Spin Rotations with Wigner D-Functions

Hello! Welcome back. In our last lesson, we built the entire theory of angular momentum from its fundamental commutation relations, deriving the quantized spectrum of the ${\hat{J}}^2$ and ${\hat{J}}_z$ operators and establishing the abstract basis states $|j,m⟩$.

This lesson continues that journey by addressing a crucial physical question: how do these angular momentum states, like the state of a spin, transform when we physically rotate the system? Answering this will lead us to the rotation operator, $\hat{R}$, and its matrix representation in the $|j,m⟩$ basis. These representation matrices are known as the Wigner D-matrices.

By the end of this lesson, you will be able to apply these rotation matrices and the associated Wigner D-functions to describe how spin states and spin operators transform under arbitrary rotations. This is a vital tool, especially in your field of condensed matter physics, for handling anisotropic interactions that depend on orientation, such as spin-orbit coupling or dipolar interactions in a crystal lattice.

1. The Rotation Operator in Quantum Mechanics

You'll recall from your mechanics and previous quantum courses that symmetries lead to conservation laws. Rotational symmetry, specifically the isotropy of space, implies the conservation of angular momentum. The angular momentum operator, $\hat{\vec{J}}$, is the generator of rotations.

For an infinitesimal rotation by an angle $d\varphi$ about an axis defined by the unit vector $\vec{n}$, the rotation operator $\hat{R}$ is:

$$\hat{R}(\vec{n},d\varphi )=\hat{1}-\frac{i}{\hslash }d\varphi (\vec{n}\cdot \hat{\vec{J}})$$

A finite rotation by an angle $\varphi$ can be constructed by composing an infinite number of these infinitesimal rotations. This process leads to the exponential form of the rotation operator:

$$\hat{R}(\vec{n},\varphi )=\exp \left(-\frac{i}{\hslash }\varphi (\vec{n}\cdot \hat{\vec{J}})\right)$$

Since $\hat{\vec{J}}$ is a Hermitian operator, $\hat{R}$ is a unitary operator (${\hat{R}}^{†}\hat{R}=\hat{1}$), which ensures that the norm of a quantum state is preserved under rotation.

2. Euler Angles and the Structure of the D-matrix

Any arbitrary rotation in three dimensions can be parameterized by three angles. A common and standard choice in quantum mechanics is the Z-Y-Z Euler angle convention. An arbitrary rotation is decomposed into three successive rotations about fixed laboratory axes:

1. A rotation by angle $\alpha$ about the $z$-axis.
2. A rotation by angle $\beta$ about the $y$-axis.
3. A rotation by angle $\gamma$ about the $z$-axis.

The corresponding rotation operator is the product of these three operations:

$$\hat{R}(\alpha ,\beta ,\gamma )={\hat{R}}_z(\alpha ){\hat{R}}_y(\beta ){\hat{R}}_z(\gamma )=e^{-i\alpha {\hat{J}}_z/\hslash }{e}^{-i\beta {\hat{J}}_y/\hslash }{e}^{-i\gamma {\hat{J}}_z/\hslash }$$

Note: There are many different Euler angle conventions. This Z-Y-Z convention is common in nuclear and particle physics literature (and often used in NMR, which is relevant to your interest in experimental condensed matter).

Now, let's consider how a general angular momentum state $|j,m⟩$ transforms under this rotation. The rotated state, which we'll call $|j,m{⟩}^{\prime }$, is:

$$|j,m{⟩}^{\prime }=\hat{R}(\alpha ,\beta ,\gamma )|j,m⟩$$

To understand this new state, we can express it as a linear combination of the original basis states $|j,m^{\prime }⟩$. We do this by inserting a completeness relation, ${\sum }_{m^{\prime }}|j,m^{\prime }⟩⟨j,m^{\prime }|=\hat{1}$:

$$|j,m{⟩}^{\prime }=\sum _{m^{\prime }=-j}^j|j,m^{\prime }⟩⟨j,m^{\prime }|\hat{R}(\alpha ,\beta ,\gamma )|j,m⟩$$

The coefficients in this expansion are the matrix elements of the rotation operator in the $|j,m⟩$ basis. These are the celebrated Wigner D-functions:

$$D_{m^{\prime }m}^j(\alpha ,\beta ,\gamma )\equiv ⟨j,m^{\prime }|\hat{R}(\alpha ,\beta ,\gamma )|j,m⟩$$

So, the transformation rule for the basis states is:

$$\hat{R}(\alpha ,\beta ,\gamma )|j,m⟩=\sum _{m^{\prime }=-j}^j|j,m^{\prime }⟩D_{m^{\prime }m}^j(\alpha ,\beta ,\gamma )$$

Let's analyze the structure of the D-function by substituting the Euler angle decomposition of $\hat{R}$:

$$D_{m^{\prime }m}^j(\alpha ,\beta ,\gamma )=⟨j,m^{\prime }|e^{-i\alpha {\hat{J}}_z/\hslash }{e}^{-i\beta {\hat{J}}_y/\hslash }{e}^{-i\gamma {\hat{J}}_z/\hslash }|j,m⟩$$

Since $|j,m⟩$ and $|j,m^{\prime }⟩$ are eigenstates of ${\hat{J}}_z$ (with eigenvalues $m\hslash$ and $m^{\prime }\hslash$), the exponential operators involving ${\hat{J}}_z$ act simply as phase factors:

$$D_{m^{\prime }m}^j(\alpha ,\beta ,\gamma )=⟨j,m^{\prime }|e^{-i\alpha {\hat{J}}_z/\hslash }\left(e^{-i\beta {\hat{J}}_y/\hslash }|j,m⟩e^{-im\gamma }\right)=e^{-i{m}^{\prime }\alpha }⟨j,m^{\prime }|e^{-i\beta {\hat{J}}_y/\hslash }|j,m⟩e^{-im\gamma }$$

The central part, which involves the rotation about the y-axis, is called the Wigner small d-matrix:

$$d_{m^{\prime }m}^j(\beta )\equiv ⟨j,m^{\prime }|e^{-i\beta {\hat{J}}_y/\hslash }|j,m⟩$$

The full D-function is then elegantly expressed as:

$$D_{m^{\prime }m}^j(\alpha ,\beta ,\gamma )=e^{-i{m}^{\prime }\alpha }{d}_{m^{\prime }m}^j(\beta )e^{-im\gamma }$$

All the complexity of the rotation is encapsulated in the real-valued $d_{m^{\prime }m}^j(\beta )$ matrix, while the $\alpha$ and \gamma) rotations contribute simple phase factors.

3. The Wigner Matrix for Spin-1/2

The most important case for many applications, including the description of electrons, protons, and neutrons, is $j=1/2$. Let's construct the D-matrix for this case explicitly.

The basis states are $|1/2,1/2⟩$ and $|1/2,-1/2⟩$, often denoted as "spin up" $|\uparrow ⟩$ and "spin down" $|\downarrow ⟩$. The angular momentum operators are given by the Pauli matrices, $\hat{\vec{J}}=\frac{\hslash }{2}\vec{\sigma }$, where:

$${\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$$

To find $d^{1/2}(\beta )$, we need to compute the matrix exponential $e^{-i\beta {\hat{J}}_y/\hslash }=e^{-i(\beta /2){\sigma }_y}$. We can use the Taylor series expansion of the exponential. A key property of Pauli matrices is that ${\sigma }_y^2=\hat{1}$ (the identity matrix). This simplifies the expansion greatly:
\begin{align*}
e^{-i(\beta/2)\sigma_y} &= \sum_{n=0}^{\infty} \frac{(-i\beta/2)^n}{n!} \sigma_y^n \
&= \left(\sum_{k=0}^{\infty} \frac{(-1)^k(\beta/2)^{2k}}{(2k)!}\right)\hat{1} - i \left(\sum_{k=0}^{\infty} \frac{(-1)^k(\beta/2)^{2k+1}}{(2k+1)!}\right)\sigma_y \
&= \cos(\beta/2)\hat{1} - i\sin(\beta/2)\sigma_y
\end{align*}
Substituting the matrix for ${\sigma }_y$, we get:

$$e^{-i\beta {\hat{J}}_y/\hslash }=\cos (\beta /2)\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)-i\sin (\beta /2)\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)=\left(\begin{array}{cc}\cos (\beta /2)& -\sin (\beta /2)\\ \sin (\beta /2)& \cos (\beta /2)\end{array}\right)$$

This is the matrix for $d^{1/2}(\beta )$. Its elements are:

$$d^{1/2}(\beta )=\left(\begin{array}{cc}{d}_{1/2,1/2}^{1/2}& d_{1/2,-1/2}^{1/2}\\ d_{-1/2,1/2}^{1/2}& d_{-1/2,-1/2}^{1/2}\end{array}\right)=\left(\begin{array}{cc}\cos (\beta /2)& -\sin (\beta /2)\\ \sin (\beta /2)& \cos (\beta /2)\end{array}\right)$$

Now we can write the full Wigner D-matrix $D^{1/2}(\alpha ,\beta ,\gamma )$ by multiplying the phase factors:

$$D^{1/2}(\alpha ,\beta ,\gamma )=\left(\begin{array}{cc}{e}^{-i\alpha /2}& 0\\ 0& e^{i\alpha /2}\end{array}\right)\left(\begin{array}{cc}\cos (\beta /2)& -\sin (\beta /2)\\ \sin (\beta /2)& \cos (\beta /2)\end{array}\right)\left(\begin{array}{cc}{e}^{-i\gamma /2}& 0\\ 0& e^{i\gamma /2}\end{array}\right)$$ $$D^{1/2}(\alpha ,\beta ,\gamma )=\left(\begin{array}{cc}{e}^{-i\alpha /2}\cos (\beta /2)e^{-i\gamma /2}& -e^{-i\alpha /2}\sin (\beta /2)e^{i\gamma /2}\\ e^{i\alpha /2}\sin (\beta /2)e^{-i\gamma /2}& e^{i\alpha /2}\cos (\beta /2)e^{i\gamma /2}\end{array}\right)$$

This $2\times 2$ unitary matrix with determinant +1 is an element of the group SU(2), which you may know is the "double cover" of the rotation group SO(3). The half-angles $(\alpha /2,\beta /2,\gamma /2)$ are a characteristic feature of spinor transformations. For example, a rotation by $2\pi$ ($\beta =2\pi$, say) results in a D-matrix of $-\hat{1}$, meaning the spinor state vector picks up a minus sign, a purely quantum mechanical effect.

Test your understanding!

An electron is in a spin-up state along the z-axis, $|\psi ⟩=|\uparrow {⟩}_z=\left(\begin{array}{c}1\\ 0\end{array}\right)$. The system is then rotated by an angle of $\pi$ around the y-axis. What is the new state $|{\psi }^{\prime }⟩$? What is the probability of measuring the electron to be spin-down along the z-axis after the rotation?

Show answer

The rotation is described by Euler angles $\alpha =0,\beta =\pi ,\gamma =0$. The new state is given by $|{\psi }^{\prime }⟩=D^{1/2}(0,\pi ,0)|\psi ⟩$.

The D-matrix for this rotation is:

$$D^{1/2}(0,\pi ,0)=\left(\begin{array}{cc}\cos (\pi /2)& -\sin (\pi /2)\\ \sin (\pi /2)& \cos (\pi /2)\end{array}\right)=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$$

Applying this to the initial state:

$$|{\psi }^{\prime }⟩=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\left(\begin{array}{c}1\\ 0\end{array}\right)=\left(\begin{array}{c}0\\ 1\end{array}\right)$$

This is the spin-down state along the z-axis, $|\downarrow {⟩}_z$.
The probability of measuring spin-down is the square of the amplitude of the $|\downarrow {⟩}_z$ component, which is $|1{|}^2=1$. So, the probability is 100%. This makes physical sense: a 180-degree rotation around the y-axis flips the z-axis, so a spin aligned with +z becomes aligned with -z.

4. Transformation of Operators

So far, we've focused on how state vectors transform (the "active" rotation picture). Equally important is how operators, particularly vector and tensor operators constructed from spin, transform. If a state transforms as $|{\psi }^{\prime }⟩=\hat{R}|\psi ⟩$, then for the expectation value of an operator $\hat{A}$ to be invariant under a coordinate system rotation, the operator must transform as:

$${\hat{A}}^{\prime }=\hat{R}\hat{A}{\hat{R}}^{†}$$

This describes the operator ${\hat{A}}^{\prime }$ in the new, rotated coordinate system.

For a vector operator like spin $\hat{\vec{S}}=({\hat{S}}_x,{\hat{S}}_y,{\hat{S}}_z)$, its components in the rotated frame $({\hat{S}}_x^{\prime },{\hat{S}}_y^{\prime },{\hat{S}}_z^{\prime })$ are given by:

$${\hat{S}}_i^{\prime }=\hat{R}{\hat{S}}_i{\hat{R}}^{†}$$

More generally, this concept extends to irreducible spherical tensor operators, $T_q^{(k)}$. These are sets of $2k+1$ operators that transform under rotation in exactly the same way as the spherical harmonics $Y_{kq}$. Their transformation rule is a direct generalization of the state transformation:

$$\hat{R}{T}_q^{(k)}{\hat{R}}^{†}=\sum _{q^{\prime }=-k}^k{T}_{q^{\prime }}^{(k)}{D}_{q^{\prime }q}^k(\alpha ,\beta ,\gamma )$$

This transformation property is the formal definition of a spherical tensor operator. It is fundamental in condensed matter physics for describing anisotropic interactions (e.g., crystal field effects, magnetic anisotropy), as it allows you to express an interaction defined in a local (e.g., molecular) frame in the laboratory frame.

The following resources provide a more formal and detailed look at Wigner D-matrices, their properties, and their application to rotating tensors.

Conclusion

In this lesson, we have developed the machinery to handle rotations in quantum mechanics. We moved from the abstract generators of rotation ($\hat{\vec{J}}$) to concrete matrix representations for any given angular momentum $j$.

Key Takeaways:

• The rotation operator for a finite rotation is given by $\hat{R}(\vec{n},\varphi )=\exp (-\frac{i}{\hslash }\varphi (\vec{n}\cdot \hat{\vec{J}}))$.
• The Wigner D-matrix, $D_{m^{\prime }m}^j(\alpha ,\beta ,\gamma )$, is the matrix representation of the rotation operator $\hat{R}(\alpha ,\beta ,\gamma )$ in the basis of angular momentum eigenstates $|j,m⟩$.
• The D-matrix can be factored into phase factors for the $z$-rotations and the Wigner small d-matrix for the $y$-rotation: $D_{m^{\prime }m}^j(\alpha ,\beta ,\gamma )=e^{-i{m}^{\prime }\alpha }{d}_{m^{\prime }m}^j(\beta )e^{-im\gamma }$.
• We explicitly calculated the $D^{1/2}$ matrix, which describes the transformation of spin-1/2 states (spinors) and is an element of SU(2).
• Operators transform under rotation as ${\hat{A}}^{\prime }=\hat{R}\hat{A}{\hat{R}}^{†}$. Spherical tensor operators $T_q^{(k)}$ transform via a linear combination involving the D-matrices, which is their defining property.

This framework is not just a mathematical formality; it is a practical tool. When dealing with a spin in a magnetic field that is not aligned with the z-axis, you use these rotation matrices to rotate your operators or states into a more convenient coordinate system.

Preview of the next lesson:
Now that we have established the transformation properties of angular momentum states and operators, we can combine them. The Wigner-Eckart Theorem, which we will cover next, is a powerful result that separates the geometric aspects of a system (related to rotations and orientations) from its dynamic aspects. It dramatically simplifies the calculation of matrix elements for tensor operators, which is essential for calculating transition rates and energy level shifts in atoms and solids.