# Physics Speedrun - Quantum Theory

## Photon Theory of Light

The light ought to be emitted, transported, and absorbed as tiny particles, or photons. 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.

### Photoelectric Effect Circuit of photoelectic effect

Photoelectric effect: electron emitted under light. Relationships in photoelectic effect

The photocurrent changes with voltage

1. When is high, there is saturated photocurrent, related to the intensity of light
2. When is low, there is a stopping potential / voltage , which is independent of the intensity of light. and changes over the frequency of light.
3. 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:

1. 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

1. J.J Thomtons's plum-pudding model
2. Rutherfords's planetary model
3. 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

1. Stationary states: All electrons are in stable and discrete energy level
2. Quantum transition: An electron jumps to another energy level by emit or absorb a photon
3. 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

Orbital kinetic energy:

Electric potential energy:

Total energy:

is also quantized.

• : 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

1. Each solution represents a stationary state
2. The system may be in a superposition state
3. 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

1. Principle quantum number
2. Orbital quantum number is the magnitude of orbital angular momentum
3. 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 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:

1. Lowest energy principle: At the ground state, each electron tends to occupt the lowest energy level. An empirical formula of energy:
2. Pauli exclusion principle: No two electrons in an atom can occupy the same quantum state. Shell structure of electrons

1. No interaction with outside, has spatial symmetry↩︎