The light ought to be emitted, transported, and absorbed as tiny particles, or photons.
Blackbody Radiation
All object emit thermal radiation. The total intensity of radiation .
Differnent theories on the spectrum of blackbody radiation
Wien's law predicts the peak wavelength as
Planck's formula compeletely fits the experimental data.
Planck's hypothesis: The energy of any molecular vibration could be only some whole number multiply of , where the Planck constant
and is the frequency of oscillation.
Quantum of radiation:
Photoelectric Effect
Circuit of photoelectic effect
Photoelectric effect: electron emitted under light.
Relationships in photoelectic effect
The photocurrent changes with voltage
When is high, there is saturated photocurrent, related to the intensity of light
When is low, there is a stopping potential / voltage, which is independent of the intensity of light. and changes over the frequency of light.
When is low, there is a cutoff frequency, under which there will not be any photoelectrons.
An electron is ejected out of the metal atom by an inelastic collition with a single photon. To get out of the atom, the electron need to absorb a constant amount of energy from the photon, and the rest of the photon's energy transforms into the electron's kinetic energy . The minimum energy to get out depends on the atom's type, and is called work function. There is the photoelectric equation:
From the equation we know:
has a linerar relationship
Compton Effect
Compton's X-ray scattering experiment
Scattering means light propagate in different directions when passing through material. In classical theory, EM waves are forced vibration, so their frequency (wavelength) should remain the same after being scattered. However, contradictionary experimental results have been observed:
Wavelength changed after scattering
depends on the scattering angle. In the view of photon theory, in Compton scattering, a single photon hits an electroin and knocks it out of the atom, which is an elastic collision procedure. Energy is conserved during the procedure:
Compton scattering
The photon loses energy, causing . With the conservation of momentum:
we can solve the Compton shift:
and the Compton wavelength:
Wave-Partical Duality
Not only light has the property of wave-partial duality, but all matter does, called de Borilie wave or matter-wave. For a partical with momentum , it has a wavelength
Bohr Model
Early Models of Atom
J.J Thomtons's plum-pudding model
Rutherfords's planetary model
Bohr model
The Spectrum of Hydrogen
The spectrum of hydrogen
Balmer's formula for visible lines:
Rydberg constant:
General formula for other series in UV and IR regions:
Lyman series (ultraviolet)
Balmer series (visible)
Paschen series (infrared)
Bohr's Three Postulates
Stationary states: All electrons are in stable and discrete energy level
Quantum transition: An electron jumps to another energy level by emit or absorb a photon
Quantum condition for angular momentum: The electron's possible angular momentum is also discrete
Orbital Properties
Rutherford's model
The orbital radius of electron is quantized
where is called Bohr radius:
Orbital kinetic energy:
Electric potential energy:
Total energy:
is also quantized.
Transition and Radiation
: Ground state,
: First excited state,
: Second excited state,
The energy are all negative, called bound state. The binding / ionization energy:
Jumping from upper state to lower state :
where is the Rydberg constant:
Energy level digram
From de Borglie's hypothesis, we may consider the stable orbit for electron as a standing wave. For de Broglie wave: and for a circular standing wave: We can get the quantum condition by Bohr:
Quantum Mechanics
Wave Function
The wave function is the displacement of wave. The energy density of EM wave satisfies . In the view of particle theory, the number density of photon should satisfy . As discrete particles we can consider the probability density of finding photon, and let . As a probability distribution, the wave function should satisfy the normallization condition:
Therefore we can treat the de Broglie wave as a probability wave. at a certain point represents the probability of finding the particle within volume aboud the given position and time.
Conherent wave functions
There is interference between coherent wave functions.
De-conherence of wave functions
When de-conherence occurs,
Uncertainty Principle
Measurement disturbs the state of the particle.
When we observe an electron by a photon, increasing causes larger , and decreasing causes larger . There is always an uncertainty in position or momentum. The Heisenberg uncertainty principle says that,
The position and the momentum of a particle can not be precisely determined simultaneously.
Other forms:
The principle indicates that
Microscopic particles will not stay at rest.
Diffraction of electron
The central bright fringe satisfies the uncertainty relation
Schrodinger Equation
The Schrodinger equaiton is an equation to determine the wave function . A free particle1 with moves along axis. Consider a wave function in general complex value form
For nonrelativistic free particle:
And consider the potentioal energy, we get its Schrodinger equation:
3D time-dependent Schrodinger equation:
is the Hamilton operator
Time-independent Schrodinger equation:
Solve the equation
Each solution represents a stationary state
The system may be in a superposition state
The wave function of the system should be continuous, finite and normalized.
An infinitely deep well potential
The Schrodinger equation:
where
General solution:
must be continous the energy is quantized:
Notice that the minimum energy is not zero (zero point energy)
The wave function:
Figure of wave function
de Broglie wavelength:
A finite potential well
The paritcle can get out even if (Quantum tunneling)
Tunneling probability
where
Atoms
Schrodinger equation for hydrogen atom
Solution can be labeled with 3 quantum numbers
Principle quantum number
Orbital quantum number is the magnitude of orbital angular momentum
Magnetic quantum number Space quantization:
Except the orbital motion, the electron also has a spin and the spin angular momentum. Every elementary particle has a spin. Spin in a relativistic effect. Spin quantum number can be integers, and such kind of particles are called boson, such as photons. It can also be half-integers, called fermion, like electrons:
Each electron occupies a state
Different possile states for an electron with n=2
In a hydrogen atom, the wave function for ground state
which produces a radial probability distribution
Radial probability distribution
This tells that there is no orbit for the electron in atom. It's a probabality distribution related to different wave functions, like an electron cloud.
The energy of an electron in an atom depends on and . In complex atoms where atomic number , there are two principles for the configuration of electrons:
Lowest energy principle: At the ground state, each electron tends to occupt the lowest energy level. An empirical formula of energy:
Pauli exclusion principle: No two electrons in an atom can occupy the same quantum state.
Shell structure of electrons
No interaction with outside, has spatial symmetry↩︎