Struct WignerDMatrix

Source

pub struct WignerDMatrix { /* private fields */ }

Expand description

Precomputed helper for Wigner rotation matrix elements.

Stores all coefficients needed for repeated evaluation of

$d^j_{m' m}(\beta)$ and $D^j_{m' m}(\alpha,\beta,\gamma)$.

§Definitions

$D^j_{m' m}(\alpha,\beta,\gamma) = e^{-i m' \alpha} d^j_{m' m}(\beta) e^{-i m \gamma}$

§Notes

Designed for reuse across many angle evaluations.

Implementations§

Source §

impl WignerDMatrix

Source

pub fn new( j: impl TryInto<J>, mp: impl TryInto<M>, m: impl TryInto<M>, ) -> LadduPhysicsResult<Self>

Constructs a Wigner small-$d$/full-$D$ matrix element helper for fixed quantum numbers $j$, $m'$, and $m$.

All angular momenta and projections are passed as strongly typed quantum numbers.

The constructed value precomputes the combinatorial factors and summation bounds needed for repeated evaluation of:

the reduced Wigner matrix element $d^j_{m' m}(\beta)$, and
the full Wigner matrix element $D^j_{m' m}(\alpha,\beta,\gamma)$.

§Errors

This method will return an error result if:

|m'| > j
|m| > j
j and m' do not have matching integer/half-integer parity
j and m do not have matching integer/half-integer parity
the internally derived summation bounds are inconsistent

§Notes

The returned struct is intended for reuse when evaluating the same matrix element for many angles. This avoids recomputing factorial-based prefactors on every call.

§Examples

Integer angular momentum:

let w = WignerDMatrix::new(j!(1), m!(1), m!(0)).unwrap(); // j = 1, m' = 1, m = 0

Half-integer angular momentum:

let w = WignerDMatrix::new(j!(1/2), m!(1/2), m!(-1/2)).unwrap(); // j = 1/2, m' = 1/2, m = -1/2

Source

pub fn d(&self, beta: impl Into<Expr>) -> Expr

Evaluates the reduced Wigner small-$d$ matrix element $d^j_{m' m}(\beta)$.

The quantum numbers $j$, $m'$, and $m$ are those fixed when the WignerDMatrix was constructed.

§Parameters

beta: expression for the middle Euler angle $\beta$, in radians

§Returns

Returns an expression graph for the real-valued reduced Wigner matrix element $d^j_{m' m}(\beta)$.

§Notes

This method builds the standard finite sum in powers of $\cos(\beta/2)$ and $\sin(\beta/2)$.

§Examples

let w = WignerDMatrix::new(j!(1), m!(1), m!(0)).unwrap(); // j = 1, m' = 1, m = 0
let expr = w.d(event_scalar("beta"));

Source

pub fn D( &self, alpha: impl Into<Expr>, beta: impl Into<Expr>, gamma: impl Into<Expr>, ) -> Expr

Evaluates the full Wigner $D$ matrix element $D^j_{m' m}(\alpha,\beta,\gamma)$.

The implemented convention is

$D^j_{m' m}(\alpha,\beta,\gamma) = e^{-\imath m' \alpha} d^j_{m' m}(\beta) e^{-\imath m \gamma}$,

§Parameters

alpha: expression for the first Euler angle $\alpha$, in radians
beta: expression for the middle Euler angle $\beta$, in radians
gamma: expression for the third Euler angle $\gamma$, in radians

§Returns

Returns an expression graph for the complex Wigner $D$ matrix element.

§Notes

Since $d^j_{m' m}(\beta)$ is real for real $\beta$, the complex phase comes entirely from the $\alpha$ and $\gamma$ dependence.

§Examples

let w = WignerDMatrix::new(j!(1), m!(1), m!(0)).unwrap(); // j = 1, m' = 1, m = 0
let expr = w.D(event_scalar("alpha"), event_scalar("beta"), event_scalar("gamma"));