chapter-3.-symmetry-group-and-generator

Chapter 3. Symmetry, Group, and Generators

In chapters 1 and 2, we learned about the static “stage” called Hilbert space and the “states” and “operators” defined on it. Now, to move forward to the “dynamics” of part 2, we must explore the principles that cause “change” and “motion” on this stage. The key to this is Symmetry.

In quantum mechanics, symmetry goes beyond mere geometric beauty, serving as a fundamental principle that determines the dynamics of a system and governs physical laws. Symmetry transformations follow a mathematical structure called a Group, and the core engine that creates this “group”—the command that causes “infinitesimally small changes”—is the Generator.

1. Fundamental Concepts

Symmetry: It refers to the situation where the physical properties of a system (especially, measurement probabilities) remain unchanged when a transformation is applied to it. According to Wigner’s theorem, such symmetry transformations must be represented by a Unitary (Unitary) operator \(U\) (or anti-Unitary).
- When transformed as \(|\psi'\rangle = U|\psi\rangle\), \(\langle\phi'|\psi'\rangle = \langle\phi|\psi\rangle\) (inner product preservation).
Detailed Explanation: Why is ⟨ϕ’|ψ’⟩ = ⟨ϕ|ψ⟩ necessary?

“Measurement probability” is always determined by the relationship between two states, namely, the “current state of the system \(|\psi\rangle\)” and the “target (question) state to be measured \(|\phi\rangle\)”, through their inner product. (Probability \(P = |\langle\phi|\psi\rangle|^2\))

Since the symmetry transformation \(U\) must be applied to the entire laboratory, both the system and the measurement device (target state) must be transformed identically.
- Transformed system: \(|\psi'\rangle = U|\psi\rangle\)
- Transformed target state: \(|\phi'\rangle = U|\phi\rangle\)
“Symmetry” means that the physical results (probabilities) must remain the same after this transformation. Because \(U\) is unitary (\(U^\dagger U = \mathbf{1}\)), this preservation is mathematically guaranteed.

\(\langle\phi'|\psi'\rangle = (U|\phi\rangle)^\dagger (U|\psi\rangle) = (\langle\phi| U^\dagger) (U|\psi\rangle) = \langle\phi| (U^\dagger U) |\psi\rangle = \langle\phi| \mathbf{1} |\psi\rangle = \langle\phi|\psi\rangle\).

Therefore, the \(U\) transformation preserves the fundamental inner product (probability amplitude) between two states, and this is the mathematical definition of symmetry.
Group: It is a set of symmetry transformations. This set is closed under operations such as “applying transformations sequentially” (multiplication), “not applying a transformation” (identity element), and “reversing a transformation” (inverse element).
- A group that deals with continuous transformations such as time translation, spatial translation, and rotation is called a Lie Group (Lie Group). > Detailed Explanation: Lie Group - From a Mathematical Tool to the Language of Physics > > 1. Mathematical Motivation (Sophus Lie, Sophus Lie) > > Lie groups were not born with the purpose of physics (time, space symmetry). At the end of the 19th century, the Norwegian mathematician Sophus Lie aimed to develop a general theory that could systematically solve “Differential Equations”. > > His idea was inspired by Galois, who analyzed the roots of “algebraic equations” (e.g., 5th-degree equations) using “discrete symmetry groups” (finite groups). Lie thought similarly that by studying the “continuous symmetry groups” (continuous groups) that the solutions of differential equations possess, one could understand the structure of the equation and find the solution. > > A Lie group is a very special mathematical structure that is both a set of “smooth” transformations and simultaneously forms a geometric space known as a “differentiable manifold”. Lie’s key achievement was revealing that the local structure of this continuous group is completely determined by “infinitesimal transformations”, and the algebraic structure of these infinitesimal transformations is exactly the Lie Algebra. (The “generators” in Chapter 3 of the main text correspond precisely to the elements of this Lie algebra.) > > 2. Adoption in Physics (Why Physics Needs Lie Groups) > > At the beginning of the 20th century, physicists (especially after Einstein and Noether) realized that the fundamental laws of nature are inextricably linked to symmetry. And the most fundamental symmetry that physics deals with is mostly continuous. > > * Time Symmetry: Physical laws are the same whether it is “now” or “one second later.” (Continuous time translation) > * Space Symmetry: Physical laws are the same whether it is “here” or “one meter away.” (Continuous space translation) > * Rotation Symmetry: Physical laws are the same whether you look “this way” or “that way.” (Continuous rotation) > > Physicists discovered that the perfect mathematical tool to describe these continuous symmetries (time translation, space translation, rotation) had already been completed in the late 19th century by Sophus Lie for the purpose of pure mathematics. > > Noether’s theorem perfectly connected these two. “Continuous symmetry in physics (Lie group, Lie group)” gives rise to “conservation laws (Lie algebra/generators, Lie algebra).” > > After that, Lie groups became the core language for describing special relativity theory (Lorentz group \(SO(1,3)\)), spin in quantum mechanics (rotation group \(SO(3)\) and \(SU(2)\)), and the standard model of modern particle physics (gauge group \(U(1) \times SU(2) \times SU(3)\)). > > Generator: A Hermitian operator corresponding to the “seed” or “engine” that generates continuous symmetry transformations (Lie groups). > * The generator of time translation (\(t\)) is the Hamiltonian (\(H\)). > * The generator of space translation (\(x\)) is the momentum operator (\(P_x\)). > * The generator of rotation about the \(z\)-axis (\(\theta\)) is the \(z\)-axis angular momentum operator (\(J_z\)).
Noether’s Theorem (Quantum Version): Just as in classical mechanics, in quantum mechanics as well, symmetry and conservation laws are deeply connected.
- “If the system’s dynamics (Hamiltonian \(H\)) has some continuous symmetry (\(U\)), then the generator (\(G\)) corresponding to that symmetry is a conserved quantity.” Detailed Explanation: Why does symmetry guarantee conservation? (Mathematical Proof)

Noether’s theorem shows that when the two physical concepts of “symmetry” and “conservation” are translated into mathematical language, the two expressions are in fact the same equation.

1. Mathematical Meaning of “Symmetry”: \([H, G] = 0\)

If a system is symmetric under a transformation \(U(\epsilon) = e^{-iG\epsilon/\hbar}\) (where \(\epsilon\) is the magnitude of the transformation, \(G\) is the generator), it means that the Hamiltonian \(H\) does not change when the transformation is applied.
That is, the transformed Hamiltonian \(U(\epsilon)HU(\epsilon)^\dagger\) must be equal to the original \(H\): \(H = U(\epsilon)HU(\epsilon)^\dagger\).
Multiplying both sides by \(U(\epsilon)\) gives \(HU(\epsilon) = U(\epsilon)H\), which means that \(H\) and \(U(\epsilon)\) commute: \([H, U(\epsilon)] = 0\).
Since this must hold for all \(\epsilon\) (especially for infinitely small \(\epsilon\)), \(H\) must necessarily commute with the generator \(G\) as well.
Condition of Symmetry (Mathematically): \([H, G] = 0\)

2. Mathematical Meaning of “Conservation”: \(\frac{d}{dt}\langle G \rangle = 0\)

When a physical quantity \(G\) is conserved, it means that its expectation value (measurement average) \(\langle G \rangle\) does not change over time: \(\frac{d}{dt}\langle G \rangle = 0\).
According to Heisenberg’s equation of motion (or Ehrenfest’s theorem), when \(G\) itself does not change over time (\(\partial G/\partial t = 0\)), the time derivative of the expectation value is given by the commutator with \(H\).
Condition of Conservation (Mathematically): \(\frac{d}{dt}\langle G \rangle = \frac{1}{i\hbar}\langle [G, H] \rangle\)

3. Linking the Two Concepts: \([H, G] = 0 \iff \frac{d}{dt}\langle G \rangle = 0\)

Now, by connecting the above two equations, Noether’s theorem is completed.

Step 1 (Symmetry Assumption): Assume the system has a symmetry generated by \(G\). Mathematically, this means \([H, G] = 0\).
Step 2 (Proof of Conservation): Let’s check whether the physical quantity \(G\) is conserved. \(\frac{d}{dt}\langle G \rangle = \frac{1}{i\hbar}\langle [G, H] \rangle\)
However, from the assumption in Step 1, \([H, G] = 0\), so \([G, H] = -[H, G] = 0\).
Substituting this, we get \(\frac{d}{dt}\langle G \rangle = \frac{1}{i\hbar}\langle 0 \rangle = 0\).

Conclusion: The “symmetry” that \(H\) commutes with \(G\) inevitably leads to the “conservation” that the expectation value of \(G\) does not change over time.

💡 Specific Examples

Spatial Translation Symmetry \(\implies\) Conservation of Momentum If a system is spatially uniform (all positions are equal), the symmetry is spatial translation. The generator of this symmetry is the momentum operator (\(P_x\)). Noether’s theorem: \([H, P_x] = 0\), so \(\frac{d}{dt}\langle P_x \rangle = 0\). (Conservation of momentum)
Rotational Symmetry \(\implies\) Conservation of Angular Momentum If a system is spatially isotropic (all directions are equal), the symmetry is rotation. The generator of this symmetry is the angular momentum operator (\(J_z\)). Noether’s theorem: \([H, J_z] = 0\), so \(\frac{d}{dt}\langle J_z \rangle = 0\). (Conservation of angular momentum)
Time Translation Symmetry \(\implies\) Conservation of Energy If a system is temporally uniform (physical laws do not change over time), the symmetry is time translation. The generator of this symmetry is the Hamiltonian (\(H\)) itself. Noether’s theorem: \([H, H] = 0\), so \(\frac{d}{dt}\langle H \rangle = 0\). (Conservation of energy) > 💡 Why is it called ‘Generator’ instead of ‘Operator’? > > This is a very profound and abstract perspective on the dynamics of quantum mechanics. The name “Generator” is given to this operator because it is the agent that causes ‘infinitesimal transformations’. > > 1. ‘Generator’ as a ‘Rate of Change’: > Let’s consider \(H\) (the Hamiltonian). Saying that \(H\) is the “generator of time translation” means that \(H\) is the “velocity vector” or “rate of change” that determines the direction in which the system state starts to change at the point where time \(t\) is 0. \(G = i\hbar \frac{dU}{dt}\Big|_{t=0}\) > > 2. ‘Recipe’ for ‘Infinitesimal Steps’: > The transformation operator \(U(dt)\) over a very short time \(dt\) is just slightly deviated from the identity operator (\(\mathbf{1}\)). > \(U(dt) \approx \mathbf{1} - \frac{i}{\hbar} H dt\) > Here, \(H\) is the recipe that determines the direction and magnitude of this “infinitely small step”. > > 3. Meaning of ‘Generating’ (Exponential as Compounding): > Then, how do we obtain the transformation \(U(t)\) for a finite time \(t\)? It is by compounding (repeating and multiplying) this “infinitely small step” \(U(dt)\) \(N\) times (with \(t=N \cdot dt\)). > > Since \(U(dt)\) is a matrix (operator) acting on the state vector, applying the transformation continuously corresponds to matrix multiplication rather than function composition. > > * After time \(dt\): \(|\psi(dt)\rangle = U(dt)|\psi(0)\rangle\) > * After time \(2dt\): \(|\psi(2dt)\rangle = U(dt)|\psi(dt)\rangle = \big(U(dt) \cdot U(dt)\big)|\psi(0)\rangle\) > > Therefore, the total transformation operator at time \(t=N\cdot dt\) is expressed as a matrix power (operator power). > \(U(t) = (U(dt))^N = \left(\mathbf{1} - \frac{i}{\hbar} H \frac{t}{N}\right)^N \quad (\text{as } N \to \infty)\) > > This limit generalizes the definition of the exponential function for a scalar \(x\), \(e^x = \lim_{N\to\infty} (1+x/N)^N\), to matrices (operators). > > In mathematics, the Matrix Exponential \(e^A\) is defined in the same form as the Taylor series expansion of the scalar exponential function. Let \(A = \frac{-iHt}{\hbar}\), then, > \[ > e^A \equiv \sum_{k=0}^{\infty} \frac{A^k}{k!} = \mathbf{1} + A + \frac{A^2}{2!} + \frac{A^3}{3!} + \cdots > \] > The limit \(\lim_{N \to \infty} (\mathbf{1} + A/N)^N\) converges exactly to this Taylor series \(e^A\). > > Therefore, the finite transformation \(U(t)\) can be perfectly expressed as the exponential function of the generator. > \[ > U(t) = e^{-iHt/\hbar} > \] > > In other words, a single operator called the generator \(H\) generates the entire ‘group’ of time translation (all values of \(U(t)\)) through the operation of the exponential function (repeatedly applying infinitesimal transformations in a compounding manner). > > This perspective moves beyond the one-dimensional view of seeing the Hamiltonian, momentum, and angular momentum operators as just “some physical quantity,” providing a much more powerful and abstract view of them as “fundamental engines” that cause specific symmetry transformations (translations, rotations). ### 2. Symbols and Core Relations
Symmetry:
- Symmetry is a transformation that preserves all measurement probabilities (i.e., the inner product in Hilbert space).
- Wigner’s Theorem: These probability-preserving transformations must be represented by a Unitary (Unitary) operator \(U\) (\(U^\dagger U = \mathbf{1}\)) or an Anti-Unitary operator (e.g., time reversal). (Most continuous transformations in physics are unitary.)
- State transformation: \(|\psi'\rangle = U |\psi\rangle\)
- Operator transformation (Heisenberg): \(A' = U A U^\dagger\)
1-Parameter Lie Group:
- It refers to a family of unitary transformations \(U(\lambda)\) dependent on a continuous real parameter \(\lambda\) such as time \(t\) or angle \(\theta\).
- Every 1-parameter Lie group can be expressed in the form of an Exponential Map using some Hermitian operator \(G\) (generator). \[ U(\lambda) = e^{-iG\lambda/\hbar} = \exp\left(-\frac{iG\lambda}{\hbar}\right) \]
- (\(\hbar\) is a constant used to match the units of \(G\) and \(\lambda\) physically, although in mathematical definitions, \(\hbar=1\) is also used.)
💡 Deep Dive: Why must \(G\) be Hermitian?

The condition that \(U\) is a unitary operator (\(U^\dagger = U^{-1}\)) is equivalent to the condition that \(G\) is a Hermitian operator (\(G=G^\dagger\)).
1. Hermitian conjugate of \(U\): \(U(\lambda)^\dagger = \left(e^{-iG\lambda/\hbar}\right)^\dagger = \left(\sum_k \frac{1}{k!}\left(\frac{-iG\lambda}{\hbar}\right)^k\right)^\dagger = \sum_k \frac{1}{k!}\left(\frac{+iG^\dagger\lambda}{\hbar}\right)^k = e^{+iG^\dagger\lambda/\hbar}\)
2. Inverse of \(U\): \(U(\lambda)^{-1} = \left(e^{-iG\lambda/\hbar}\right)^{-1} = e^{+iG\lambda/\hbar}\)
3. From the condition \(U^\dagger = U^{-1}\): \(e^{+iG^\dagger\lambda/\hbar} = e^{+iG\lambda/\hbar}\)
4. Since this equation must hold for all \(\lambda\), the operators in the exponent must be equal. \(G = G^\dagger\) (i.e., \(G\) is a Hermitian operator.)
This implies an important physical fact: “The generator of a symmetry transformation (Unitary) is a physical observable (Hermitian).”
Generators and Infinitesimal Transformations:
- \(\lambda \to d\lambda\) (a very small value), the first-order Taylor expansion approximation of the exponential function is as follows. \[ U(d\lambda) \approx \mathbf{1} - \frac{i}{\hbar}G\,d\lambda \]
- Conversely, the generator \(G\) is defined as the ‘rate of change’ of \(U(\lambda)\) at \(\lambda=0\). \[ G = i\hbar \frac{dU(\lambda)}{d\lambda}\Big|_{\lambda=0} \]
Symmetry and Conservation Laws (Noether’s Theorem):
- “Symmetry” definition: A transformation \(U\) is a symmetry of the system’s dynamics (Hamiltonian \(H\)) if the transformed Hamiltonian \(H' = U H U^\dagger\) is equal to the original \(H\). \(U H U^\dagger = H \implies UH = HU \implies [U, H] = 0\).
- Connection to generators: The fact that \(U = e^{-iG\lambda/\hbar}\) commutes with \(H\) (\([U,H]=0\)) means that \(H\) also commutes with all the ‘components’ that make up \(U\). (Considering the exponential expansion, this implies that \(H\) commutes with all powers of \(G\) and \(G^n\).) Therefore, \([U, H] = 0 \iff [G, H] = 0\).
- “Conservation” definition: A physical quantity \(G\) being conserved means that its expectation value \(\langle G \rangle\) does not change over time (\(\frac{d}{dt}\langle G \rangle = 0\)).
- Ehrenfest’s Theorem: (When the operator \(G\) is time-independent) the time rate of change of the expectation value of \(G\) is given by the commutator with \(H\) as follows. \[ \frac{d}{dt}\langle G \rangle = \frac{1}{i\hbar}\langle [G, H] \rangle \]
- Noether’s Theorem (Proof):
  1. Assume the system has a symmetry generated by \(G\) \(\implies [G, H] = 0\).
  2. Substitute this into Ehrenfest’s Theorem \(\implies \frac{d}{dt}\langle G \rangle = \frac{1}{i\hbar}\langle 0 \rangle = 0\).
  3. Conclusion: Symmetry with respect to \(G\) necessarily guarantees that \(G\) is a conserved quantity.
Lie Algebra:
- When a system has more than one continuous symmetry (e.g., 3D rotation), the set of generators \(\{G_i\}\) forms a special algebraic structure called a Lie algebra (\(\mathfrak{g}\)).
- The core of this algebra is the Commutation Relation. The commutator of two generators must be “closed” under a linear combination of the generators. \[ [G_i, G_j] = i\hbar \sum_k f_{ijk} G_k \]
- Here, \(f_{ijk}\) is a real constant that defines the structure of the group and is called the structure constants.
- Example (Lie algebra \(\mathfrak{so}(3)\) of the rotation group \(SO(3)\)): The angular momentum operators \(\{J_x, J_y, J_z\}\), which generate 3D rotations, satisfy the following Lie algebra: \[ [J_x, J_y] = i\hbar J_z \quad (\text{and cyclic permutations}) \]
- This commutation relation itself defines the local (infinitesimal transformation) structure of the “group” and forms the basis for handling more abstract symmetry groups such as \(SU(2)\) (spin).

3. Easy Examples (Examples with Deeper Insight)

Here, we first examine Discrete Groups (groups with a finite or countable number of elements) to become familiar with the concept of a “group,” and then we analyze in detail the key examples of Lie Groups (continuous groups), which are directly connected to Noether’s theorem.

3.1. Basic Example (1): Discrete Groups (Discrete Groups)

🧐 Discrete groups are groups without ‘continuous’ parameters (\(\lambda\)). These examples help understand key group properties such as non-Abelian.

Example 1: \(Z_n\) - “Clock” group (Cyclic Group)
- Physical Situation: 90° rotational symmetry of a square lattice.
- Group: \(Z_4 = \{0, 1, 2, 3\}\). (4 elements)
- Operation: Modular 4 addition (e.g., \(2+3 = 5 \equiv 1 \pmod 4\)).
- Identity: 0 (rotation by 0° = no rotation).
- Inverse: The inverse of 1 (90°) is 3 (270°). (\(1+3=4\equiv 0\)).
- Feature: Since \(a+b = b+a\) always holds, it is an Abelian Group. This can be thought of as a discrete version of a continuous group like \(U(1)\).
Example 2: \(S_3\) - “Swapping Three Objects” Group (Permutation Group)
- Physical Situation: Transformation that swaps the positions of three identical particles.
- Group: Set of all \(3! = 6\) ways to shuffle the three elements \(\{1, 2, 3\}\).
- Operation: Successive application of transformations (function composition).
- Key Feature (Non-Commutativity): The order of operations matters.
  - \(a = (1 \leftrightarrow 2)\) : Swap 1 and 2.
  - \(b = (1 \to 2, 2 \to 3, 3 \to 1)\) : Cycle.
  - \(a \cdot b\) (do \(b\) first then \(a\)): \((1 \to 2 \to 1)\), \((2 \to 3 \to 3)\), \((3 \to 1 \to 2)\). Result: \((2 \leftrightarrow 3)\).
  - \(b \cdot a\) (do \(a\) first then \(b\)): \((1 \to 2 \to 3)\), \((2 \to 1 \to 2)\), \((3 \to 3 \to 1)\). Result: \((1 \leftrightarrow 3)\).
- Conclusion: Since \(a \cdot b \neq b \cdot a\), \(S_3\) is a Non-Abelian Group.
- Physical Significance: This is a simple model showing that most important symmetries in quantum mechanics, such as \(SU(2)\) (rotation group), are non-commutative. Non-commutativity is the source of the commutator \([A, B] \neq 0\) and the uncertainty principle.

3.2. Key Example (2): Lie Groups and Noether’s Theorem

🧐 Let’s explore how continuous symmetries (Lie Groups) are connected to physical conservation laws (generators) with specific quantum mechanics examples.

Example 3: \(U(1)\) - Time Translation
- Symmetry: Physical laws are the same today and tomorrow (uniformity of time).
- Group: \(U(1)\), transformation \(U(t)\) that evolves the system by time \(t\).
- Generator: \(G = H\) (Hamiltonian).
- Group Representation: \(U(t) = e^{-iHt/\hbar}\) (time evolution operator).
- Noether’s Theorem:
  - “Symmetry” means \(H\) does not change over time (\(\partial H/\partial t=0\)).
  - According to Ehrenfest’s theorem, \(\frac{d}{dt}\langle H \rangle = \frac{1}{i\hbar}\langle [H, H] \rangle + \langle \frac{\partial H}{\partial t} \rangle\).
  - Since \([H, H] = 0\) and \(\partial H/\partial t = 0\), \(\frac{d}{dt}\langle H \rangle = 0\).
- Conserved Quantity: \(H\), that is, Energy (Energy).
Example 4: \(U(1)\) - Spatial Translation
- Symmetry: Physical laws are the same here and there (e.g., moving by \(a\)) (uniformity of space). This corresponds to ‘free space’ where the potential \(V(x)\) is constant.
- Group: \(U(1)\), transformation \(T(a)\) that shifts by distance \(a\).
- Generator: \(G = P_x\) (x-axis momentum operator). \(P_x = -i\hbar\frac{\partial}{\partial x}\).
- Group Representation: \(T(a) = e^{-iP_x a/\hbar}\). (This operator shifts the wavefunction as \(\psi(x) \to \psi(x-a)\)).
- Noether’s Theorem: “Symmetry” means that \(H\) commutes with \(P_x\) (\([H, P_x] = 0\)).
- Conserved Quantity: \(P_x\), i.e., Momentum.
- Quantum Mechanics Example:
  - Conserved Case (Symmetry O): Free Particle \(H = \frac{P_x^2}{2m}\).
    \([H, P_x] = \left[\frac{P_x^2}{2m}, P_x\right] = \frac{1}{2m}[P_x^2, P_x] = 0\). (The operator always commutes with itself).
    \(\implies\) Momentum is conserved because symmetry exists.
  - Non-conserved Case (Symmetry X): Harmonic Oscillator \(H = \frac{P_x^2}{2m} + \frac{1}{2}m\omega^2 X^2\).
    \([H, P_x] = \left[\frac{1}{2}m\omega^2 X^2, P_x\right] = \frac{1}{2}m\omega^2 [X^2, P_x]\).
    \([X^2, P_x] = X[X, P_x] + [X, P_x]X = X(i\hbar) + (i\hbar)X = 2i\hbar X \neq 0\).
    \(\implies\) Momentum is not conserved because symmetry is broken (potential depends on position \(X\)). (The particle is reflected back by the spring).
Example 5: \(SU(2)\) / \(SO(3)\) - Rotation
- Symmetry: Physical laws are the same regardless of the direction of rotation (isotropy of space). This corresponds to the situation where the potential depends only on distance, not direction, \(V(r)\) (central force).
- Group: \(SU(2)\) (spin 1/2) or \(SO(3)\) (3D space rotation).
- Generator: \(G = \{J_x, J_y, J_z\}\) (three angular momentum operators).
- Group Representation: \(R_z(\theta) = e^{-iJ_z \theta/\hbar}\) (rotation by \(\theta\) around the Z-axis).
- Noether’s Theorem: “Symmetry” means that \(H\) commutes with all generators (\([H, J_x]=[H, J_y]=[H, J_z]=0\), or simply \([H, \vec{J}]=0\)).
- Conserved Quantity: \(\vec{J}\), i.e., Angular Momentum.
- Quantum Mechanics Example (Non-Commutativity):
  - The rotation group is a Non-Abelian group (Non-Abelian), like \(S_3\) in Example 2. (Rotation around the X-axis followed by rotation around the Y-axis \(\neq\) rotation around the Y-axis followed by rotation around the X-axis).
  - This is manifested by the fact that the generators do not commute with each other. \[ [J_x, J_y] = i\hbar J_z \neq 0 \]
  - Physical Meaning:
    1. \(J_x\) and \(J_y\) cannot be measured simultaneously with exact precision (uncertainty principle).
    2. Even if the system has rotational symmetry and \([H, \vec{J}]=0\), the moment \(|\psi\rangle\) becomes an eigenstate of \(J_z\) when \(J_z\) is measured and its value is determined, \(J_x\) and \(J_y\) become uncertain.
    3. (Example: Hydrogen atom) The electron energy commutes with \(H\), \(J_z\), and \(\vec{J}^2\), so we can simultaneously know three quantum numbers: energy (\(n\)), total angular momentum (\(l\)), and z-axis angular momentum (\(m_l\)).

4. Practice Problems

Basic Concepts of Generators (1~7)

(Basic) Hermiticity of Generators: Using the unitary condition \(U(\lambda)^\dagger U(\lambda) = \mathbf{1}\) and the relation \(U(\lambda) = e^{-iG\lambda}\), prove that \(G\) must be a Hermitian operator (\(G=G^\dagger\)). (Hint: Use \(U(d\lambda) \approx \mathbf{1} - iG d\lambda\) and \(U(d\lambda)^\dagger \approx \mathbf{1} + iG^\dagger d\lambda\) to expand \(U^\dagger U = \mathbf{1}\))
(Basic) Finding Generators: The rotation operator for a spin-1/2 particle around the z-axis is \(R_z(\theta) = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}\). Directly calculate the generator \(G_z\) of this group using the formula \(G_z = i \frac{dR_z(\theta)}{d\theta}\Big|_{\theta=0}\), and show that it is equal to \(\frac{1}{2}\sigma_z\) (i.e., \(J_z/\hbar\)).
(Concept) Meaning of “Generator”: From the perspective of “infinitesimal transformations,” explain why operators like \(H\) or \(P_x\) are called “observables” rather than “generators.” (Hint: Consider the relationship between \(e^{-iG\lambda}\) and \((\mathbf{1} - iG\lambda/N)^N\))
(Calculation) Spatial Translation Generator: The spatial translation operator \(T(a)\) for a one-dimensional wave function is defined as \(T(a)\psi(x) = \psi(x-a)\). Expand \(\psi(x-a)\) in a Taylor series around \(a=0\), and compare it with the form \(T(a) \approx \mathbf{1} - iG_x a\) to determine the relationship between the spatial translation generator \(G_x\) and the momentum operator \(P_x = -i\hbar\frac{d}{dx}\). (Assume \(\hbar=1\))
(Concept) Group and Generator: In the unitary transformation \(U(t) = e^{-iHt/\hbar}\), \(H\) is the “generator”. Then, what is \(U(t)\) called, and what is the relationship between \(H\) and \(U(t)\)? (Hint: \(H\) is the engine, \(U(t)\) is the “movement” of the car.)
(Application) Construction of Transformation Operator: The \(y\)-axis angular momentum operator \(J_y\) is the generator of rotation about the \(y\)-axis. Express the unitary operator \(R_y(\phi)\) that rotates by an angle \(\phi\) about the \(y\)-axis in the exponential form using \(J_y\).
(Application) Phase Transformation: There is a transformation \(U(\phi) = e^{i\phi}\mathbf{1}\) that multiplies the state vector by a global phase \(e^{i\phi}\). What is the generator of this transformation, and which physical quantity is it related to? (Hint: Try rewriting \(U(\phi)\) in the form \(e^{-iG\phi}\). \(G\) is related to the identity operator.)

Symmetry and Conservation Laws (8–14)

(Basic) Proof of Conservation Law: In the Heisenberg picture, the time evolution of an operator \(A\) is given by \(\frac{dA}{dt} = \frac{i}{\hbar}[H, A]\). If an operator \(G\) commutes with the Hamiltonian \(H\) (\([G, H]=0\)), show that \(G\) is a conserved quantity (\(\frac{dG}{dt}=0\)).
(Concept) Symmetry and Commutator: When a transformation \(U\) is said to be a “symmetry” of a system, it means that \(U\) commutes with the Hamiltonian \(H\) (\([U, H] = 0\)). Explain why this implies that “the physical laws are invariant under that transformation”. (Hint: After the \(U\) transformation, the time evolution (\(HU\)) and the time evolution followed by the \(U\) transformation (\(UH\)) must be the same.)
(Concept) Noether’s Theorem: Restate Noether’s theorem, which states “continuous symmetries give rise to conserved quantities”, using the terminology from Chapter 3 (generator, commutator).
(Application) Free Particle: The Hamiltonian \(H = P_x^2 / (2m)\) describes a one-dimensional free particle.
1. Show whether \(H\) commutes with the momentum operator \(P_x\) (\([H, P_x]=?\)).
2. From the result in (a), explain which physical quantity is conserved and what symmetry this corresponds to.
(Application) Central Force Problem: A three-dimensional particle is in a central force potential \(V(r) = V(\sqrt{x^2+y^2+z^2})\) that depends only on the distance from the origin. The Hamiltonian \(H\) of this system is symmetric under rotation about the \(z\)-axis (\(R_z(\theta)\)).
1. What physical quantity can be predicted to be conserved due to this symmetry?
2. What is the generator of that physical quantity?
(Advanced) Non-Conservation: If an operator \(Q\) does not commute with the Hamiltonian \(H\) (\([Q, H] \neq 0\)), predict how the expectation value \(\langle Q \rangle(t) = \langle\psi(t)|Q|\psi(t)\rangle\) of the corresponding physical quantity evolves over time.
(Concept) Discrete Symmetry: The ‘Parity’ operator \(P\) that flips space satisfies \(P=P^\dagger\) and \(P^2=\mathbf{1}\).

Is \(P\) a unitary operator?
Does \(P\) belong to a continuous group expressed as \(U(\lambda) = e^{-iG\lambda}\)? Does this ‘discrete symmetry’ have a generator?

Lie Algebra (15–17)

(Basic) Lie Algebra Calculation: Using the Pauli matrices \(\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\), directly calculate the commutator \([\sigma_x, \sigma_y]\) and show that it equals \(2i\sigma_z\).
(Concept) What is a Lie Algebra?: The relation \([J_x, J_y] = i\hbar J_z\) satisfied by angular momentum generators is called a ‘Lie Algebra’. Explain why this relation is important and relate it to the ‘structure of the group’. (Hint: This relation determines how \(R_x(\theta_1)R_y(\theta_2)\) differs from \(R_y(\theta_2)R_x(\theta_1)\) (non-commutativity).)
(Application) Cyclic Relations: Given the Lie algebra relations \([J_x, J_y] = i\hbar J_z\) and \([J_y, J_z] = i\hbar J_x\), use the properties of commutators (e.g., \([A, B] = -[B, A]\)) to find \([J_z, J_x]\).

Comprehensive Applications (18–20)

(Comprehensive) Harmonic Oscillator: The Hamiltonian of the one-dimensional quantum harmonic oscillator is \(H = \frac{P^2}{2m} + \frac{1}{2}m\omega^2 X^2\).

Is this \(H\) symmetric under spatial translation (generator \(P\))? (\([H, P]\) calculation)
Is this \(H\) symmetric under parity (operator \(P_{op}\) transforms \(X \to -X, P \to -P\))?
From the results of (a) and (b), explain which physical quantities are conserved and which are not.

(Proof) Unitary Invariance: When \(U\) is a unitary transformation, prove that the inner product (overlap) \(\langle\phi|\psi\rangle\) between two states \(|\psi\rangle, |\phi\rangle\) is preserved after transformation, i.e., \(\langle\phi'|\psi'\rangle = \langle\phi|\psi\rangle\). (Here, \(|\psi'\rangle = U|\psi\rangle\))
(Advanced) Baker–Campbell–Hausdorff (BCH): When the generators \(G_1, G_2\) do not commute (\([G_1, G_2] \neq 0\)), \(e^{A}e^{B} \neq e^{A+B}\). When expanding \(e^{-iG_1 \lambda} e^{-iG_2 \lambda}\) into the form \(e^{-i(G_1+G_2)\lambda + \dots}\), predict what the next term is proportional to. (Hint: Refer to the commutator notation \([A, B]\) in Chapter 10 of this book.)

5. Explanations

Since \(U(\lambda) = e^{-iG\lambda}\), we have \(U(\lambda)^\dagger = (e^{-iG\lambda})^\dagger = e^{iG^\dagger\lambda}\).
\(U^\dagger U = e^{iG^\dagger\lambda} e^{-iG\lambda} = \mathbf{1}\).
For this equation to hold for all \(\lambda\), the exponential part must be zero, so \(G^\dagger = G\), i.e., \(G\) must be Hermitian.
Infinitesimal proof: \(U(d\lambda) \approx \mathbf{1} - iG d\lambda\). \(U^\dagger \approx \mathbf{1} + iG^\dagger d\lambda\).
\(U^\dagger U \approx (\mathbf{1} + iG^\dagger d\lambda)(\mathbf{1} - iG d\lambda) \approx \mathbf{1} - iG d\lambda + iG^\dagger d\lambda + O(d\lambda^2) = \mathbf{1} + i(G^\dagger - G)d\lambda\).
For this value to be \(\mathbf{1}\), the term in parentheses must be zero, i.e., \(G=G^\dagger\).
\(G_z = i \frac{d}{d\theta}\Big|_{\theta=0} \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix} = i \begin{pmatrix} -i/2 \cdot e^{-i\theta/2} & 0 \\ 0 & i/2 \cdot e^{i\theta/2} \end{pmatrix}\Big|_{\theta=0}\)
\(= i \begin{pmatrix} -i/2 & 0 \\ 0 & i/2 \end{pmatrix} = \begin{pmatrix} 1/2 & 0 \\ 0 & -1/2 \end{pmatrix} = \frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = \frac{1}{2}\sigma_z\).
‘Generator’ is the recipe for creating a finite transformation (e.g., moving for \(t\) seconds) from an infinitesimal transformation (e.g., moving for \(dt\) seconds). As seen in \(U(t) = (\mathbf{1} - \frac{i}{\hbar} H \frac{t}{N})^N\) (\(N \to \infty\)), \(H\) defines a very small step, and repeating this step infinitely (exponentiating) generates the finite transformation \(t\).
\(\psi(x-a) \approx \psi(x) - a \frac{d\psi(x)}{dx} = (\mathbf{1} - a \frac{d}{dx})\psi(x)\).
Comparing with \(T(a)\psi(x) = (\mathbf{1} - i G_x a)\psi(x)\), we get
\(-i G_x = - \frac{d}{dx} \implies G_x = -i \frac{d}{dx}\).
Since \(P_x = -i\hbar\frac{d}{dx}\), in the unit system where \(\hbar=1\), \(G_x = P_x\). That is, the momentum operator is the generator of spatial translations.
\(H\) is a Generator. \(U(t)\) is an element of the Unitary Group (more precisely, a 1-parameter group) generated by the generator \(H\). If \(H\) is the ‘engine’, then \(U(t)\) represents the ‘actual driving’ for time \(t\).
\(R_y(\phi) = e^{-i J_y \phi / \hbar}\).
\(U(\phi) = e^{i\phi}\mathbf{1} = e^{-i(-\mathbf{1})\phi}\). Therefore, the generator \(G = -\mathbf{1}\) (or \(-c\mathbf{1}\) depending on the unit). This is proportional to the identity operator, and corresponds to the ‘global phase’ symmetry of the entire wavefunction rather than being directly related to any specific physical quantity.
\(\frac{dG}{dt} = \frac{i}{\hbar}[H, G]\). If \([G, H] = 0\), then \([H, G] = -[G, H] = 0\). Therefore, \(\frac{dG}{dt} = 0\), so \(G\) is constant over time, i.e., a conserved quantity.
\(H\) is the generator of time evolution. The statement that \([U, H] = 0\) (i.e., \(UH = HU\)) means that applying “(\(U\) transformation) and (time evolution)” is the same as applying “(time evolution) and (\(U\) transformation)”. This implies that time evolution itself is invariant under the transformation \(U\), i.e., symmetric.
“If the Hamiltonian \(H\) of a system commutes with a continuous unitary transformation \(U(\lambda)=e^{-iG\lambda}\) (\([H, U]=0\)), then the generator \(G\) of that transformation is a conserved quantity (\([H, G]=0\) and \(dG/dt=0\)).”
1. \(H\) depends only on \(P_x\), so it obviously commutes with itself. \([H, P_x] = [P_x^2/(2m), P_x] = 0\).
2. Momentum \(P_x\) is conserved. This means the Hamiltonian has spatial translation symmetry (generator \(P_x\)), and physically it implies “space is uniform and physical laws are the same regardless of position \(x\).”
1. Since the Hamiltonian is symmetric under rotation about the \(z\)-axis, the generator of \(z\)-axis rotation, \(z\)-axis angular momentum (\(J_z\) or \(L_z\)), is conserved.
2. The generator is \(J_z\) (or \(L_z\)).
\(\frac{d\langle Q \rangle}{dt} = \frac{d}{dt}\langle\psi(t)|Q|\psi(t)\rangle = \dots = \frac{i}{\hbar}\langle\psi(t)|[H, Q]|\psi(t)\rangle = \frac{i}{\hbar}\langle[H, Q]\rangle\). Since \([H, Q] \neq 0\), generally the expectation value \(\langle Q \rangle(t)\) has a non-zero time derivative, i.e., it is not conserved and changes over time.
1. \(P=P^\dagger\) and \(PP^\dagger = P^2 = \mathbf{1}\), so it is unitary.
2. No. \(P\) cannot be expressed in the form \(e^{-iG\lambda}\) (if expressed as an exponential, \(G\) is not Hermitian). This is a Discrete Symmetry with no infinitesimal transformation near \(\lambda=0\). Therefore, discrete symmetry (generally) does not have a Hermitian generator.
\([\sigma_x, \sigma_y] = \sigma_x \sigma_y - \sigma_y \sigma_x\) \(= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} - \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\) \(= \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} - \begin{pmatrix} -i & 0 \\ 0 & i \end{pmatrix} = \begin{pmatrix} 2i & 0 \\ 0 & -2i \end{pmatrix} = 2i \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = 2i\sigma_z\).
Lie algebra defines the commutation relations between generators. This relation determines at the infinitesimal level how the multiplication order of finite transformations (group elements) (e.g., \(R_x R_y\) vs \(R_y R_x\)) differs. That is, the algebra of generators determines the local geometric structure of the group.
The property of the commutator \([A, B] = -[B, A]\) is used. \([J_z, J_x] = -[J_x, J_z]\). Using the cyclic relation (\(x \to y \to z \to x\)), from \([J_y, J_z] = i\hbar J_x\) we can deduce that \([J_z, J_x] = i\hbar J_y\) by cycling the indices.
1. \([H, P] = [\frac{P^2}{2m} + \frac{1}{2}m\omega^2 X^2, P] = [\frac{P^2}{2m}, P] + \frac{1}{2}m\omega^2 [X^2, P]\). \([P^2, P] = 0\) and \([X^2, P] = X[X, P] + [X, P]X = X(i\hbar) + (i\hbar)X = 2i\hbar X \neq 0\). Therefore, since \([H, P] \neq 0\), it is not symmetric under spatial translation.
2. Under the parity transformation \(P_{op}\), \(X \to -X\), \(P \to -P\). \(H \to \frac{(-P)^2}{2m} + \frac{1}{2}m\omega^2 (-X)^2 = \frac{P^2}{2m} + \frac{1}{2}m\omega^2 X^2 = H\). Since the Hamiltonian is invariant, it is symmetric under parity.
3. From (a), the momentum \(P\) is not conserved (in the harmonic oscillator, the particle oscillates back and forth, so the momentum keeps changing). From (b), parity is conserved (the wave function maintains even or odd function properties).
\(|\psi'\rangle = U|\psi\rangle, |\phi'\rangle = U|\phi\rangle\). \(\langle\phi'|\psi'\rangle = (U|\phi\rangle)^\dagger (U|\psi\rangle) = \langle\phi| U^\dagger U |\psi\rangle\). \(U\) is unitary, so \(U^\dagger U = \mathbf{1}\). Therefore, \(\langle\phi'|\psi'\rangle = \langle\phi| \mathbf{1} |\psi\rangle = \langle\phi|\psi\rangle\).
(Second-order term of the BCH formula) \(e^A e^B = e^{A+B + \frac{1}{2}[A, B] + \dots}\). Substituting \(A = -iG_1 \lambda\), \(B = -iG_2 \lambda\), we get \(e^{-iG_1 \lambda} e^{-iG_2 \lambda} = \exp\left( -i(G_1+G_2)\lambda + \frac{1}{2}[-iG_1 \lambda, -iG_2 \lambda] + \dots \right)\) \(= \exp\left( -i(G_1+G_2)\lambda - \frac{\lambda^2}{2}[G_1, G_2] + \dots \right)\). Therefore, the next term is proportional to the commutator(commutator) \([G_1, G_2]\) of the two generators.