# Physics Speedrun - Quantum Theory

## Photon Theory of Light

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

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

and

Quantum of radiation:

### Photoelectric Effect

Photoelectric effect: electron emitted under light.

The photocurrent

- 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 **work function** **photoelectric equation**:

From the equation we know:

has a linerar relationship

### Compton Effect

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:

**scattering angle** *single* photon hits an electroin and knocks it out of the atom, which is an elastic collision procedure. Energy is conserved during the procedure:

The photon loses energy, causing

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

## Bohr Model

### 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:

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

The orbital radius of electron is quantized

where **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

where

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 *displacement* of wave. The energy density of EM wave satisfies **probability density** of finding photon, and let **normallization condition**:

Therefore we can treat the de Broglie wave as a probability wave.

There is interference between **coherent** wave functions.

When **de-conherence** occurs,

### Uncertainty Principle

Measurement disturbs the state of the particle.

When we observe an electron by a photon, increasing **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.

The central bright fringe satisfies the uncertainty relation

### Schrodinger Equation

The Schrodinger equaiton is an equation to determine the wave function **free particle**^{1} with

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:

where

General solution:

Notice that the minimum energy is not zero (zero point energy)

The wave function:

de Broglie wavelength:

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

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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