Spin

Stern-Gerlach Experiment

Bohr Magneton

Bohr magneton is a physical constant that is defined as:

Def Stern-Gerlach Experiment
The Stern-Gerlach experiment is the experiment showing that the

Intrinsic Spin

Definition

Def Spin
Spin is an intrinsic property of particles that is not associated with any spatial coordinates. It is a form of angular momentum that is not due to the motion of the particle. The spin quantum number is a half-integer or integer value that determines the magnitude of the spin angular momentum.

Def Spin Angular Momentum Operator
The spin angular momentum operators are denoted by , , and ,

Prop The spin angular momentum operators satisfy the same commutation relations as the orbital angular momentum operators:
Def Spin Ladder Operators
The spin ladder operators are defined as:

Prop The action of the operators on states of definite spin is given by:

Definition

Def Eigenspinors
The eigenfunctions of the spin operator are called eigenspinors. The eigenspinors of the spin angular momentum operators are denoted by .

e.g. In matrix representation, and . Thus an arbitrary spin state for such a particle can then be represented by .

Pauli Spin Matrices

The Pauli spin matrices are complex matrices defined as:

e.g If a particle has intrinsic spin , then the spin operator can be represented by the Pauli matrices:

The Hydrogen Atom

Proposition

Prop In spherical polar coordinates, the squared angular momentum operator can be expressed as: Proof Simply because .

Bohr Radius

In an atomic system, the Bohr radius is a natural length scale:

Rydberg Constant

The Rydberg constant is a scale for energy that defined as

Definition

Def Effective Potential
The effective potential in a hydrogen system is defined as:

Proposition

Prop The minimum of the effective potential is achieved at where is the Bohr radius. And the minimum value of the effective potential is given by

Theorem

Thrm Bohr Expression
The eigenvalues of the hydrogen system is given by where is defined as the principle quantum number.

Spectroscopic Notation

In spectroscopic notation, we use combination of a number and a letter to denote the principle quantum number and orbital quantum number respectively. If , we use ; If , we use ; If , we use .
e.g.

• stands for , ;
• stands for , ;
• The atom Trusconium has protons, neutrons and electrons, then its shell structure is