Quantum numbers j and m

j is the total angular momentum quantum number (0, 1/2, 1, ...). For fixed j, m takes values −j, −j+1, ..., j, giving 2j+1 states.

A quantum system with intrinsic spin s = 1/2, having exactly two spin states, often written as |↑⟩ and |↓⟩. It is the prototype of a two-level system and a qubit.

Three 2×2 matrices σ_x, σ_y, σ_z that represent spin-1/2 operators up to a factor of ħ/2. They form a basis for 2×2 Hermitian matrices (with identity).

Stern–Gerlach experiment

A 1922 experiment where a beam of atoms in a non-uniform magnetic field splits into discrete spots, demonstrating quantized spin components and two-level outcomes.

Intrinsic form of angular momentum carried by quantum particles, not due to literal spatial rotation, characterized by a spin quantum number s.

The basic unit of quantum information, represented by a normalized superposition of two basis states, mathematically equivalent to a spin-1/2 system.

A system with spin quantum number s = 1/2, having exactly two allowed spin projection values m_s = ±1/2.

What is Pauli matrices?

Three 2×2 matrices σ_x, σ_y, σ_z used to represent spin-1/2 operators and qubit operations.

Angular Momentum, Spin, and Two-Level Quantum Systems — Quantum Mechanics Mastery: From Foundations to Modern Frontiers

From Classical to Quantum Angular Momentum

Classical Angular Momentum

In classical mechanics, angular momentum is L = r × p. It describes how objects rotate around a point, like planets orbiting the Sun or a spinning top.

Quantum Upgrade

In quantum mechanics, angular momentum becomes an operator. Its components do not commute, and only certain quantized values of angular momentum are allowed.

Orbital vs Spin

We distinguish: 1) Orbital angular momentum L from motion in space, and 2) Spin S, an intrinsic angular momentum not due to literal spinning. The total is J = L + S.

Module Roadmap

We will: write angular momentum commutators, obtain allowed quantum numbers, introduce spin-1/2 and Pauli matrices, and connect to real two-level systems like qubits and atomic spins.

Angular Momentum Algebra and Quantization

Commutation Relations

For any angular momentum J: [Jx, Jy] = i ħ J_z, and cyclic permutations. These non-zero commutators mean you cannot know all three components exactly.

Diagonalizing J² and Jz

We choose a basis where J² and Jz are diagonal: J²|j,m⟩ = ħ² j(j+1)|j,m⟩ and Jz|j,m⟩ = ħ m |j,m⟩.

Allowed Quantum Numbers

The total quantum number j can be 0, 1/2, 1, 3/2, .... For each j, m runs from -j to +j in steps of 1, giving 2j+1 possible m values.

Example Values

For j = 1, m = -1, 0, 1. For j = 1/2, m = -1/2, +1/2. This discreteness comes from the algebra and the requirement that J² has a lower bound.

Ladder Operators and Building States

Ladder Operators

Define J+ = Jx + i Jy and J- = Jx - i Jy. These operators raise or lower the m quantum number for fixed j.

Action on States

J+|j,m⟩ = ħ√[j(j+1) − m(m+1)]|j,m+1⟩, and J-|j,m⟩ = ħ√[j(j+1) − m(m−1)]|j,m−1⟩.

Constructing the Multiplet

Start from the highest state |j,j⟩ and apply J_- repeatedly to generate |j,j−1⟩, ..., |j,−j⟩. The ladder stops when the factor under the square root is zero.

Analogy with Oscillator

Like the harmonic oscillator ladder operators raise and lower energy, J_± raise and lower m. The ladder idea is a recurring pattern in quantum systems.

Spin: Intrinsic Angular Momentum

What is Spin?

Spin is an intrinsic angular momentum of particles. It is not a tiny ball spinning; it is a purely quantum property with no exact classical analogue.

Spin Algebra

Spin operators Sx, Sy, Sz satisfy the same commutation relations as orbital angular momentum: [Sx, Sy] = i ħ Sz, and cyclic permutations.

Spin-1/2 Basics

For spin-1/2: s = 1/2, S²|s,ms⟩ = (3/4)ħ²|s,ms⟩. The allowed m_s values are −1/2 and +1/2, giving exactly two spin states.

Up and Down States

We write |↑⟩ for ms = +1/2 and |↓⟩ for ms = −1/2. This two-dimensional space is the same mathematical structure used for qubits.

Stern–Gerlach and Two-Level Outcomes

Stern–Gerlach Setup

A beam of silver atoms passes through a non-uniform magnetic field pointing roughly along z. Classically, we expect a continuous spread of deflections.

Discrete Outcomes

Experimentally, the beam splits into exactly two spots: one up, one down. This shows the z-component of spin only takes two values, +ħ/2 and −ħ/2.

Two-Level Interpretation

We interpret this as atoms emerging in either |↑⟩ (ms = +1/2) or |↓⟩ (ms = −1/2). This is a concrete physical realization of a two-level system.

Modern Technologies

MRI uses proton spin states; atomic clocks use transitions between spin or hyperfine levels; qubits in many platforms are engineered analogues of spin-1/2 systems.

Pauli Matrices and Spin-1/2 Operators

Spin-1/2 Basis Vectors

Represent |↑⟩ = (1, 0)^T and |↓⟩ = (0, 1)^T. Spin states become 2-component vectors, making calculations concrete.

Pauli Matrices

σx = [[0,1],[1,0]], σy = [[0,−i],[i,0]], σ_z = [[1,0],[0,−1]]. They form a basis for 2×2 Hermitian matrices (up to identity).

Spin Operators

Sx = (ħ/2)σx, Sy = (ħ/2)σy, Sz = (ħ/2)σz. These obey the angular momentum commutation relations and act on spinors.

Physical Interpretation

σz measures up vs down along z. σx and σ_y mix |↑⟩ and |↓⟩, corresponding to spin components along x and y, central in qubit control.

Thought Exercise: Spin Measurements Along Different Axes

Work through these questions step by step. You only need basic algebra and the Pauli matrices.

Spin up along z, measure along z

State: `|ψ⟩ = |↑⟩`.
Observable: `Sz = (ħ/2) σz`.
Question: What are the possible outcomes and their probabilities?
Hint: `|↑⟩` is an eigenstate of `S_z`.

Spin up along z, measure along x

Same initial state: `|ψ⟩ = |↑⟩`.
Now measure `Sx = (ħ/2) σx`.
First, write the eigenstates of `σ_x`:
`|+x⟩ = (1/√2)(|↑⟩ + |↓⟩)`
`|-x⟩ = (1/√2)(|↑⟩ - |↓⟩)`
Express `|↑⟩` as a combination of `|+x⟩` and `|-x⟩`.
Question: What are the probabilities to get `+ħ/2` and `-ħ/2` when measuring `S_x`?

Concept check

If you measure `Sz` and get `+ħ/2`, then immediately measure `Sx`, do you always get `+ħ/2`?
What does your answer say about the compatibility of `Sx` and `Sz` measurements?

Try to write down the expansions and probabilities explicitly before checking with a textbook or notes.

Python Demo: Pauli Matrices and a Two-Level Hamiltonian

You can use Python with NumPy to experiment with spin-1/2 systems numerically. This mirrors what many research codes and teaching demos use in 2026.

The Hamiltonian of a spin-1/2 in a static magnetic field `B` along `z` is often written as:

`H = - γ B Sz = - (γ B ħ / 2) σz`

Here `γ` is the gyromagnetic ratio (depends on the particle, e.g., proton vs electron).

Below is a minimal Python script:

```python

import numpy as np

Pauli matrices

sigma_x = np.array([[0, 1], [1, 0]], dtype=complex)

sigma_y = np.array([[0, -1j], [1j, 0]], dtype=complex)

sigma_z = np.array([[1, 0], [0, -1]], dtype=complex)

Parameters (set ħ = 1 for simplicity)

gamma = 1.0 # gyromagnetic ratio (arbitrary units)

B = 0.5 # magnetic field along z

Hamiltonian H = - (gamma B / 2) sigma_z (with ħ = 1)

H = - (gamma B / 2.0) sigma_z

Diagonalize H to find energy eigenvalues and eigenvectors

energies, states = np.linalg.eigh(H)

print("Hamiltonian H:\n", H)

print("\nEigenvalues (energies):", energies)

print("\nEigenvectors (columns):\n", states)

```

Try this:

Change the sign of `B` and see how the energy levels swap.
Replace `sigmaz` with `sigmax` to simulate a field along `x`.
Think about which physical states (|↑⟩ or |↓⟩) correspond to the lower energy for a given sign of `γ` and `B`.

Check Understanding: Spin-1/2 and Pauli Matrices

Answer this quick question to test your understanding of spin-1/2 systems.

For a spin-1/2 particle, which statement is correct about the operators S_x, S_y, S_z?

All three components commute, so you can know them exactly at the same time.
They satisfy [S_x, S_y] = i ħ S_z (and cyclic), so you can diagonalize at most S² and one component simultaneously.
S_x, S_y, S_z all have continuous spectra, similar to position.

Show Answer

Answer: B) They satisfy [S_x, S_y] = i ħ S_z (and cyclic), so you can diagonalize at most S² and one component simultaneously.

Spin components obey the angular momentum commutation relations: [S_x, S_y] = i ħ S_z and cyclic permutations. This means you can choose a basis where S² and one component (usually S_z) are diagonal, but the other components are incompatible observables and cannot be known simultaneously with arbitrary precision.

Key Term Review: Angular Momentum and Two-Level Systems

Use these flashcards to review the core ideas from this module.

Angular momentum commutation relations: For any angular momentum J: [J_x, J_y] = i ħ J_z and cyclic permutations. They imply quantization and mutual incompatibility of different components.
Quantum numbers j and m: j is the total angular momentum quantum number (0, 1/2, 1, ...). For fixed j, m takes values −j, −j+1, ..., j, giving 2j+1 states.
Spin-1/2 system: A quantum system with intrinsic spin s = 1/2, having exactly two spin states, often written as |↑⟩ and |↓⟩. It is the prototype of a two-level system and a qubit.
Pauli matrices: Three 2×2 matrices σ_x, σ_y, σ_z that represent spin-1/2 operators up to a factor of ħ/2. They form a basis for 2×2 Hermitian matrices (with identity).
Stern–Gerlach experiment: A 1922 experiment where a beam of atoms in a non-uniform magnetic field splits into discrete spots, demonstrating quantized spin components and two-level outcomes.
Two-level Hamiltonian in a magnetic field: For a spin-1/2 in a field B along z: H = −γ B S_z = −(γ B ħ / 2) σ_z. It produces two energy levels separated by γ B ħ.
Connection to qubits: A qubit is mathematically identical to a spin-1/2 system: any pure qubit state can be written as a superposition of two basis states, analogous to |↑⟩ and |↓⟩.

Key Terms

spin: Intrinsic form of angular momentum carried by quantum particles, not due to literal spatial rotation, characterized by a spin quantum number s.
qubit: The basic unit of quantum information, represented by a normalized superposition of two basis states, mathematically equivalent to a spin-1/2 system.
spin-1/2: A system with spin quantum number s = 1/2, having exactly two allowed spin projection values m_s = ±1/2.
Pauli matrices: Three 2×2 matrices σ_x, σ_y, σ_z used to represent spin-1/2 operators and qubit operations.
ladder operators: Operators (J_+, J_−) that raise or lower the magnetic quantum number m for a fixed total angular momentum j.
two-level system: A quantum system with exactly two accessible energy or basis states, such as a spin-1/2 particle or a qubit.
gyromagnetic ratio: Constant γ relating a particle's magnetic moment to its angular momentum; appears in spin Hamiltonians in magnetic fields.
magnetic quantum number: The quantum number m specifying the component of angular momentum along a chosen axis, with allowed values from −j to +j in steps of 1.
angular momentum operator: A quantum mechanical operator (L, S, or J) that generalizes classical angular momentum and obeys specific commutation relations.
Stern–Gerlach experiment: An experiment where atoms in a non-uniform magnetic field split into discrete beams, revealing quantized spin projections.