Understanding Planck’s Constant

Max Planck, a German physicist in the late 1800s and early 1900s, worked intensely on a concept called black-body radiation. A black body is both the ideal absorber and the ideal emitter of light energy (electromagnetic radiation), not unlike the sun or a black-hole. To make his math work, he had to propose that light energy did not exist along a continuum, but in quanta, or discrete amounts. This notion was treated with deep skepticism at the time, but it ultimately became a foundation of quantum mechanics, and Planck won a Nobel Prize in Physics in 1918.

The derivation of the Planck constant, ‌h‌, involved combining this idea of quantum levels of energy with three recently developed concepts: the Stephen-Boltzmann law, Wein’s displacement law and the Rayleigh-James law. This led Planck produce the relationship

(E = hnu)

Where ‌E‌ is the energy and ν is the particle’s oscillation frequency. This is known as Planck’s equation or the Planck-Einstein equation, and the value of ‌**‌h‌‌‌, Planck’s constant, is 6.626 × 10‌‌-34‌ ‌J s (joule-seconds)**‌.

The Physics Behind Planck’s Constant

Planck’s constant is a fundamental physical constant (or universal constant) which helps to describe many properties of electromagnetic waves and matter waves (as described in quantum physics). Planck’s constant ‌h‌, is used in a variety of wave equations and relationships to describe everything from the energy of a photon to the electron shells of a hydrogen atom (from the Bohr Model).

In SI units, the value of Planck’s constant is described as 6.626 × 10-34 kg⋅m2⋅s−2, which also happens to be the same units as angular momentum (a relationship that helps describe electrons around an atom).

Using Planck’s Constant in the Planck-Einstein’s equation

Given light with a wavelength of 525 nanometers (nm), calculate the energy.

Since ‌c‌ = ‌ν‌ × ‌λ,‌ where ‌c‌ is the speed of light and ‌λ‌ is wavelength, we can solve for the frequency:

(nu = frac{c}{lambda} = frac{3times10^8 text{ m/s}}{525times10^{-9} text{ m}}= 5.71times10^{14} text{ Hz})

Then we can use the frequency and the Planck-Einstein equation to find the amount of energy in a single photon with a wavelength of 525 nm:

(E = (5.71times10^{14} text{ Hz}) cdot(6.626times10^{-34}text{ J}cdottext{s}) = 3.7863times10^{-19} text{ J})

TL;DR (Too Long; Didn’t Read)

The units of ‌_ν_‌ are in hertz (Hz) or events per second (1/s). The units of energy are joules (J).

The Photoelectric Effect

The photoelectric effect was a monumental experiment in physics in the early 20th century. Albert Einstein was even awarded his Nobel Prize for his development of the law of the photoelectric effect.

When light impacts on a surface, it tries to impart some energy E onto the surface (as described by the Planck-Einstein equation). If the energy is high enough, it results in the emission of electrons from the metal’s surface, as they are freed from their bonds, which is then measured as an electric current. At the time of Einstein’s work, the nature of light and how it imparts energy to other surfaces was not fully understood.

Einstein developed the idea that if light is quantized, then regardless of the intensity (energy per area) of the light, electrons would only be freed from the metal if the individual energy of each electron was large enough. This is described by the relationship:

(text{Electron Kinetic Energy} rightarrow text{KE} = hnu-phi)

where ‌h‌ is Planck’s constant, ‌ν‌ is frequency (sometimes also ‌f‌), and ‌ϕ‌ is the work function of the surface. The work function simply describes the minimum amount of energy needed to free an electron from a (metal) surface; it is dependent on the material.

De Broglie Wavelength

The de Broglie wavelength describes some fundamental properties of particles through the use of waves and the fundamental constant ‌h‌. The core component of the de Broglie wavelength that is true for any particle (massive or massless) relates momentum and wavelength:

(lambda = frac{h}{p})

Planck’s Constant and the Uncertainty Principle

A quantity called “h-bar,” or is defined as ‌ħ‌.

(hslash = frac{h}{2pi}= 1.054 times 10^{-34} text{ J }cdottext{s})

Heisenberg’s uncertainty principle states that the product of the uncertainty of the location of a particle (‌Δx‌) and the uncertainty of its momentum (‌Δp‌) must be greater than one-half of h-bar. Thus:

(Delta x Delta p geq frac{hslash}{2} = frac{h}{4pi})

Given a particle for which ‌Δp‌ = 3.6 × 10-35 kg m/s, find the standard deviation of the uncertainty in its position.

(Delta x geq frac{hslash}{2Delta p} rightarrow Delta x geq frac{1.054times10^{-34}}{2(3.6times10^{-35})} rightarrow Delta x geq 1.463 text{ m})