P.Chem
In 1925, Erwin Schrodinger and Weiner Heisenberg
(and Warner Heisenberg) independently formulated a general quantum theory. The two methods appeared different because Heisenberg's method is formulated in terms of matrices whereas schrodinger's method is formulated in terms of partial differential equations. pChem student use schrodinger equation bc they not familiar with matrix
dv =
-c dλ / λ^2 (b.c λv =c)
Experimentally
1) the KE of the ejected electrons is independent of intensity of the incident radiation. Classical predicts that the photoelectric effect should occur for any frequency of light as long as intensity is sufficiently high. 2) But the experiment fact that there is a threshold frequency! regardless of the intensity (amplitude).
wien and Ralieh-Jeans assumption based on classical physics
1)electronic oscillators responsible for emission off radiation, so electro-magnetic radiation is wavelike 2)energies of electronic oscillators can have any value
Max Planck
1. Born in Germany, 1858-1948 2. talent in both music and science 3. Obtained Ph.D in theoretical physics in 1879 at university of Munich & dissertation on 2nd law of thermodynamic 3. father of quantum theory 4. warded Nobel Prize in physics in 1918 "in recognition of services he rendered to the advancement of physics by his discovery of energy quanta."
dispersion of white light into its spectrum by a prism
1. Dispersion is the splitting up of white light into seven colors on passing through a transparent medium like a glass prism. When a white light beam is passed through a prism, a band of seven colors are formed is known as spectrum of white light.The seven colors in the spectrum are red, orange, yellow, green, blue, indigo and violet respectively. 2. Example: The rainbow is an arch of seven colors visible in the sky which is produced by the dispersion of sun's light by rainbow drops in the atmosphere. The rainbow is actually a natural spectrum of sunlight in the sky. A rainbow is produced by the dispersion of white sunlight by raindrops (or water drops) in the atmosphere. Each raindrop acts as a tiny glass prism splitting the sunlight into a spectrum
De Broglie Waves are observed experimentally
1. X-ray diffraction 2. Electron diffraction
Sir George P.Thomson
1. among the first to show experimentally in 1926 that the electron could act as a wave. 2. won nobel prize in 1937 for showing that it is a wave.
Assumptions of explaining the spectrum of H-atom
1. atom can be pictured as a central, rather massive nucleus with one associated electron. 2. because has a much higher mass than the electron, nucleus is considered to be fixed and the electron to be revolving about it.
explaining the spectrum of H-atom (classically)
1. electron is constantly being accelerated, it should emit electro-magnetic radiation and lose energy 2. the revolving electron around the nucleus will lose energy and spiral into the nucleus. 3. stable orbitals for the electron is classical forbidden
wave-particle duality of light
1. light appears wavelike in some instances and particle-like in others 2. dispersion of white light into spectrum and photolectric effect.
Photoelectric effect
1. shows that light behave as a stream of photons 2. observation that many metals emit electrons when light shines upon them. 3.According to classical electromagnetic theory, this effect can be attributed to the transfer of energy from the light to an electron in the metal. From this perspective, an alteration in either the intensity or wavelength of light would induce changes in the rate of emission of electrons from the metal. Furthermore, according to this theory, a sufficiently dim light would be expected to show a time lag between the initial shining of its light and the subsequent emission of an electron. However, the experimental results did not correlate with either of the two predictions made by classical theory. Instead, electrons are only dislodged by the impingement of photons when those photons reach or exceed a threshold frequency. Below that threshold, no electrons are emitted from the metal regardless of the light intensity or the length of time of exposure to the light. To make sense of the fact that light can eject electrons even if its intensity is low, Albert Einstein proposed that a beam of light is not a wave propagating through space, but rather a collection of discrete wave packets (photons), each with energy hf. This shed light on Max Planck's previous discovery of the Planck relation (E = hf) linking energy (E) and frequency (f) as arising from quantization of energy. The factor h is known as the Planck constant
What is a wave function?
1. solution to the differential equation of classical wave equation 2. solution to SQ 3. give complete discribtion of any system
electron diffraction pattern of aluminum foil
1. wave-particle duality which states that a particle of matter (in this case the incident electron) can be described as a wave. For this reason, an electron can be regarded as a wave much like sound or water waves. 2. the wavelike property of electrons is used in electron microscopes. In contrast to electromagnetic radiation (UV and X-rays), Electron beam can be readily focused by using electric and magnetic fields, generating sharper images. 3.
English Physicist Sir Joseph J. Thomson
1. won nobel price in 1906 for showing that the electron is a particle 2. Was first person to show that the electron was a subatomic particle in 1895 3. cathode ray tube
1 eV =
1.602x10^-19 J
Planck Constant
6.626x10^-34 Joule second (J*s)
Erwin Schrodinger
Austrian physicist
Kb
Boltzmann constant = R / Avogadro constant. Units J / K
Richard Feynman
Born in new york American theoretical physicist noble prize in physics in 1965 "for fundamental work in quantum electrodynamics"
ideal body
Called a Blackbody; absorbs and emits all frequencies
Bohr's frequency condition
E = h*v, simple and beautiful
blackbody radiation
EM-Radiation given off by materials bodies when they are heated.
Explanation of photoelectric based on classical
Effect can be attributed to the transfer of energy from the light to an electron in metal energy of wave depends only on amplitude (not on its frequency)
In 1905
Einstein used the very same idea (that energies of the oscillator are quantized, integral multiple of hv) to explain the photoelectric effect
Photoelectric effect
Ejection of electrons from the surface of metal by radiation. In 1886 and 1887, Heinrich Hertz discovered that ultraviolet light (radiation) causes electrons to be emitted from a metallic surface.
remove
Equation 1.2
what is a blackbody
If an object is not black then it gives off less thermal radiation and more reflected light. A body is a black body to the degree that it is black, that is, to the degree that it absorbs light. A black body in thermal equilibrium (that is, at a constant temperature) emits electromagnetic radiation called black-body radiation. The radiation is emitted according to Planck's law, meaning that it has a spectrum that is determined by the temperature alone (see figure at right), not by the body's shape or composition. In the late nineteenth century, physicists noticed that the spectrum emitted by hot objects did not depend on the chemical makeup or the shape of the object; all that mattered was temperature. If you can identify the wavelength at which an object emits the most energy per unit wavelength (the peak of the spectrum), then you can estimate its temperature
K = 0
K is Separation constant. yield out a linear equation, X(x) = ax + b and T(t) =at +b. But this is a trivial solution because for the linear equation to satiny the boundary condition, u(0,t) = and u(l, t) = 0, the constant must be 0. a = b = 0.
A plot of the intensity of blackbody radiation versus frequency
Many physicists tried to derive expressions consistent with these experimental curves of intensity verses frequency, but all failed.
How to explain spectrum of H-atom (Niels Bohrs assumptions)
Niels Bohr (1885-1962) Danish physicist 1. existence of stationary electron orbits 2. specified these orbitals by the equivalent of assuming that the de Brohlie waves of the orbiting electrons must "match" as the electron makes the complete revolution. (without such matching, cancellation of some amplitude occurs during each revolution so the wave will disappear)
is moon a blackbody?
No. Because it reflects light that is emitted from the sun
difference between planck and Einstein
Planck: once the light energy is emitted, it behaved like a classical wave. Einstein: radiation itself existed as small packets of energy known as photons. He showed KE of an ejected electron is equal to the energy of the incident photon minus the minimum energy required to remove an electron from the surface of metal.
Diffraction pattern from a slit experiment
Shows that electrons and light behave as wave.
Wien displacement law
The blackbody radiation curve for different temp. peaks at a wavelength incersally proportional to the temp. λ(max) T = 2.90x10^-3 (m K)
Max Planck's quantum hypothesis
The quantum hypothesis, first suggested by Max Planck (1858-1947) in 1900, postulates that light energy can only be emitted and absorbed in discrete bundles called quanta. Planck came up with the idea when attempting to explain blackbody radiation, work that provided the foundation for his quantum theory.
What is Quantum Theory?
The study of small components that compromise the matter. Quantum theory is the theoretical basis of modern physics that explains the nature and behavior of matter and energy on the atomic and subatomic level. The nature and behavior of matter and energy at that level is sometimes referred to as quantum physics and quantum mechanics. In 1900, physicist Max Planck presented his quantum theory to the German Physical Society. Planck had sought to discover the reason that radiation from a glowing body changes in color from red, to orange, and, finally, to blue as its temperature rises. He found that by making the assumption that energy existed in individual units in the same way that matter does, rather than just as a constant electromagnetic wave - as had been formerly assumed - and was therefore quantifiable, he could find the answer to his question. The existence of these units became the first assumption of quantum theory.
is sun a blackbody
The sun is a blackbody because it emits a continuous spectrum. The light from the Sun is not due to transitions between atomic energy levels (in fact, the Sun is mostly a plasma). Instead, it is a thermodynamic system, including a photon gas of the same temperature as the surroundings.
X-ray diffraction
X rays are directed at a crystalline substance, the beam is scattered in a definite manner characteristic of the atomic structure of the crystalline substance. This occurs because the interatomic spacings in the crystal are about the same as the wavelength of the X-rays. The X-rays scatter from the foil in rings of different diameters. The distances between the rings are determined by the interatomic spacing in the metal foil.
Kirchhoff's First law
a hot solid, liquid or dense gas produces a continuous spectrum
Diffraction of waves
a long slit of infinitesimal width which is illuminated by light diffracts the light into a series of circular waves and the wave front which emerges from the slit is a cylindrical wave of uniform intensity. (spacing of the slits has about the same dimensions as the wavelength of the incident wave.) results: diffraction pattern from destructive interference of the circular waves.
emission line spectrum (bright-line spectrum)
a spectrum with bright spectral lines placed close together against dark background
kirchhoff's third law
a thin, cool gas infront of a source of continuous spectrum will form an absorption line spectrum
Kirchhoff's second law
a thin, hot gas produces an emission line spectrum
the second harmonic
also called the first overtone. vibrates harmonically with frequency v/l. The midpoint (node) of this harmonic is fixed at zero for all t. the second harmonic oscillates with twice the frequency of the first harmonic
for the wave pattern around an orbit to be stable
an integral number of complete wavelengths must fit around the circumference of the orbit
what is a wave?
an oscillation accompanied by a transfer of energy that travels through space or mass. Any vibrating body that is connected to its environment will transfer energy to its environment. The vibrations are then transferred though the environment from neighbour to neighbour. This energy transfer is called wave motion. Waves move energy through a medium without moving the whole medium. Leonardo di Vinci "waves made in a field of grain by the wind, ... we see the waves running across the field while the grain remains in place."
Rayleigh-Jeans laws p(T)dv
attempts to describe the spectral radiance of electromagnetic radiation at all wavelengths from a blackbody at a given temp. through classical argument. only agrees with small frequency but strongly disagree at short wavelength.
why should we care about classical wave equations?
because if matter behaves like a wave, then their must exists a wave equation the governs it. Classical wave equation is a good start.
linear
because the unknowns and their derivatives appear only to the first power.
u(0) and u(l) = 0
because they are fixed at boundary condition. they are not nodes!
the radiation emitted by a blackbody is called
blackbody radiation
First term,
called the fundamental mode or first harmonic. represents a harmonic time dependence of frequency v/2l of the motion
Limitations of Bohr's Frequency condition
can not successfully extended to two-electron system (Such as helium) can not explain spectra that arise when a magnetic field is applied to the system can not predict intensities of the spectral lines
the force holding the electron in a circular orbit is supplied by
coulomb force (attraction of proton and electron). Coulomb force is balanced by the centrifugal forces.
absorption line spectrum (dark line spectrum)
dark spectral lines interspersed on a continuous spectrum
According to classical physics
electromagnetic radiation is an electric field oscillating perpendicular to its direction of propagation, and the intensity of the radiation is proportional to the square of the amplitude of the electric field. As intensity increases, so does the amplitude, the electron oscillate more and break of with KE that depends on amplitude. This was in disagreement with the experimental observations. classical physicists expected that when using very dim light, it would take some time for enough light energy to build up to eject an electron from a metallic surface. WRONG!! Experiments show that if light of a certain frequency can eject electrons from a metal, it makes no difference how dim the light is. There is never a time delay.
Radiation
emission of energy as electromagnetic wave or moving subatomic particle
work function of metal (@)
energy required to remove an electron from the surface. unit, electron volts (eV).
states of higher energy are called
excited states and are generally unstable with respect to the ground state
at ordinary temp, H-atoms are found
exclusively in their ground state
black body radiation
given off by material bodies when they are heated.
the third harmonic (or second overtone)
has two nodes number of nodes = n - 1.
learning about the classical wave equation
helps to learn the language of general quantum theory
theory of blackbody radiation is used
in astronomy to estimate the surface temp of stars by obtaining the maximum wavelength using a spectrum and then using the theory.
is incandescent light blackbody?
incandescent light bulb or the burner element on an electric stove. As you increase the setting on the stove from low to high, you can observe it produce blackbody radiation; the element will go from nearly black to glowing red hot.
Ordinary derivatives
is a differential equation containing a function or functions of one independent variable and its derivatives. The term "ordinary" is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.
de Broglie wavelength
is the wavelength, λ, associated with a massive particle and is related to its momentum, p, through the Planck constant, h: De Broglie, in his 1924 PhD thesis, proposed that just as light has both wave-like and particle-like properties, electrons also have wave-like properties, λ = h /mv The concept that matter behaves like a wave
k > 0
let y(x) = e^(ax). (a^2 - k^2)y(x) = 0. If y(x) = 0, it's a trivial solution. and so (a^2 - k^2) = 0. Thus, a = + or -k.
substitute de Broglie wavelength formula ( λ = h / p) we get,
m*v*R = n*h/(2pi) in other words, Bohr: the angular momentum of the electron must be quantized
circumference, 2pi*r =
nλ, n =1, 2, 3
What does the wave equation describe?
propagation of disturbances out from the region at a fixed speed in one or in all spatial directions, as do physical waves from plane or localized sources; the constant c is identified with the propagation speed of the wave.
Finding the Radius of first bohr orbital,
r = ... (equation 1. 17) the equation gives the only vlues of r!
Like R-J, Planck assumed
radiation emitted by the blackbody was caused by the oscillations of the electrons in the constituent particles of the material body,
As body heated to higher temp.
red -> white -> blue
atoms/molecules in excited states will usually
relax back to the ground state and give of this energy as electromagnetic radiation
How to solve differential equations such as the classical wave equation?
separation of variables
n = 1 corresponds to the lowest energy (most stable situation):
so called "ground state"
Planck's revolutionary assumption
that energies of the oscillators were discrete and had to be proportional to an integral multiple of the frequency or in equation form E = n h v. (quantization of energy) h (planks constant) v (characteristic frequency of Oscillator) E = ne (Elementary energy "Quantum")
Ultraviolet Catastrophe
the error at short wavelengths in the Rayleigh-Jeans law for energy emitted by an ideal black body. At high frequencies, the Rayleigh-Jeans law predicts that the radiant energy density diverges as v^2 because the frequency increases as the radiation enters the ultraviolet region.
amplitude
the maximum displacement of the string from its equilibrium horizontal position
standing waves
the position of the nodes are fixed in time.
In term of frequency
the radiation emitted goes from a lower frequency to a higher frequency as the temp increases. (b.c red is lower frequency)
partial differential equation
the unknown is in partial derivatives. such as u(x,t).
Key step in the method of separation of variables is
to assume that u(x,t) factors into a function of x, X(x), times a function of t, T(t), or that u(x,t) = X(x)T(t). (look at equation 2.5) the left side is function of x only and the right side is a function of t only. b.c x and t are independent variables, each side can be varied independently. the only way for equality of the two sides to be preserved under any variation of x and t is to be equal to a constant, K (separation constant).
Energy of Bohr orbitals
total energy of the electron in an atom is equal to the sum of its kinetic energy and potential energy. (look at equation 1.22, read the part) negative sign: the described energies are less than when the proton and electron are infinitely separated. "binding" is energetically attractive.
wilhelm wien
used classical argument to derive wien's displacement law.
threshold frequency
v0 = @/h
how to describe the electron wave
vibrating string ocean waves acoustic waves vibrating drum head
Model
we need 1)assumption, 2)limitation
