wave function and a time-modulation factor.
The Schrӧdinger equation can be applied to many physical situations.

Energy states of a quantum particle in a box are found by solving the time-independent Schrӧdinger equation.
To solve the time-independent Schrӧdinger equation for a particle in a box and find the stationary states and allowed energies, we require that the wave function terminate at the box wall.
Energy states of a particle in a box are quantized and indexed by principal quantum number.
The quantum picture differs significantly from the classical picture when a particle is in a low-energy state of a low quantum number.
In the limit of high quantum numbers, when the quantum particle is in a highly excited state, the quantum description of a particle in a box coincides with the classical description, in the spirit of Bohr’s correspondence principle.

The quantum harmonic oscillator is a model built in analogy with the model of a classical harmonic oscillator. It models the behavior of many physical systems, such as molecular vibrations or wave packets in quantum optics.
The allowed energies of a quantum oscillator are discrete and evenly spaced. The energy spacing is equal to Planck’s energy quantum.
The ground state energy is larger than zero. This means that, unlike a classical oscillator, a quantum oscillator is never at rest, even at the bottom of a potential well, and undergoes quantum fluctuations.
The stationary states (states of definite energy) have nonzero values also in regions beyond classical turning points. When in the ground state, a quantum oscillator is most likely to be found around the position of the minimum of the potential well, which is the least-likely position for a classical oscillator.
For high quantum numbers, the motion of a quantum oscillator becomes more similar to the motion of a classical oscillator, in accordance with Bohr’s correspondence principle.

A quantum particle that is incident on a potential barrier of a finite width and height may cross the barrier and appear on its other side. This phenomenon is called ‘quantum tunneling.’ It does not have a classical analog.
To find the probability of quantum tunneling, we assume the energy of an incident particle and solve the stationary Schrӧdinger equation to find wave functions inside and outside the barrier. The tunneling probability is a ratio of squared amplitudes of the wave past the barrier to the incident wave.
The tunneling probability depends on the energy of the incident particle relative to the height of the barrier and on the width of the barrier. It is strongly affected by the width of the barrier in a nonlinear, exponential way so that a small change in the barrier width causes a disproportionately large change in the transmission probability.
Quantum-tunneling phenomena govern radioactive nuclear decays. They are utilized in many modern technologies such as STM and nano-electronics. STM allows us to see individual atoms on metal surfaces. Electron-tunneling devices have revolutionized electronics and allow us to build fast electronic devices of miniature sizes.