Physics Speedrun - Quantum Theory
Photon Theory of Light
The light ought to be emitted, transported, and absorbed as tiny particles, or photons.
All object emit thermal radiation. The total intensity of 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
Quantum of radiation:
Photoelectric effect: electron emitted under light.
is high, there is saturated photocurrent, related to the intensity of light
is low, there is a stopping potential / voltage , which is independent of the intensity of light. and changes over the frequency of light.
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
From the equation we know:
has a linerar relationship
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:
The photon loses energy, causing
we can solve the Compton shift:
and the Compton wavelength:
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
Early Models of Atom
- J.J Thomtons's plum-pudding model
- Rutherfords's planetary model
- Bohr model
The Spectrum of Hydrogen
Balmer's formula for visible lines:
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
The orbital radius of electron is quantized
Orbital kinetic energy:
Electric potential 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
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:
The wave function
Therefore we can treat the de Broglie wave as a probability wave.
There is interference between coherent wave functions.
When de-conherence occurs,
Measurement disturbs the state of the particle.
When we observe an electron by a photon, increasing
The position and the momentum of a particle can not be precisely determined simultaneously.
The principle indicates that
Microscopic particles will not stay at rest.
The central bright fringe satisfies the uncertainty relation
The Schrodinger equaiton is an equation to determine the wave function
For nonrelativistic free particle:
And consider the potentioal energy, we get its Schrodinger equation:
3D time-dependent Schrodinger equation:
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.
The Schrodinger equation:
Notice that the minimum energy is not zero (zero point energy)
The wave function:
de Broglie wavelength:
The paritcle can get out even if
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
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
In a hydrogen atom, the wave function for ground state
which produces a 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
- 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.
No interaction with outside, has spatial symmetry↩︎
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