This page fixes the phase-space conventions used in the Dynamics volume. It is a reference page for signs, factors of 2 π ℏ 2\pi\hbar 2 π ℏ Weyl Transform , Wigner Function , Star Product , and Moyal Bracket .
The conventions are chosen so that
⟨ x ∣ p ⟩ = 1 2 π ℏ e i p x / ℏ , \langle x|p\rangle
=
\frac{1}{\sqrt{2\pi\hbar}}e^{ipx/\hbar}, ⟨ x ∣ p ⟩ = 2 π ℏ 1 e i p x /ℏ , and
x ⋆ p − p ⋆ x = i ℏ . x\star p-p\star x=i\hbar. x ⋆ p − p ⋆ x = i ℏ. Different books may place signs or factors in different places. Translate definitions before comparing Wigner functions, Weyl symbols, or star products.
The position-momentum convention is
ψ ( x ) = 1 2 π ℏ ∫ − ∞ ∞ d p e i p x / ℏ ϕ ( p ) , \psi(x)
=
\frac{1}{\sqrt{2\pi\hbar}}
\int_{-\infty}^{\infty}dp\,
e^{ipx/\hbar}\phi(p), ψ ( x ) = 2 π ℏ 1 ∫ − ∞ ∞ d p e i p x /ℏ ϕ ( p ) , and
ϕ ( p ) = 1 2 π ℏ ∫ − ∞ ∞ d x e − i p x / ℏ ψ ( x ) . \phi(p)
=
\frac{1}{\sqrt{2\pi\hbar}}
\int_{-\infty}^{\infty}dx\,
e^{-ipx/\hbar}\psi(x). ϕ ( p ) = 2 π ℏ 1 ∫ − ∞ ∞ d x e − i p x /ℏ ψ ( x ) . Consequently,
⟨ x ∣ p ⟩ = 1 2 π ℏ e i p x / ℏ . \langle x|p\rangle
=
\frac{1}{\sqrt{2\pi\hbar}}e^{ipx/\hbar}. ⟨ x ∣ p ⟩ = 2 π ℏ 1 e i p x /ℏ . This sign choice fixes the sign in the Wigner and Weyl transforms below. The site-wide convention page is Fourier Transform Conventions .
For an operator A A A
A W ( x , p ) = ∫ − ∞ ∞ d y e − i p y / ℏ ⟨ x + y 2 | A | x − y 2 ⟩ . A_W(x,p)
=
\int_{-\infty}^{\infty}dy\,
e^{-ipy/\hbar}
\left\langle
x+\frac{y}{2}
\middle|
A
\middle|
x-\frac{y}{2}
\right\rangle. A W ( x , p ) = ∫ − ∞ ∞ d y e − i p y /ℏ ⟨ x + 2 y A x − 2 y ⟩ . Equivalently, if
A ( x 1 , x 2 ) = ⟨ x 1 ∣ A ∣ x 2 ⟩ , A(x_1,x_2)
=
\langle x_1|A|x_2\rangle, A ( x 1 , x 2 ) = ⟨ x 1 ∣ A ∣ x 2 ⟩ , then
A W ( x , p ) = ∫ d y e − i p y / ℏ A ( x + y 2 , x − y 2 ) . A_W(x,p)
=
\int dy\,
e^{-ipy/\hbar}
A\left(
x+\frac{y}{2},
x-\frac{y}{2}
\right). A W ( x , p ) = ∫ d y e − i p y /ℏ A ( x + 2 y , x − 2 y ) . The inverse transform is
⟨ x 1 | A | x 2 ⟩ = ∫ d p 2 π ℏ e i p ( x 1 − x 2 ) / ℏ A W ( x 1 + x 2 2 , p ) . \left\langle x_1\middle|A\middle|x_2\right\rangle
=
\int\frac{dp}{2\pi\hbar}\,
e^{ip(x_1-x_2)/\hbar}
A_W\left(
\frac{x_1+x_2}{2},
p
\right). ⟨ x 1 ∣ A ∣ x 2 ⟩ = ∫ 2 π ℏ d p e i p ( x 1 − x 2 ) /ℏ A W ( 2 x 1 + x 2 , p ) . The Weyl symbol is tied to symmetric ordering. For example,
( 1 2 ( x ^ p ^ + p ^ x ^ ) ) W = x p . \left(
\frac12(\hat x\hat p+\hat p\hat x)
\right)_W
=
xp. ( 2 1 ( x ^ p ^ + p ^ x ^ ) ) W = x p . The Wigner function of a density operator ρ \rho ρ
W ρ ( x , p ) = 1 2 π ℏ ρ W ( x , p ) . W_\rho(x,p)
=
\frac{1}{2\pi\hbar}\rho_W(x,p). W ρ ( x , p ) = 2 π ℏ 1 ρ W ( x , p ) . Explicitly,
W ρ ( x , p ) = 1 2 π ℏ ∫ − ∞ ∞ d y e − i p y / ℏ ⟨ x + y 2 | ρ | x − y 2 ⟩ . W_\rho(x,p)
=
\frac{1}{2\pi\hbar}
\int_{-\infty}^{\infty}dy\,
e^{-ipy/\hbar}
\left\langle
x+\frac{y}{2}
\middle|
\rho
\middle|
x-\frac{y}{2}
\right\rangle. W ρ ( x , p ) = 2 π ℏ 1 ∫ − ∞ ∞ d y e − i p y /ℏ ⟨ x + 2 y ρ x − 2 y ⟩ . For a pure state,
W ψ ( x , p ) = 1 2 π ℏ ∫ d y e − i p y / ℏ ψ ( x + y 2 ) ψ ∗ ( x − y 2 ) . W_\psi(x,p)
=
\frac{1}{2\pi\hbar}
\int dy\,
e^{-ipy/\hbar}
\psi\left(x+\frac{y}{2}\right)
\psi^*\left(x-\frac{y}{2}\right). W ψ ( x , p ) = 2 π ℏ 1 ∫ d y e − i p y /ℏ ψ ( x + 2 y ) ψ ∗ ( x − 2 y ) . With this normalization,
∫ d x d p W ρ ( x , p ) = Tr ρ . \int dx\,dp\,W_\rho(x,p)
=
\operatorname{Tr}\rho. ∫ d x d p W ρ ( x , p ) = Tr ρ . For a normalized state, the integral is 1 1 1
The position marginal is
∫ d p W ρ ( x , p ) = ⟨ x ∣ ρ ∣ x ⟩ . \int dp\,W_\rho(x,p)
=
\langle x|\rho|x\rangle. ∫ d p W ρ ( x , p ) = ⟨ x ∣ ρ ∣ x ⟩ . The momentum marginal is
∫ d x W ρ ( x , p ) = ⟨ p ∣ ρ ∣ p ⟩ . \int dx\,W_\rho(x,p)
=
\langle p|\rho|p\rangle. ∫ d x W ρ ( x , p ) = ⟨ p ∣ ρ ∣ p ⟩ . These are ordinary probability densities. The full Wigner function can be negative. Even when nonnegative, it is not the probability distribution of a simultaneous sharp position–momentum measurement; sharp joint projectors do not exist.
The trace measure for one degree of freedom is
d Γ = d x d p 2 π ℏ . d\Gamma
=
\frac{dx\,dp}{2\pi\hbar}. d Γ = 2 π ℏ d x d p . For suitable trace-class operators,
Tr A = ∫ d Γ A W ( x , p ) . \operatorname{Tr}A
=
\int d\Gamma\,A_W(x,p). Tr A = ∫ d Γ A W ( x , p ) . For products of operators,
Tr ( A B ) = ∫ d Γ A W ( x , p ) B W ( x , p ) . \operatorname{Tr}(AB)
=
\int d\Gamma\,A_W(x,p)B_W(x,p). Tr ( A B ) = ∫ d Γ A W ( x , p ) B W ( x , p ) . Since W ρ = ρ W / ( 2 π ℏ ) W_\rho=\rho_W/(2\pi\hbar) W ρ = ρ W / ( 2 π ℏ ) 1 / ( 2 π ℏ ) 1/(2\pi\hbar) 1/ ( 2 π ℏ )
Tr ( ρ A ) = ∫ d x d p W ρ ( x , p ) A W ( x , p ) . \operatorname{Tr}(\rho A)
=
\int dx\,dp\,
W_\rho(x,p)A_W(x,p). Tr ( ρ A ) = ∫ d x d p W ρ ( x , p ) A W ( x , p ) . This formula resembles a classical phase-space average, but the Wigner function may be negative and operator products use the star product.
For n n n
q = ( q 1 , … , q n ) , p = ( p 1 , … , p n ) . \mathbf q=(q^1,\ldots,q^n),
\qquad
\mathbf p=(p_1,\ldots,p_n). q = ( q 1 , … , q n ) , p = ( p 1 , … , p n ) . The Wigner function is
W ρ ( q , p ) = 1 ( 2 π ℏ ) n ∫ d n y e − i p ⋅ y / ℏ × ⟨ q + y 2 | ρ | q − y 2 ⟩ . W_\rho(\mathbf q,\mathbf p)
=
\frac{1}{(2\pi\hbar)^n}
\int d^n\mathbf y\,
e^{-i\mathbf p\cdot\mathbf y/\hbar}
\times
\left\langle
\mathbf q+\frac{\mathbf y}{2}
\middle|
\rho
\middle|
\mathbf q-\frac{\mathbf y}{2}
\right\rangle. W ρ ( q , p ) = ( 2 π ℏ ) n 1 ∫ d n y e − i p ⋅ y /ℏ × ⟨ q + 2 y ρ q − 2 y ⟩ . The trace measure is
d Γ n = d n q d n p ( 2 π ℏ ) n . d\Gamma_n
=
\frac{d^n\mathbf q\,d^n\mathbf p}
{(2\pi\hbar)^n}. d Γ n = ( 2 π ℏ ) n d n q d n p . Again,
∫ d n q d n p W ρ ( q , p ) = Tr ρ . \int d^n\mathbf q\,d^n\mathbf p\,
W_\rho(\mathbf q,\mathbf p)
=
\operatorname{Tr}\rho. ∫ d n q d n p W ρ ( q , p ) = Tr ρ . For non-Cartesian coordinates or constrained phase spaces, this flat measure is not automatically correct.
The Weyl symbol of an operator product is the star product of Weyl symbols:
( A B ) W = A W ⋆ B W . (AB)_W
=
A_W\star B_W. ( A B ) W = A W ⋆ B W . The derivative exponential is an exact terminating expression for suitable polynomials and a formal or asymptotic expansion for general symbols; the operator product defines the star product even when this derivative series does not converge. For one degree of freedom,
( A ⋆ B ) ( x , p ) = A ( x , p ) exp [ i ℏ 2 ( ∂ x ← ∂ p → − ∂ p ← ∂ x → ) ] B ( x , p ) . (A\star B)(x,p)
=
A(x,p)
\exp\left[
\frac{i\hbar}{2}
\left(
\overleftarrow{\partial_x}\overrightarrow{\partial_p}
-
\overleftarrow{\partial_p}\overrightarrow{\partial_x}
\right)
\right]
B(x,p). ( A ⋆ B ) ( x , p ) = A ( x , p ) exp [ 2 i ℏ ( ∂ x ∂ p − ∂ p ∂ x ) ] B ( x , p ) . The first terms are
A ⋆ B = A B + i ℏ 2 { A , B } P B + O ( ℏ 2 ) . A\star B
=
AB
+
\frac{i\hbar}{2}\{A,B\}_{\rm PB}
+
O(\hbar^2). A ⋆ B = A B + 2 i ℏ { A , B } PB + O ( ℏ 2 ) . The basic check is
x ⋆ p = x p + i ℏ 2 , p ⋆ x = x p − i ℏ 2 . x\star p
=
xp+\frac{i\hbar}{2},
\qquad
p\star x
=
xp-\frac{i\hbar}{2}. x ⋆ p = x p + 2 i ℏ , p ⋆ x = x p − 2 i ℏ . Therefore,
x ⋆ p − p ⋆ x = i ℏ . x\star p-p\star x=i\hbar. x ⋆ p − p ⋆ x = i ℏ. For n n n
z = ( q 1 , … , q n , p 1 , … , p n ) , \mathbf z=(q^1,\ldots,q^n,p_1,\ldots,p_n), z = ( q 1 , … , q n , p 1 , … , p n ) , and
Ω = ( 0 I − I 0 ) . \Omega
=
\begin{pmatrix}
0&I\\
-I&0
\end{pmatrix}. Ω = ( 0 − I I 0 ) . Then the compact convention is
A ⋆ B = A exp [ i ℏ 2 ∂ z ← T Ω ∂ z → ] B . A\star B
=
A
\exp\left[
\frac{i\hbar}{2}
\overleftarrow{\partial_{\mathbf z}}^{\,T}
\Omega
\overrightarrow{\partial_{\mathbf z}}
\right]
B. A ⋆ B = A exp [ 2 i ℏ ∂ z T Ω ∂ z ] B . The Poisson bracket convention is
{ A , B } P B = ∂ A ∂ x ∂ B ∂ p − ∂ A ∂ p ∂ B ∂ x . \{A,B\}_{\rm PB}
=
\frac{\partial A}{\partial x}
\frac{\partial B}{\partial p}
-
\frac{\partial A}{\partial p}
\frac{\partial B}{\partial x}. { A , B } PB = ∂ x ∂ A ∂ p ∂ B − ∂ p ∂ A ∂ x ∂ B . For many Cartesian degrees of freedom,
{ A , B } P B = ∑ a = 1 n ( ∂ A ∂ q a ∂ B ∂ p a − ∂ A ∂ p a ∂ B ∂ q a ) . \{A,B\}_{\rm PB}
=
\sum_{a=1}^{n}
\left(
\frac{\partial A}{\partial q^a}
\frac{\partial B}{\partial p_a}
-
\frac{\partial A}{\partial p_a}
\frac{\partial B}{\partial q^a}
\right). { A , B } PB = a = 1 ∑ n ( ∂ q a ∂ A ∂ p a ∂ B − ∂ p a ∂ A ∂ q a ∂ B ) . This sign convention gives Hamilton’s equations in the form
q ˙ a = { q a , H } P B = ∂ H ∂ p a , \dot q^a=\{q^a,H\}_{\rm PB}
=
\frac{\partial H}{\partial p_a}, q ˙ a = { q a , H } PB = ∂ p a ∂ H , and
p ˙ a = { p a , H } P B = − ∂ H ∂ q a . \dot p_a=\{p_a,H\}_{\rm PB}
=
-\frac{\partial H}{\partial q^a}. p ˙ a = { p a , H } PB = − ∂ q a ∂ H . The Moyal bracket is
{ A , B } M = 1 i ℏ ( A ⋆ B − B ⋆ A ) . \{A,B\}_M
=
\frac{1}{i\hbar}
\left(
A\star B-B\star A
\right). { A , B } M = i ℏ 1 ( A ⋆ B − B ⋆ A ) . It obeys
{ A , B } M = { A , B } P B + O ( ℏ 2 ) \{A,B\}_M
=
\{A,B\}_{\rm PB}
+O(\hbar^2) { A , B } M = { A , B } PB + O ( ℏ 2 ) for smooth functions in the formal small-ℏ \hbar ℏ
For a closed system with density operator ρ \rho ρ
∂ ρ ∂ t = 1 i ℏ [ H , ρ ] . \frac{\partial\rho}{\partial t}
=
\frac{1}{i\hbar}[H,\rho]. ∂ t ∂ ρ = i ℏ 1 [ H , ρ ] . With
W ρ = 1 2 π ℏ ρ W , W_\rho
=
\frac{1}{2\pi\hbar}\rho_W, W ρ = 2 π ℏ 1 ρ W , the phase-space evolution equation is
∂ W ∂ t = { H W , W } M . \frac{\partial W}{\partial t}
=
\{H_W,W\}_M. ∂ t ∂ W = { H W , W } M . For
H W ( x , p ) = p 2 2 m + V ( x ) , H_W(x,p)=\frac{p^2}{2m}+V(x), H W ( x , p ) = 2 m p 2 + V ( x ) , the leading classical part is
∂ W ∂ t = − p m ∂ W ∂ x + V ′ ( x ) ∂ W ∂ p + O ( ℏ 2 ) . \frac{\partial W}{\partial t}
=
-
\frac{p}{m}\frac{\partial W}{\partial x}
+
V'(x)\frac{\partial W}{\partial p}
+
O(\hbar^2). ∂ t ∂ W = − m p ∂ x ∂ W + V ′ ( x ) ∂ p ∂ W + O ( ℏ 2 ) . This equals the classical Liouville equation with the Poisson-bracket sign convention above:
∂ W ∂ t = { H W , W } P B + O ( ℏ 2 ) . \frac{\partial W}{\partial t}
=
\{H_W,W\}_{\rm PB}
+
O(\hbar^2). ∂ t ∂ W = { H W , W } PB + O ( ℏ 2 ) . For quadratic Hamiltonians, the Moyal bracket equals the Poisson bracket exactly.
Item
Convention here
Common alternative
Wigner prefactor
1 / ( 2 π ℏ ) n 1/(2\pi\hbar)^n 1/ ( 2 π ℏ ) n 1 / h n 1/h^n 1/ h n h = 2 π ℏ h=2\pi\hbar h = 2 π ℏ
Separation phase
e − i p ⋅ y / ℏ e^{-i\mathbf p\cdot\mathbf y/\hbar} e − i p ⋅ y /ℏ opposite sign if ⟨ x ∣ p ⟩ \langle x\vert p\rangle ⟨ x ∣ p ⟩
Trace measure
d n q d n p / ( 2 π ℏ ) n d^nq\,d^np/(2\pi\hbar)^n d n q d n p / ( 2 π ℏ ) n absorb measure into symbol definitions
Star product
exp [ ( i ℏ / 2 ) Λ ] \exp[(i\hbar/2)\Lambda] exp [( i ℏ/2 ) Λ ] opposite sign with opposite Fourier convention
Moyal bracket
( A ⋆ B − B ⋆ A ) / ( i ℏ ) (A\star B-B\star A)/(i\hbar) ( A ⋆ B − B ⋆ A ) / ( i ℏ ) definitions that absorb i ℏ i\hbar i ℏ
Wigner function
W ρ = ρ W / ( 2 π ℏ ) n W_\rho=\rho_W/(2\pi\hbar)^n W ρ = ρ W / ( 2 π ℏ ) n calling ρ W \rho_W ρ W
Forgetting the prefactor in the Wigner transform.
Mixing a Weyl symbol from one convention with a Wigner function from another.
Treating W ( x , p ) W(x,p) W ( x , p )
Using ordinary multiplication for symbols of noncommuting operator products.
Reversing the Poisson-bracket sign when translating Hamilton’s equations.
Dropping the phase-space trace measure 1 / ( 2 π ℏ ) n 1/(2\pi\hbar)^n 1/ ( 2 π ℏ ) n
Assuming flat Cartesian formulas apply unchanged to spin, angle variables, curved spaces, constrained systems, or gauge systems.
T. L. Curtright and C. K. Zachos, “Quantum Mechanics in Phase Space,” Asia Pacific Physics Newsletter 1 , 37–46, 2012, arXiv:1104.5269 .
M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: Fundamentals,” Physics Reports 106, 121-167, 1984.
J. E. Moyal, “Quantum mechanics as a statistical theory,” Proceedings of the Cambridge Philosophical Society 45, 99-124, 1949.
W. P. Schleich, Quantum Optics in Phase Space , Wiley-VCH, 2001.
E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Physical Review 40, 749-759, 1932.
C. K. Zachos, D. B. Fairlie, and T. L. Curtright, eds., Quantum Mechanics in Phase Space , World Scientific, 2005.
Starting from the Wigner definition, show that ∫ d x d p W ρ ( x , p ) = Tr ρ \int dx\,dp\,W_\rho(x,p)=\operatorname{Tr}\rho ∫ d x d p W ρ ( x , p ) = Tr ρ
Solution Use
W ρ ( x , p ) = 1 2 π ℏ ∫ d y e − i p y / ℏ ⟨ x + y 2 | ρ | x − y 2 ⟩ . W_\rho(x,p)
=
\frac{1}{2\pi\hbar}
\int dy\,
e^{-ipy/\hbar}
\left\langle
x+\frac{y}{2}
\middle|
\rho
\middle|
x-\frac{y}{2}
\right\rangle. W ρ ( x , p ) = 2 π ℏ 1 ∫ d y e − i p y /ℏ ⟨ x + 2 y ρ x − 2 y ⟩ . Integrating over p p p
∫ d p 2 π ℏ e − i p y / ℏ = δ ( y ) . \int\frac{dp}{2\pi\hbar}e^{-ipy/\hbar}
=
\delta(y). ∫ 2 π ℏ d p e − i p y /ℏ = δ ( y ) . Therefore
∫ d p W ρ ( x , p ) = ⟨ x ∣ ρ ∣ x ⟩ . \int dp\,W_\rho(x,p)
=
\langle x|\rho|x\rangle. ∫ d p W ρ ( x , p ) = ⟨ x ∣ ρ ∣ x ⟩ . Integrating over x x x
∫ d x ⟨ x ∣ ρ ∣ x ⟩ = Tr ρ . \int dx\,\langle x|\rho|x\rangle
=
\operatorname{Tr}\rho. ∫ d x ⟨ x ∣ ρ ∣ x ⟩ = Tr ρ .
Use the star product to compute x ⋆ p − p ⋆ x x\star p-p\star x x ⋆ p − p ⋆ x
Solution For x ⋆ p x\star p x ⋆ p
x ⋆ p = x p + i ℏ 2 ( ∂ x ∂ x ∂ p ∂ p ) = x p + i ℏ 2 . x\star p
=
xp
+
\frac{i\hbar}{2}
\left(
\frac{\partial x}{\partial x}
\frac{\partial p}{\partial p}
\right)
=
xp+\frac{i\hbar}{2}. x ⋆ p = x p + 2 i ℏ ( ∂ x ∂ x ∂ p ∂ p ) = x p + 2 i ℏ . Similarly,
p ⋆ x = p x − i ℏ 2 = x p − i ℏ 2 . p\star x
=
px
-
\frac{i\hbar}{2}
=
xp-\frac{i\hbar}{2}. p ⋆ x = p x − 2 i ℏ = x p − 2 i ℏ . Subtracting gives
x ⋆ p − p ⋆ x = i ℏ . x\star p-p\star x=i\hbar. x ⋆ p − p ⋆ x = i ℏ.
Check the Poisson-bracket sign for H = p 2 / ( 2 m ) + V ( x ) H=p^2/(2m)+V(x) H = p 2 / ( 2 m ) + V ( x )
Solution With
{ A , B } P B = A x B p − A p B x , \{A,B\}_{\rm PB}
=
A_xB_p-A_pB_x, { A , B } PB = A x B p − A p B x , one finds
{ x , H } P B = 1 ⋅ p m − 0 = p m , \{x,H\}_{\rm PB}
=
1\cdot\frac{p}{m}-0
=
\frac{p}{m}, { x , H } PB = 1 ⋅ m p − 0 = m p , and
{ p , H } P B = 0 − 1 ⋅ V ′ ( x ) = − V ′ ( x ) . \{p,H\}_{\rm PB}
=
0-
1\cdot V'(x)
=
-V'(x). { p , H } PB = 0 − 1 ⋅ V ′ ( x ) = − V ′ ( x ) . These are Hamilton’s equations for the particle.
A text defines the Wigner function with e + i p y / ℏ e^{+ipy/\hbar} e + i p y /ℏ 1 / ( 2 π ℏ ) 1/(2\pi\hbar) 1/ ( 2 π ℏ )
Solution The separation Fourier sign changed. This is usually tied to using the conjugate position-momentum convention, or equivalently to replacing p p p − p -p − p
