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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). The
exchange 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 the
distribution of intensities of electromagnetic energy and power and also
the 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 absorb
only 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 detected
1.) 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 sodium
a.) 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)
Solution
Using 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?
Solution
Nothing 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
Problem
What is the energy of an electron with
light)?
Solution
= 600 nm (orange