The Poisson Algebra and the Canonical Bridge to Quantum Mechanics

Derivation. Derives Theorem 25.3. Write the bracket in the phase-space notation of Equation (22.55). With \(\pp_{c}:=\pp/\pp\eta^{c}\) and \(J^{cd}\) the components of the symplectic matrix Equation (22.56) — constants, so the position of the index carries no meaning here — the definition Equation (22.41) reads

Unpacking the blocks of Equation (22.56) returns \(\pp_{q}u\,\pp_{p}v-\pp_{p}u\,\pp_{q}v\), so Equation (25.1) is the same bracket and not a new one.

Bilinearity is immediate, since \(u\) and \(v\) enter through one derivative each. Antisymmetry follows from \(J^{cd}=-J^{dc}\) after relabelling the two summed indices, which is Equation (22.43).

For the Jacobi identity, differentiate once more. Because \(J\) is constant it passes through \(\pp_{d}\), and

Every term of the cyclic sum \(\pb{u}{\pb{v}{w}}+\pb{v}{\pb{w}{u}}+\pb{w}{\pb{u}{v}}\) therefore carries exactly one second derivative, of one of the three functions, and first derivatives of the other two. Collect the two terms carrying \(\pp\pp u\). They come from the second and third brackets and are

In \(B\) interchange the names \(e\leftrightarrow f\), which turns \(J^{ef}\) into \(J^{fe}=-J^{ef}\), and then interchange the pairs \((c,d)\leftrightarrow(e,f)\), which merely renames dummies:

so the two cancel. The cyclic sum is invariant under \(u\to v\to w\to u\), so the same cancellation removes the \(\pp\pp v\) and \(\pp\pp w\) terms, and nothing is left. This is Equation (22.46).

Two features of the computation are worth naming, because they are what the identity really rests on. Antisymmetry of \(J\) is used once, and constancy of \(J\) is used once — and the latter is the coordinate shadow of the closedness of the symplectic form (Definition 13.105), which is why the Jacobi identity survives the change to arbitrary coordinates of Theorem 24.12 even though \(J^{cd}\) does not stay constant there. A bracket built from an antisymmetric \(J^{cd}(\eta)\) that is not the inverse of a closed form is bilinear and antisymmetric but fails Jacobi; the Leibniz rule alone does not save it.

Derivation. Derives Proposition 25.4. For the Leibniz rule, put the product \(vw\) into the first slot of Equation (25.1) and use the product rule on the single derivative that acts on it:

which is Equation (24.24). A linear map obeying Equation (25.3) is by definition a derivation of the commutative product, and since it involves exactly one derivative of its argument it is a first-order differential operator: reading off the coefficient,

which is Equation (24.6). Taking \(u=\Ham\) makes \(X_{\Ham}^{c}=J^{cd}\pp_{d}\Ham\) the right-hand side of Equation (22.57): the Hamiltonian vector field is the velocity field of the motion in phase space.

For the second statement, let \(w\) be arbitrary and compute the commutator of the two vector fields on it, using Equation (25.4) twice:

Now write the Jacobi identity Equation (22.46) with \(w\) in the outer slot and move each inner bracket outwards by antisymmetry, \(\pb{u}{\pb{v}{w}}=\pb{\pb{w}{v}}{u}\) and \(\pb{v}{\pb{w}{u}}=-\pb{\pb{w}{u}}{v}\):

The first term is \(X_{\pb{u}{v}}[w]\) and the remaining two are \(\comm{X_{u}}{X_{v}}[w]\), so \(X_{\pb{u}{v}}[w]=-\comm{X_{u}}{X_{v}}[w]\) for every \(w\), which is Equation (24.8). The sign is a consequence of the conventions, not a defect: it says that \(u\mapsto X_{u}\) is an antihomomorphism in the convention of Equation (22.41), and it is why some texts define the Hamiltonian vector field with the opposite sign.

Derivation. Derives Proposition 25.7. Substitute \(F_{2}=q^{a}P_{a}+\varepsilon G(q,P)\) into the two transformation relations of Equation (22.21):

Setting \(\varepsilon=0\) gives \(P_{a}=p_{a}\) and \(Q^{a}=q^{a}\): the first term alone generates the identity, which is the check the construction needs, since a transformation infinitesimally close to the identity must be generated by a function infinitesimally close to that one.

For \(\varepsilon\neq0\) the relations are implicit, \(G\) being a function of \((q,P)\) rather than of \((q,p)\); but the difference between \(P_{a}\) and \(p_{a}\) is itself of order \(\varepsilon\) by the first of Equation (25.6), so replacing \(P\) by \(p\) inside a derivative of \(G\) costs only \(O(\varepsilon^{2})\). To first order, therefore,

These are exactly Equation (25.5): by Equation (22.41), \(\pb{q^{a}}{G}=\pp G/\pp p_{a}\) and \(\pb{p_{a}}{G}=-\pp G/\pp q^{a}\). The statement for an arbitrary observable follows by the chain rule,

so a single function \(G\) moves every observable at once, and it does so through the bracket alone.

Derivation. Derives Proposition 25.8. Put \(G=\Ham\) and \(\varepsilon=\dd t\) in the last member of Equation (25.5). For an observable \(u\) carrying no explicit time dependence this gives \(\delta u=\dd t\,\pb{u}{\Ham}\), that is \(\dd u/\dd t=\pb{u}{\Ham}\), which is Equation (22.51). If \(u\) does depend explicitly on the time, the transformation still contributes only \(\dd t\,\pb{u}{\Ham}\) — it acts on the phase-space arguments — and the remaining \(\pp u/\pp t\) of Equation (22.50) comes from the explicit slot, which the transformation does not touch. Applying the result to \(u=q^{a}\) and \(u=p_{a}\) and using Equation (22.44) returns Hamilton's equations Equations (22.3) and (22.4), so the family generated by \(\Ham\) is the motion itself and not merely a transformation resembling it.

Derivation. Derives Theorem 25.9. \((1)\Leftrightarrow(2)\). With \(\pp G/\pp t=0\) the evolution equation Equation (22.50) reads \(\dd G/\dd t=\pb{G}{\Ham}\). If the bracket vanishes identically then \(G\) is constant along every motion. Conversely, suppose \(G\) is constant along every motion. The bracket \(\pb{G}{\Ham}\) is a function on phase space, and through each point of phase space there passes a motion — take that point as the initial condition of Equations (22.3) and (22.4) — along which \(\dd G/\dd t\) vanishes at that point. So \(\pb{G}{\Ham}\) vanishes at every point. The quantifier is what does the work: constancy along a single trajectory is a weaker statement and does not give (2).

\((2)\Leftrightarrow(3)\). By Proposition 25.7 the family generated by \(G\) moves any observable by \(\delta u=\varepsilon\pb{u}{G}\), so it moves the Hamiltonian by

the second equality being the antisymmetry Equation (22.43). The right-hand side vanishes for all \(\varepsilon\) exactly when \(\pb{G}{\Ham}=0\), which is (2); and its vanishing is the statement that the family leaves \(\Ham\) invariant, which is (3).

Derivation. Derives Proposition 25.12. Throughout, the bracket is the one of Definition 22.29 in the Cartesian coordinates \((x_{i},p_{i})\), whose fundamental values are \(\pb{x_{i}}{p_{j}}=\delta_{ij}\) and \(\pb{x_{i}}{x_{j}}=\pb{p_{i}}{p_{j}}=0\); the last of these is the third entry of Equation (25.12), and the time \(t\) is a parameter, not a phase-space variable, so it passes through every bracket as a constant.

Rotations on a vector. Because the bracket is a derivation in each slot, the action of \(L_{i}\) on any function is fixed by its action on \(x_{j}\) and \(p_{j}\). From the definition Equation (25.10),

which is the second entry of Equation (25.12): every vector built from \(\vect{x}\) and \(\vect{p}\) rotates as a vector.

Rotations among themselves. Applying the derivation property to \(L_{j}=\epsilon_{jab}x_{a}p_{b}\) and using the two results just obtained,

the two contractions of Levi-Civita symbols being the standard identity \(\epsilon_{iab}\epsilon_{icd}=\delta_{ac}\delta_{bd} -\delta_{ad}\delta_{bc}\) used twice after a cyclic relabelling. Since

the two agree, which is the first entry of Equation (25.12).

Rotations on the boost. The boost generator Equation (25.11) is a linear combination of \(x_{i}\) and \(p_{i}\) with coefficients that are not phase-space functions, so it rotates as a vector by linearity:

the first entry of Equation (25.13).

Boosts among themselves. Expanding both slots and using the fundamental brackets,

the two surviving terms cancelling by the antisymmetry \(\pb{p_{i}}{x_{j}}=-\delta_{ij}\). This is the third entry of Equation (25.13).

Boosts and the Hamiltonian. Since \(\pb{x_{i}}{\Ham}=p_{i}/m\) and \(\pb{p_{i}}{\Ham}=0\),

the second entry of Equation (25.13).

The remaining brackets with the Hamiltonian. The momenta commute among themselves, so \(\pb{p_{i}}{\Ham}=0\); and

because a symmetric product is contracted with an antisymmetric symbol. Together with \(\pb{\Ham}{\Ham}=0\), which holds by antisymmetry, this is Equation (25.14): energy, momentum and angular momentum are constants of the motion.

Boosts and translations. Finally, and this is the bracket the proposition is about,

which is Equation (25.15).

Every bracket therefore closes on the span of \(\set{\Ham,p_{i},L_{i},K_{i}}\) together with the constant function \(m\), and by Theorem 22.38 the ten generators are constants of the motion. For \(K_{i}\) this last claim needs the explicit time dependence, and is worth checking separately: by the evolution equation Equation (22.50),

so that \(K_{i}=m\,x_{i}(0)\) is conserved. The conservation law carried by the boost generator is the statement that the centre of mass moves uniformly.

Derivation. Derives Proposition 25.14. That \(c\) takes constant values is read off the brackets computed above: every one of Equations (25.12), (25.13) and (25.14) reproduces the abstract Galilei bracket exactly, so \(c\) vanishes on those pairs, and Equation (25.15) gives

a constant.

Now suppose the generators are redefined by constants, \(J_{\xi}\mapsto\tilde{J}_{\xi}=J_{\xi}+b(\xi)\) with \(b\) linear. Constants have vanishing bracket with everything, so the left-hand term of Equation (25.17) is unchanged while the right-hand term shifts, and

Removing the defect Equation (25.18) would therefore require

But in the Galilei algebra \(\comm{K_{i}}{P_{j}}=0\), so the left-hand side is \(b(0)=0\) by linearity, while the right-hand side is \(m\,\delta_{ij} \neq0\) for a massive particle. No choice of \(b\) works, and the extension is not removable.

Derivation. Derives Theorem 25.19. Vary Equation (25.20) in both arguments at once:

Only one term carries a derivative of a variation. Integrating it by parts once,

and the boundary term vanishes because \(\delta q^{a}\) was required to vanish at the endpoints. Collecting,

The variations \(\delta q^{a}\) and \(\delta p_{a}\) are independent and otherwise arbitrary, so by the fundamental lemma Lemma 16.18 applied to each in turn, \(\delta S=0\) for all of them if and only if both coefficients vanish identically, which is Equations (22.3) and (22.4). The converse direction is the same computation read backwards.

The asymmetry of the endpoint conditions is not an oversight. The integrand contains \(\dot{q}^{a}\) but no \(\dot{p}_{a}\), so the integration by parts is needed only for the coordinates, and only they generate a boundary term to be killed. Requiring \(\delta p_{a}\) to vanish at the ends as well would be harmless but redundant; requiring it of \(\delta q^{a}\) is not optional.

Derivation. Derives Theorem 25.22. The Euler–Lagrange equation of Equation (25.22) is

since \(\pp\left(a_{d}\dot{\xi}^{d}\right)/\pp\dot{\xi}^{c}=a_{c}\) and \(\dd a_{c}/\dd t=\pp_{d}a_{c}\,\dot{\xi}^{d}\). Rearranged, this is Equation (25.24). Note that \(a_{c}\) enters the equation of motion only through the antisymmetric combination Equation (25.23): adding a gradient, \(a_{c}\mapsto a_{c}+\pp_{c}\chi\), changes the integrand of Equation (25.22) by a total time derivative and leaves the dynamics alone. The potential is a gauge-dependent bookkeeping device; the form is the physics.

If \(f\) is invertible, contracting Equation (25.24) with the inverse gives \(\dot{\xi}^{c}=\left(f^{-1}\right)^{cd}\pp_{d}\Ham\), and for any observable \(\dd u/\dd t=\pp_{c}u\,\dot{\xi}^{c} =\left(f^{-1}\right)^{cd}\pp_{c}u\,\pp_{d}\Ham\), which is \(\pb{u}{\Ham}\) in the bracket Equation (25.25). The inverse of an antisymmetric matrix is antisymmetric, so the bracket is antisymmetric, and it is a derivation in each slot because it carries one derivative per slot.

For the Jacobi identity the proof of Theorem 25.3 does not apply verbatim, since \(\left(f^{-1}\right)^{cd}\) is not constant; what replaces constancy is that \(f=\dd a\) is exact, hence closed (Definition 13.105), \(\pp_{c}f_{de}+\pp_{d}f_{ec} +\pp_{e}f_{cd}=0\) by inspection of Equation (25.23). Differentiating \(\left(f^{-1}\right)^{ce}f_{ed}=\delta^{c}_{d}\) gives \(\pp_{b}\left(f^{-1}\right)^{cd} =-\left(f^{-1}\right)^{ce}\left(f^{-1}\right)^{fd}\pp_{b}f_{ef}\), and substituting this into the cyclic sum turns the three second-derivative groups into three copies of the closedness relation contracted with three inverses; they cancel in the same pattern as before.

Derivation. Derives Proposition 25.25. Equation (25.29) is Equation (25.31) applied to Equation (25.28). Two operators obeying a commutation relation of the canonical form \(\comm{\hat{A}}{\hat{B}}=\ii\lambda\) obey the uncertainty relation \(\Delta A\,\Delta B\ge\abs{\lambda}/2\) of Section 77.6.3, so here \(\Delta x_{1}\,\Delta x_{2}\ge\ell_{B}^{2}/2\): a state cannot be localized in the plane to better than an area \(2\pi\ell_{B}^{2}/2\pi\), and the count of orthogonal states per unit area is bounded by \(1/2\pi\ell_{B}^{2}\). Substituting \(\ell_{B}^{2}=\hbar/eB\) and \(\hbar=h/2\pi\) gives Equation (25.30), the flux quantum \(\Phi_{0}=h/e\) being exact in the SI, since \(h\) and \(e\) are both defining constants (Physical Constants and SI Units). The bound is an inequality because an uncertainty relation is one; that it is saturated — that a Landau level holds exactly \(eB/h\) states per unit area and not fewer — does not follow from this argument and requires the explicit eigenfunctions, which are obtained in Experiment: The Quantum Hall Effect. What the bracket settles here is the scale, and the scale is what the experiment measures.

Numerically, at \(B=10\,\mathrm{T}\) — a field routinely available in a superconducting laboratory magnet — \(\ell_{B}=8.11\,\mathrm{nm}\) and \(n=2.418\times 10^{15}\,/\mathrm{m}^{2}\), that is \(2.418\times 10^{11}\,/\mathrm{cm}^{2}\), which is the carrier density scale at which the integer quantum Hall plateaux appear (Experiment: The Quantum Hall Effect).

Derivation. Derives Proposition 25.29. Read Equation (22.44) through Equation (25.31) term by term. The first two brackets vanish, so the corresponding commutators do. The third is \(\pb{q^{a}}{p_{b}}=\delta^{a}_{b}\), a multiple of the constant function \(1\), whose operator image is \(\identity\) by the second requirement of Theorem 25.38; hence \(\comm{\hat{q}^{a}}{\hat{p}_{b}}=\ii\hbar\,\delta^{a}_{b}\identity\), the identity being suppressed in Equation (25.33) as usual. Both sides carry the SI unit \(\mathrm{J}\,\mathrm{s}\), as they must.

Derivation. Derives Corollary 25.30. In finite dimension take the trace of Equation (25.33): the trace of a commutator vanishes, while the trace of \(\ii\hbar\identity\) is \(\ii\hbar\,\dim\), which is non-zero. The general bounded statement is Theorem 12.75, and the consequences for domains are Corollary 12.76. This is the first place where the quantum theory demands more than linear algebra, and it is why the Hilbert-space apparatus of Hilbert Spaces is not optional decoration.

Derivation. Derives Proposition 25.31. The evolution equation Equation (22.50) reads \(\dd U/\dd t=\pb{U}{\Ham}+\pp U/\pp t\). Its first term is a Poisson bracket and goes over into \(\comm{\hat{A}}{\hat{\Ham}}/\ii\hbar\) by Equation (25.31); its second is a derivative with respect to the explicit time slot, which the correspondence does not touch, since \(t\) is a parameter and not an observable. This gives Equation (25.34). Setting \(\pp\hat{A}/\pp t=0\), the operator is constant in time precisely when it commutes with \(\hat{\Ham}\), which is the operator form of Proposition 22.37.

Two remarks keep the parallel honest. First, this is a transfer of a postulate, not a derivation: Equation (25.34) is equivalent to the Schrödinger equation of Section 77.1.6 and stands or falls with it. Second, the transfer is legitimate here, and not merely suggestive, because \(\Ham\) enters only through a single bracket — so Theorem 25.38 has nothing to obstruct as long as \(\hat\Ham\) has been fixed by some other means.

Derivation. Derives Theorem 25.32. Both statements are Equation (25.34) evaluated in a state, with \(\pp\hat{A}/\pp t=0\) for \(\hat{A}=\hat{x}_{i}\) and \(\hat{A}=\hat{p}_{i}\). Two commutators are needed. For the first, the commutator with a fixed operator is a derivation of the operator product, \(\comm{\hat{A}}{\hat{B}\hat{C}} =\hat{B}\comm{\hat{A}}{\hat{C}}+\comm{\hat{A}}{\hat{B}}\hat{C}\), so with \(\comm{\hat{x}_{i}}{\hat{p}_{j}}=\ii\hbar\delta_{ij}\) from Equation (25.33),

while \(\comm{\hat{x}_{i}}{V(\hat{\vect{x}})}=0\) because \(V\) is built from the \(\hat{x}_{j}\), which commute among themselves. Hence \(\dd\avg{\hat{x}_{i}}/\dd t =\avg{\comm{\hat{x}_{i}}{\hat{\Ham}}}/\ii\hbar =\avg{\hat{p}_{i}}/m\), the first of Equation (25.35).

For the second, \(\comm{\hat{p}_{i}}{\hat{\vect{p}}^{2}}=0\), and the same derivation property applied repeatedly to a monomial \(\hat{x}_{j_{1}}\cdots\hat{x}_{j_{n}}\) gives

since each factor contributes \(\comm{\hat p_i}{\hat x_j}=-\ii\hbar \delta_{ij}\) in turn, which is exactly the product rule for \(\pp/\pp x_{i}\). The identity therefore holds for polynomials, and extends to any \(V\) reachable from them — by a convergent power series, or in general through the functional calculus of Definition 12.60 for a self-adjoint \(\hat{\vect{x}}\). Then \(\dd\avg{\hat{p}_{i}}/\dd t=\avg{\comm{\hat{p}_{i}}{\hat{\Ham}}}/\ii\hbar =-\avg{\pp_{i}V}\), the second of Equation (25.35).

The final sentence is the substantive one. Expand \(\pp_{i}V\) about the mean position \(\avg{\hat{\vect{x}}}\) and take the expectation term by term. The linear term drops, because the mean of \(\hat{x}_{j}-\avg{\hat{x}_{j}}\) is zero by construction, and what is left is

with \(\operatorname{cov}(\hat{x}_{j},\hat{x}_{k}) =\tfrac{1}{2}\avg{\acomm{\hat{x}_{j}}{\hat{x}_{k}}} -\avg{\hat{x}_{j}}\avg{\hat{x}_{k}}\). If \(V\) is at most quadratic every third derivative vanishes identically and Equation (25.35) becomes the classical equation exactly, whatever the state; otherwise the leading correction is a third derivative of the potential times the variance of the position, and it is not small merely because \(\hbar\) is small — it is small only while the wave packet stays narrow.

Derivation. Derives Lemma 25.39. Throughout, hypothesis (1) is applied only to the pairs written down, and \(\comm{\hat{q}}{\hat{p}}=\ii\hbar\) from Proposition 25.29. The tool used repeatedly is:

Schur step. If an operator \(D\) commutes with both \(\hat{q}\) and \(\hat{p}\) then \(D=\lambda\identity\) for some complex \(\lambda\). This is Schur's lemma Theorem 5.160 applied to the irreducible representation of hypothesis (3); Theorem 25.34 says that representation is the Schrödinger one, in which the statement is that an operator commuting with multiplication by \(x\) and with differentiation is a constant.

Quadratics. From \(\pb{q^{2}}{q}=0\) and \(\pb{q^{2}}{p}=2q\), hypothesis (1) requires \(\comm{\widehat{q^{2}}}{\hat{q}}=0\) and \(\comm{\widehat{q^{2}}}{\hat{p}}=2\ii\hbar\,\hat{q}\). The operator \(\hat{q}^{2}\) satisfies both, so their difference passes the Schur step and \(\widehat{q^{2}}=\hat{q}^{2}+\lambda\). Symmetrically \(\widehat{p^{2}}=\hat{p}^{2}+\nu\). Writing \(\hat{S}:=\tfrac{1}{2}\acomm{\hat{q}}{\hat{p}} =\hat{q}\hat{p}-\tfrac{1}{2}\ii\hbar\), the same argument applied to \(\pb{qp}{q}=-q\) and \(\pb{qp}{p}=p\) gives \(\widehat{qp}=\hat{S}+\mu\).

The three constants are removed by three more brackets. From \(\pb{q^{2}}{p^{2}}=4qp\) and \(\comm{\hat{q}^{2}}{\hat{p}^{2}} =2\ii\hbar\acomm{\hat{q}}{\hat{p}}\), the additive constants dropping out of a commutator,

so \(\mu=0\). From \(\pb{q^{2}}{qp}=2q^{2}\) and \(\comm{\hat{q}^{2}}{\hat{S}}=2\ii\hbar\,\hat{q}^{2}\),

so \(\lambda=0\); and \(\pb{p^{2}}{qp}=-2p^{2}\) with \(\comm{\hat{p}^{2}}{\hat{S}}=-2\ii\hbar\,\hat{p}^{2}\) gives \(\nu=0\) the same way. This is Equation (25.40).

Cubics. From \(\pb{q^{3}}{q}=0\) and \(\pb{q^{3}}{p}=3q^{2}\), and using \(\widehat{q^{2}}=\hat{q}^{2}\) just established, the Schur step gives \(\widehat{q^{3}}=\hat{q}^{3}+\kappa\). Then \(\pb{q^{3}}{qp}=3q^{3}\) with \(\comm{\hat{q}^{3}}{\hat{S}}=3\ii\hbar\,\hat{q}^{3}\) forces \(3\hat{q}^{3}=3\left(\hat{q}^{3}+\kappa\right)\), so \(\kappa=0\); symmetrically \(\widehat{p^{3}}=\hat{p}^{3}\), using \(\pb{p^{3}}{qp} =-3p^{3}\). The mixed cubics need no Schur step at all, being brackets of things already fixed: \(\pb{q^{3}}{p^{2}}=6q^{2}p\) and \(\comm{\hat{q}^{3}}{\hat{p}^{2}} =3\ii\hbar\acomm{\hat{q}^{2}}{\hat{p}}\) give

and \(\pb{q^{2}}{p^{3}}=6qp^{2}\) with \(\comm{\hat{q}^{2}}{\hat{p}^{3}}=2\ii\hbar\left(\hat{q}\hat{p}^{2} +\hat{p}\hat{q}\hat{p}+\hat{p}^{2}\hat{q}\right) =6\ii\hbar\left(\hat{q}\hat{p}^{2}-\ii\hbar\hat{p}\right)\) gives \(\widehat{qp^{2}}=\hat{q}\hat{p}^{2}-\ii\hbar\hat{p} =\tfrac{1}{2}\acomm{\hat{q}}{\hat{p}^{2}}\). This is Equation (25.41).

Derivation. Derives Theorem 25.38. Consider the classical observable \(q^{2}p^{2}\). It is a Poisson bracket of cubics in two ways:

both read off Equation (22.41) directly — for the second, \(\pp_{q}\left(q^{2}p\right)\pp_{p}\left(qp^{2}\right) -\pp_{p}\left(q^{2}p\right)\pp_{q}\left(qp^{2}\right) =4q^{2}p^{2}-q^{2}p^{2}\). Hypothesis (1) applied to each of Equation (25.42), with the images supplied by Lemma 25.39, gives two expressions for the same operator \(\widehat{q^{2}p^{2}}\).

For the first route, \(\comm{\hat{q}^{3}}{\hat{p}^{3}} =3\ii\hbar\left(\hat{q}^{2}\hat{p}^{2}+\hat{p}\hat{q}^{2}\hat{p} +\hat{p}^{2}\hat{q}^{2}\right)\), and moving every \(\hat{p}\) to the right with \(\hat{p}\hat{q}^{2}=\hat{q}^{2}\hat{p}-2\ii\hbar\hat{q}\),

so that \(\comm{\hat{q}^{3}}{\hat{p}^{3}}=3\ii\hbar\left(3\hat{q}^{2}\hat{p}^{2} -6\ii\hbar\,\hat{q}\hat{p}-2\hbar^{2}\right)\) and

For the second route write the two cubics in the compact form supplied by Lemma 25.39, \(\widehat{q^{2}p}=\hat{q}^{2}\hat{p}-\ii\hbar\hat{q}\) and \(\widehat{qp^{2}}=\hat{q}\hat{p}^{2}-\ii\hbar\hat{p}\). Expanding the commutator in four pieces, and using \(\comm{\hat{q}^{2}\hat{p}}{\hat{q}\hat{p}^{2}} =3\ii\hbar\,\hat{q}^{2}\hat{p}^{2}+2\hbar^{2}\hat{q}\hat{p}\) for the leading one,

the three \(\hbar^{2}\hat{q}\hat{p}\) contributions coming one from the leading piece and one from each cross term, and

Subtracting Equation (25.44) from Equation (25.43),

which is false, \(\hbar\) being a non-zero action Equation (25.32). No map satisfying (1)–(3) exists.

Nothing in the argument used more than hypothesis (1) on the pairs listed, all of degree at most three, so the conclusion is stronger than the statement: the correspondence already fails if it is demanded only of the polynomials of degree at most three. It does not fail on the polynomials of degree at most two, which close under the bracket — the bracket of a monomial of degree \(m\) with one of degree \(n\) has degree \(m+n-2\) — and on which Equation (25.40) is not merely forced but consistent, as the brackets computed in the proof of Lemma 25.39 verify. Degree two is exactly where the correspondence is exact, and one step beyond it is where it breaks.

Derivation. Derives Proposition 25.43. The three entries of Equation (25.46) follow from \(\comm{\hat{q}}{\hat{p}}=\ii\hbar\); and \(\left(\hat{q}\hat{p}\right)^{\dagger}=\hat{p}\hat{q}\neq\hat{q}\hat{p}\), so neither of the first two is self-adjoint while the anticommutator manifestly is. For the general statement, any ordering of \(q^{m}p^{n}\) can be turned into any other by transposing adjacent factors one at a time, and each transposition replaces \(\hat{p}\hat{q}\) by \(\hat{q}\hat{p}\) at the cost of an additive \(-\ii\hbar\) multiplying the remaining \(m+n-2\) factors. Iterating, the accumulated difference is a polynomial of degree at most \(m+n-2\) with coefficients built from integers and powers of \(\ii\hbar\). Since \(\hbar\) carries the SI unit \(\mathrm{J}\,\mathrm{s}\) and \(qp\) carries the same, the correction terms are dimensionally indistinguishable from the leading one, which is why no dimensional argument can select an ordering.

Derivation. Derives Proposition 25.46. Write the classical amplitude \(\alpha=\sqrt{m\omega/2\hbar}\left(q+\ii p/m\omega\right)\), whose bracket is \(\pb{\alpha}{\bar{\alpha}}=-\ii/\hbar\) by direct substitution into Equation (22.41), so that Equation (25.31) gives \(\comm{\hat{a}}{\hat{a}^{\dagger}}=\identity\); and \(\Ham=\hbar\omega\,\bar{\alpha}\alpha\) by expanding the square. Weyl ordering of \(q^{2}\) and \(p^{2}\) gives \(\hat{q}^{2}\) and \(\hat{p}^{2}\), whose combination is \(\tfrac{1}{2}\hbar\omega\acomm{\hat{a}^{\dagger}}{\hat{a}} =\hbar\omega\left(\hat{a}^{\dagger}\hat{a}+\tfrac{1}{2}\right)\); normal ordering of \(\bar{\alpha}\alpha\) gives \(\hat{a}^{\dagger}\hat{a}\) by definition. Subtracting gives Equation (25.49).

For the last sentence, express \(\hat{q}^{2}\) in ladder operators: \(\hat{q}^{2}=\left(\hbar/2m\omega\right) \left(\hat{a}^{2}+2\hat{a}^{\dagger}\hat{a}+\identity +\hat{a}^{\dagger2}\right)\), while normal ordering of the classical \(q^{2}\) drops the \(\identity\). So normal ordering sends \(q^{2}\) to \(\hat{q}^{2}-\hbar/2m\omega\), contradicting Equation (25.40). The Dirac rule is not toothless below degree four: there it fixes the answer completely, additive constant included.

Derivation. Derives Proposition 25.48. The first sentence is immediate: \(\abs{\hat{\vect{p}}}^{2}\) involves only momenta, which commute among themselves by Equation (25.33), and \(V(\hat{\vect{x}})\) only coordinates, which do likewise.

(1) Equation (25.50) is Equation (25.37) with \(V\) replaced by \(A_{i}\) and the index summed. The two candidate cross terms \(\hat{\vect{p}}\cdot\vect{A}\) and \(\vect{A}\cdot\hat{\vect{p}}\) therefore differ by \(-\ii\hbar\,\vect{\nabla}\cdot\vect{A}\), which vanishes identically in the Coulomb gauge \(\vect{\nabla}\cdot\vect{A}=0\), and in a general gauge is the divergence of a field that a gauge transformation may set to zero without changing \(\vect{B}\).

(2) Each candidate is self-adjoint because it is built as a symmetric product of self-adjoint factors; they differ because \(m^{-1}(\hat{\vect x})\) does not commute with \(\hat{\vect{p}}\), and the difference is computed by applying Equation (25.37) twice, each application producing one factor \(\ii\hbar\) and one derivative of \(m^{-1}\). The selection rule quoted is the statement that \(\hat{p}_{i}m^{-1}\hat{p}_{i}\) is the only member of the family whose associated probability current is continuous where \(m\) jumps; the others lose or create probability at the step, which is not a matter of taste.

(3) is quoted, not derived: the curvature term is what distinguishes the Laplace–Beltrami operator from the other second-order self-adjoint operators with the same symbol, and the coefficient is a free parameter of the quantization. The point of listing it is that it is the case in which nothing in the mathematics decides, and the honest procedure is the one stated next.

Derivation. Derives Theorem 25.53. Reality. Conjugating Equation (25.53) and substituting \(\vect{y}\to-\vect{y}\) turns the kernel into \(\rho(\vect{x}-\vect{y}/2,\vect{x}+\vect{y}/2)^{\ast}\), which is \(\rho(\vect{x}+\vect{y}/2,\vect{x}-\vect{y}/2)\) because \(\hat{\rho}\) is self-adjoint, and restores the exponential; so \(W^{\ast}=W\).

Position marginal. Integrating Equation (25.53) over \(\vect{p}\) and using \(\int\ee^{-\ii\vect{p}\cdot\vect{y}/\hbar}\dd^{3}p =(2\pi\hbar)^{3}\delta^{3}(\vect{y})\), which is the Fourier inversion formula of Section 17.2.1 read as an identity between tempered distributions (Definition 17.38), the delta collapses the \(\vect{y}\) integral at \(\vect{y}=0\) and leaves \(\rho(\vect{x},\vect{x})\). This is Equation (25.54), and integrating it once more over \(\vect{x}\) gives \(\tr\hat{\rho}=1\), the normalization.

Momentum marginal. Integrate over \(\vect{x}\) and change variables to \(\vect{u}=\vect{x}+\vect{y}/2\), \(\vect{v}=\vect{x}-\vect{y}/2\), whose Jacobian has modulus \(1\) in each Cartesian pair. Then \(\vect{y}=\vect{u}-\vect{v}\) and

the double integral being exactly the momentum-representation kernel at coincident arguments, in the convention that carries one factor \((2\pi\hbar)^{-1/2}\) per coordinate. For a pure state this is \(\abs{\tilde{\psi}(\vect{p})}^{2}\).

Overlap. Take the product of two Wigner functions with integration variables \(\vect{y}\) and \(\vect{y}'\) and integrate over \(\vect{p}\) first; the same delta identity forces \(\vect{y}'=-\vect{y}\) and removes one factor \((2\pi\hbar)^{-3}\), leaving

Integrating over \(\vect{x}\) and changing to \((\vect{u},\vect{v})\) as before turns this into \((2\pi\hbar)^{-3}\int\!\!\int\rho_{1}(\vect{u},\vect{v}) \rho_{2}(\vect{v},\vect{u})\,\dd^{3}u\,\dd^{3}v\), which is \((2\pi\hbar)^{-3}\tr(\hat{\rho}_{1}\hat{\rho}_{2})\). This is Equation (25.56).

Bound. For a pure state the integrand of Equation (25.53) is \(\psi(\vect{x}+\vect{y}/2)\psi^{\ast}(\vect{x}-\vect{y}/2)\); bounding the modulus of the integral by the integral of the modulus and applying Cauchy–Schwarz (Proposition 12.4),

each integral contributing \(8=2^{3}\) from the substitution \(\vect{u}=\vect{x}\pm\vect{y}/2\) and the normalization of \(\psi\). That is Equation (25.57).

Derivation. Derives Corollary 25.54. Put \(\vect{x}=\vect{0}\) and \(\vect{p}=\vect{0}\) in Equation (25.53) with \(\rho=\psi\psi^{\ast}\). The exponential is \(1\) and the integrand is \(\psi(\vect{y}/2)\psi^{\ast}(-\vect{y}/2) =-\abs{\psi(\vect{y}/2)}^{2}\) by oddness, so

the factor \(8\) coming from \(\dd^{3}y=8\,\dd^{3}u\) under \(\vect{u}=\vect{y}/2\). Odd normalized wavefunctions exist in abundance — the first excited state of the harmonic oscillator (Section 78.4) is one — so the value Equation (25.58) is attained by real states and not only permitted.

Derivation. Derives Theorem 25.57. The von Neumann equation \(\ii\hbar\,\pp_{t}\hat{\rho}=\comm{\hat{\Ham}}{\hat{\rho}}\) becomes, on Weyl symbols, \(\pp_{t}W=\pb{\Ham}{W}_{\text{M}}\) by Definition 25.50. Evaluate the two pieces of \(\Ham\) separately in Equation (25.51).

For the kinetic term \(u=p^{2}/2m\), every \(x\) derivative of \(u\) vanishes, so only \(k=n\) survives in \(u\star W\) and only \(k=0\) in \(W\star u\):

Even \(n\) cancel; odd \(n\) beyond \(1\) vanish because \(\pp_{p}^{3}u=0\); and \(n=1\) leaves \(-\ii\hbar\left(p/m\right)\pp_{x}W\), giving the first term of Equation (25.59) after dividing by \(\ii\hbar\). The kinetic term is exact — it contributes no \(\hbar\) correction at all.

For the potential \(u=V(x)\), every \(p\) derivative vanishes, so the surviving index is the other one and the sign is reversed:

The term \(n=1\) is \(\left(\pp_{x}V\right)\left(\pp_{p}W\right)\), the classical force term. The term \(n=3\) is \(\frac{2}{\ii\hbar}\cdot\frac{1}{6} \left(\frac{\ii\hbar}{2}\right)^{3}\pp_{x}^{3}V\,\pp_{p}^{3}W =-\frac{\hbar^{2}}{24}\pp_{x}^{3}V\,\pp_{p}^{3}W\), and the next is \(O(\hbar^{4})\). This is Equation (25.59). If \(V\) has degree at most two then \(\pp_{x}^{3}V=0\) and every surviving term of the series beyond \(n=1\) vanishes with it.

Derivation. Derives Proposition 25.59. By Equation (25.54), the distribution of the quadrature \(X\) is the integral of \(W\) over the conjugate variable \(P\) — that is, the integral of \(W\) along the line \(X=\text{const}\) in the \((X,P)\) plane.

It remains to show that the same holds for every rotated quadrature. The Hamiltonian Equation (25.60) is quadratic, so by Theorem 25.57 the Wigner function evolves under it exactly by the classical Liouville flow; and the classical flow of \(\Ham=\tfrac{1}{2}\hbar\omega(X^{2}+P^{2})\) in these coordinates is the rigid rotation \(\left(X,P\right)\mapsto \left(X\cos\omega t+P\sin\omega t,\,-X\sin\omega t+P\cos\omega t\right)\), as substitution into Equations (22.3) and (22.4) confirms. So the Wigner function of the state at time \(t=\theta/\omega\) is \(W\) rotated by \(\theta\), its \(X\) marginal is the integral of \(W\) along the line \(X\cos\theta+P\sin\theta=\text{const}\), and that line integral is Equation (17.119) in two dimensions — which is Equation (25.61). In an optical realization the phase \(\theta\) is set not by waiting but by the phase of a local oscillator, which is the same rotation performed at the detector rather than in the state.

Inverting is then Theorem 17.106 verbatim, with the projections \(P_{\theta}\) of Equation (17.122) identified with \(\operatorname{pr}_{\theta}\), and Corollary 17.107 says how many phases must be sampled to resolve structure of a given fineness. Nothing in the inversion is special to quantum mechanics: it is the arithmetic of a medical tomograph, applied to a quasi-probability instead of an attenuation coefficient.