How to Use Planck's Constant

How to Use Planck's Constant
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Max Planck, a German physicist in the late 1800s and early 1900s, worked intensely on a concept called black-body radiation. He proposed that a black body was both the ideal absorber and the ideal emitter of light energy, not unlike the sun. 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 ultimately became a foundation of quantum mechanics, and Planck won a Nobel Prize in Physics in 1918.

The derivation of Planck's 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

\Delta E=h\nu

Where ​∆E​ is change in energy and ​ν​ is the the particle's oscillation frequency. This is known as the Planck-Einstein equation, and the value of ​h​, Planck's constant, is 6.626 × 10 −34 J s (joule-seconds).

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​:

    \nu = \frac{c}{\lambda}=\frac{3\times 10^8}{525\times 10^{-9}}=5.71\times 10^{14}\text{ Hz}
    \Delta E = h\nu = (6.626\times 10^{-34})(5.71\times 10^{14})=3.78\times 10^{19}\text{ J}

Planck's Constant in the Uncertainty Principle

A quantity called "h-bar," or ​h​, is defined as ​h​/2π. This has a value of 1.054 × 10 −34 J s.

Heisenberg's uncertainty principle states that the product the standard deviation of the location of a particle (​σx​) and the standard deviation of its momentum (​σp​) must be greater than one-half of h-bar. Thus

\sigma_x \sigma_p \geq\hbar /2

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

    \sigma_x \geq\hbar /2\sigma_p
    \sigma_x \geq(1.054\times 10^{-34})/2(2.6\times 10^{-35}) \\\sigma_x \geq 1.5\text{ m}

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