Quantum Physics

A blackbody is an idealized object that absorbs all electromagnetic radiation that falls on it, with no radiation reflected. A blackbody emits radiation depending on its temperature (called blackbody radiation), producing a continuous spectrum of frequencies or wavelengths, with a peak emission as seen in the figure:

The position of the peak follows Wien's Displacement Law:

Where b is Wien's constant: $b = 2.898 \cdot 10^{-3} \text{ m} \cdot \text{K}$
Classical physics predicted that the energy emitted should increase with the square of the frequency; however, experimental data showed that emission drops sharply for small wavelengths. The classical prediction only matched experimental data for large wavelengths, failing completely at small wavelengths.
This discrepancy, known as the "ultraviolet catastrophe", had no solution within classical theory. Max Planck provided a solution in 1900 by introducing a bold hypothesis.

An object cannot emit or absorb energy in arbitrary continuous amounts, but only in discrete natural multiples of minimum quantities called quanta.
The energy of each quantum is given by the expression:

Alternatively, recalling that $c = \lambda \cdot f$ for electromagnetic waves:

Where $h = 6.63 \cdot 10^{-34} \text{ J} \cdot \text{s}$ is a universal constant known as Planck's constant, and $f$ is the frequency of the radiation emitted by the body when releasing energy.
Planck proposed that just as matter is "quantized" in the form of atoms, energy is quantized in the form of quanta. This emitted radiation is what we know as electromagnetic waves.

Einstein won the Nobel Prize in Physics in 1921 for his description of the Photoelectric Effect.
The energy of an incident photon, $E = h \cdot f$, striking a metal surface can excite the material's electrons and eject them if it exceeds a certain limit, $\Phi$ (or $W_0$), called the work function, which is characteristic of each metal. The ejected electron leaves with a maximum kinetic energy $K$.

Applying Planck's hypothesis:

Where $f_0$ is the threshold frequency necessary to eject the electron, and $v$ is the velocity acquired by it.

• Increasing the intensity of the incident light increases the number of photoelectrons ejected, but not their kinetic energy.
• Increasing the frequency increases the kinetic energy of the electrons, but not their number.

Stopping Potential: is the voltage required to stop the fastest emitted electrons. Since the kinetic energy acquired by a charge is $K = e \cdot V_{\text{stop}}$, we can write:

As seen, light exhibits a dual behavior, acting as both a wave and a particle. In 1923, Louis de Broglie expanded this concept by suggesting that nature is symmetric: if a wave can exhibit particle-like behavior, a particle should have wave-like properties. He formulated the following hypothesis:

Where $p$ is the momentum of the particle.
Thus, electrons, neutrons, protons, and any particle should exhibit wave properties. However, for macroscopic masses, since $h$ is very small, $\lambda \to 0$.
Years later, Davisson and Germer managed to observe electron diffraction, confirming their wave nature.
That same year, G.P. Thomson experimentally confirmed De Broglie's relationship $\lambda = h/mv$ through the diffraction of electrons passing through thin metal foils.

A quantum object (such as an electron or a photon) is one that acts as a wave or as a particle, but will never exhibit both aspects simultaneously.
Heisenberg formulated his Uncertainty Principle as follows:

Where $\Delta x$ is the uncertainty in position, and $\Delta p$ is the uncertainty in momentum.

The uncertainty equation can also be expressed in terms of energy and time:

Because it is impossible to precisely define the trajectory of an electron, the concept of an electron orbital is defined as the region of space around the nucleus where the probability of finding an electron with a specific energy is highest.
This principle carries fundamental consequences: it abolishes scientific determinism and highlights the need for quantum systems to be expressed in terms of probability.