Statistical Temperature

Hands down, this is kind of an advanced topic so if you are not familiar with it, that is okay.

The usual way we measure temperature is through a thermometer. We have an intuitive sense of the temperature for everyday objects. In high school, you learn that temperature is related to the kinetic energy of a system: KE = 3/2kT; 1/2kT for each “degree of freedom”. However, the definition of temperature gets more refined in statistical mechanics. Here, we define temperature (actually, reciprocal of temperature) as a measure of the rate of change of entropy with respect to the energy of a system (loosely speaking). Note that we are talking about the absolute temperature (Kelvin). Entropy is not just the state of disorder of a system, rather a measure of how many ways it can get disordered (and ordered).

Sounds confusing? I don’t blame you! You can learn more about entropy here:

For more clear and technical explanation, watch this:

For more equations, see Famous equations in Physics

Planck’s Hypothesis: Quantization of Energy

When trying to explain the cavity radiation(or blackbody radiation), Planck hypothesized that the radiation is absorbed or emitted only in discrete quantities, unlike classically accepted model of continuous waves of light. Einstein extended this idea to all electromagnetic radiation and suggested that all electromagnetic radiation, from X-rays to visible light to microwaves to radio waves, all kinds of “light” exhibits the discrete (or quantum) nature. There is a lot to this story which is very interesting and shows how limited our assumptions are, regarding the basic laws of nature.

For a quirky explanation, see this:

For more equations, see Famous equations in Physics

de Broglie’s Hypothesis

Constituents of matter, such as electrons can show wave-like nature in certain experiments, e.g. electron beams shone on a crystal with appropriate kinetic energy show a diffraction pattern not unlike X-ray diffraction patterns.

The hypothesis was one of the founding concepts in Quantum Mechanics because it established the wave-particle duality: the fact that matter can behave both as wave and as a particle. It seems counter-intuitive but has been proved to be true in many experiments, just like many other concepts in Quantum Mechanics.  It helps explain a lot of natural phenomena, such as why an electron in orbit doesn’t lose its energy by radiation and fall into the nucleus and why do we have specific orbits (or energy levels, more accurately) around the nucleus.

“h” here is Planck’s constant and is equal to 6.62606957 x 10^-34 J.s

“p” here is the momentum of the entity in question.

Quantum Mechanics in a nutshell-11: Perturbation theory

Quantum Mechanics in a nutshell-10: Stern-Gerlach Experiment

Quantum Mechanics in a nutshell-9: Electrons in Magnetic Field

Quantum Mechanics in a nutshell-8: Spin

Quantum Mechanics in a nutshell-6: Hydrogen atom

We continue with the three dimensional equation encountered in the previous post. This time, we shall solve the radial part of the wave function with the potential provided by the Hydrogen atom nucleus (which contains single proton). This potential is nothing but the simple Coulomb potential:

V(r) = – e²/(4 π ε₀ r)

As before, the final solution is a product of Radial part R, and angular part Y(θ,Φ) (we have already solved for Y which turn out to be spherical harmonics). Note that the variable has been changed from R(r) to u(r) first, just to make the calculation neater:

u(r) = r x R(r)

The equation in u(r) can be solved with analytical methods used to solve linear ordinary differential equation. Here we quote only the results.

In the case of Hydrogen atom, energy only depends on the principal quantum number, n. The final wavefunction Ψ_n,l,m however depends on the three quantum numbers n(principal), l(azimuthal) and m(magnetic) as shown.

Quantum Mechanics in a nutshell-7: Angular Momentum

The Photoelectric Effect

Albert Einstein was awarded the Nobel prize for his explanation of the photoelectric effect: when light is shone on a metal surface, electrons are ejected. However, the light cannot be of just any wavelength, it has to be lower (and frequency higher) than the threshold wavelength.

In practical terms, it means that no matter how long you shine light of lower frequency (e.g. visible or infrared light for Zinc surface) on the metal surface, no change happens even if that light is very bright. However, even a faint radiation of higher frequency (e.g. UV light for Zinc surface) can cause observable effect when shone even for a few seconds.

The explanation is given in terms of photon picture of light, We assume that light is made up of photons which all have a particular frequency (and thus energy). When a photon interacts with an electron at the metal surface, it imparts this energy to the electron. However, electrons require some minimum energy to be able to escape out of the metal, and it can absorb only one photon at a time. Consequently, you cannot make it escape by providing photons of lower energy, no matter how many.

A simple demonstration:

Explanation by analogy with Elephants:

For more equations like this, see Famous equations in Physics