Quantum Mechanics

Planck’s Law

A blackbody is a hypothetical body which absorbs radiation perfectly for every wave length.

The radiation law of Rayleigh-Jeans works well for low temperatures but predicts an infinite energy for higher temperatures. This was known as the ultraviolet catastrophe and Planck's Law was a remedy for it. Planck assumed that electromagnetic radiation can be absorbed

and emitted only discretely in quants (What we now call photons). Theexchange of energy between the oscillators and the electromagnetic radiation field occurs in tiny energy packets rather than continuously as in the classical view. Planck's Radiation Law describes thedistribution of intensities of electromagnetic energy and power and alsothe density distribution of all of the photons depending on the wavelength and frequency of the emissions of a black body for a certain temperature.According to Planck's law an oscillator with a frequency v can absorbonly multiples of hº and it needs a minimal energy hº in order to be excited at all.

. Shortwaved light hits electrons out of a piece of metal

The Photoelectric Effect

The photoelectric effect deals with electrons that are emitted from a metal surface when struck by radiation of adequately high wave length (light or ultraviolet radiation).

Unoxided metal surfaces emit electrons when they are negatively charged and their surface is struck by light. With this, the following characteristics are detected1.) The kinetic energy of the emitted electron depends on the frequency (color) of the light but not on the intensity. 2.) The electrons are released as soon as the light strikes.3.) An increase in the frequency of the incident light results in an increase in the kinetic energy of the emitted electrons.

4.) The effect occurs first beneath a certain wavelength (or a certain frequency) which depends on the material. 5.) The number of emitted electrons is proportional to the intensity of the radiation.

Except for the last condition, all of these characteristics are in opposition to the classical view of light as a wave. According to Young's classical double slit experiment, light exhibits interference patterns when it is diffracted and is thus considered to be a wave. According to the photoelectric effect, light can be treated like a particle. In reality it’s neither a wave or a particle but acts as a probability wave.

for which © is the work function, the minimum energy necessary to knock an electron out of a metal.

Problem 1.) Sodium has a work function of 2.28 eV. What is the threshold frequency required to produce photoelectrons from sodium?Solution

2.) If light of the frequency 3.00 £ 1015 Hz is used to illuminate the sodiuma.) What is Kmax, the maximum kinetic energy of the ejected photoelectrons? Solution

b.) What is the maximum speed of the photoelectrons? (Electron mass = 9.11 \times 10^{-31} kg)SolutionUsing our solution for the kinetic energy, we can plug into the equation

3.) If the light wave were increased by a factor of 2 in the intensity, what would happen to the value for the maximum kinetic energy?SolutionNothing would happen to the kinetic energy. More photo electrons would be knocked out of the metal surface, but the maximum kinetic energy would be the same. In order to increase K_{max}, the frequency of the incident energy would need to increase.

Bohr’s Theory of the Atom

According to classical electrodynamics, a system of moving charges radiates electromagnet waves. Energy should be radiated and that slows down the electron. Because of the small centrifugal force, the electrons would spiral towards the nucleus in a very short time. This contradicts the stable characteristic of real atoms.

1. Electrons move in stable, circular orbits around the nucleus. In contrast to the theory of electrodynamics, the electrons don't radiate energy as they move in orbits. 2. The radius of the electron's orbit doesn't change continuously but in discrete jumps. When the electron jumps

from one orbit to another, electromagnetic radiation is emitted or absorbed which has a frequency given by Max Planck's relation between energy and frequency of light. When the energy of the exiting state is Eexit and Edestination is the energy of the destination state, then the photon is emitted with the frequency º of the emitted radiation.

3. Electron orbits are only stable when the angular momentum L of the electron is an integer multiple of the reduced Planck's constant L = mvr = n ~. Bohr's model describes the behaviour of hydrogen atoms and ions with just one electron. It describes the spectral lines of hydrogen according to the Balmer-Rydberg equation very well, but it doesn't apply to system with several electrons.

If an electron is in a circular orbit of radius rn, then the electrical attraction by the nucleus (charge = Ze where Z is the number of protons) is equal to the centripetal force:

Now assuming the Bohr hypothesis

with n = 2, 3, ...

De Broglie’s Hypothesis

An electron can be regarded as a standing wave on an orbit around the nucleus with only certain integers of wavelengths. de Broglie wrote a dissertation in which he presumed that the wave-particle dualism that was known for photons also applies to solid matter. This means that even classical particles such as electrons can have wave characteristics. De Broglie determined the wavelength of moving particles to be ¸ = h/p. This relation is known as the de Broglie equation.

A standing wave in a circle

ProblemWhat is the energy of an electron with = 600 nm (orange light)?Solution