Quantum mechanics (QM; also known as quantum physics, or quantum theory), including quantum field theory, is a fundamental branch of physics concerned with processes involving, for example, atoms and photons. In such processes, said to be quantized, the action has been observed to be only in integer multiples of the Planck constant, a physical quantity that is exceedingly, indeed perhaps ultimately, small. This is utterly inexplicable in classical physics.

Quantum mechanics gradually arose from Max Planck's solution in 1900 to the black-body radiation problem (reported 1859) and Albert Einstein's 1905 paper which offered a quantum-based theory to explain the photoelectric effect (reported 1887). Early quantum theory was significantly reformulated in the mid-1920s.

The mathematical formulations of quantum mechanics are abstract. A mathematical function, the wave function, provides information about the probability amplitude of position, momentum, and other physical properties of a particle.

Important applications of quantum mechanical theory include superconducting magnets, LEDs and the laser, the transistor and semiconductors such as the microprocessor, medical and research imaging such as MRI and electron microscopy, and explanations for many biological and physical phenomena.

The potential in this case is given by:

V(x)= \begin{cases} 0, & x < 0, \\ V_0, & x \ge 0. \end{cases}
The solutions are superpositions of left- and right-moving waves:

\psi_1(x)= \frac{1}{\sqrt{k_1}} \left(A_\rightarrow e^{i k_1 x} + A_\leftarrow e^{-ik_1x}\right)\quad x<0
\psi_2(x)= \frac{1}{\sqrt{k_2}} \left(B_\rightarrow e^{i k_2 x} + B_\leftarrow e^{-ik_2x}\right)\quad x>0
where the wave vectors are related to the energy via

k_1=\sqrt{2m E/\hbar^2}, and
k_2=\sqrt{2m (E-V_0)/\hbar^2}
with coefficients A and B determined from the boundary conditions and by imposing a continuous derivative on the solution.

Each term of the solution can be interpreted as an incident, reflected, or transmitted component of the wave, allowing the calculation of transmission and reflection coefficients. Notably, in contrast to classical mechanics, incident particles with energies greater than the potential step are partially reflected.