In the classical mechanics of conservative systems, the position and momentum evolve deterministically such that the sum of the kinetic energy and potential energy remains constant in time. This canonical trademark of energy conservation is absent in the standard presentations of quantum mechanics based on the Schrödinger picture. We present a purely canonical proof of energy conservation that focuses exclusively on the time-dependent position x(t) and momentum p(t) operators. This treatment of energy conservation serves as an introduction to the Heisenberg picture and illuminates the classical-quantum connection. We derive a quantum-mechanical work-energy theorem and show explicitly how the time dependence of x and p and the noncommutivity of x and p conspire to bring about a perfect temporal balance between the evolving kinetic and potential parts of the total energy operator.

Topics
Energy conservation, Energy policy, Classical mechanics, Schrodinger picture, Heisenberg picture
1.
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23.
Following their proof, Born and Jordan say: “the fact that nonetheless energy conservation and frequency laws could be proved in so general a context would seem to us to furnish strong grounds to hope that this theory embraces truly deep-seated physical laws.” See Ref. 21, B. L. Van Der Waerden, p. 296.
24.
The one standard application where [x,p]=iℏ plays a crucial calculational role is the ladder-operator solution to the harmonic oscillator. In this algebraic solution, [x,p]=iℏ is used explicitly to factorize the Hamiltonian and construct the energy spectrum. See Ref. 2, pp. 487–494.
25.
It should be noted that an algebraic relationship between operators has the same form in both the Schrödinger and Heisenberg pictures. For example, the operator equation, AB=C, in the Schrödinger picture becomes the operator equation, A(t)B(t)=C(t), in the Heisenberg picture. This preservation of algebraic structure is due to the unitary nature of the transformation between the two pictures: A(t)=U†AU, where U=exp(−iHt/ℏ). Thus, A(t)B(t)=C(t) is transformed into U†AUU†BU=U†CU, which is identical to AB=C because U is unitary (UU†=1).
26.
The relation between commutators and derivatives involving functions of x and p stems from iterating the fundamental commutation rule and expanding the canonical function. For example, the relation, [f(x),p]=iℏdf/dx, is derived by expanding the function f(x) in a power series involving the powers xn (assuming convergence), and then using the relation [xn,p]=iℏnxn−1, which follows from n repeated applications of [x,p]=iℏ, or alternatively from mathematical induction (see Ref. 2, pp. 171–172).
27.
Reference 1, pp. 191–196.
28.
Reference 5, pp. 323–326.
29.
Reference 9, pp. 91–94.
30.
We can derive the operator relation K̇=−V̇ more formally from the Heisenberg equations, iℏK̇=[K,H] and iℏV̇=[V,H], by noting that [K,H]=[p2/2m,V(x)] is equal to −[V,H]=−[V(x),p2/2m]. For an alternative derivation of the K̇ equation in Eq. (34), we can start with the kinematical relation, K̇=(ṗp+pṗ)/2m, and then use the dynamical relation, ṗ=F, and the commutation relation, [p,F]=−iℏdF/dx. Our derivation of the K̇ and V̇ equations in Sec. IV reveals more explicitly how the energy operators, K̇ and V̇, depend on the basic canonical operators, x and p, and on the force constants (an) that characterize the conservative system. By exhibiting the detailed canonical structure of K̇(x,p) and V̇(x,p), the derivation also reveals the explicit algebraic mechanism by which the multiplication rule, xp−px=iℏ, simplifies the energy algebra, that is, the re-ordering of x and p in K̇(x,p) and V̇(x,p), and the term-by-term cancellations that reduce K̇(x,p)+V̇(x,p) to zero.
31.
Strictly speaking, because Fp is a work rate and K̇ is an energy rate, the relation between Fp and K̇ is the work rate-energy rate theorem of quantum mechanics. From the kinematical relation, 2mK̇=ṗp+pṗ, and the dynamical relation, ṗ=F, we can express the quantum work-energy theorem in the symmetrical form, 2mK̇=Fp+pF. Our statement of this theorem in Eq. (34), or Table II, namely K̇=Fv−(iℏ/2m)dF/dx, has the advantage of exhibiting the classical and quantum parts of the theorem. One sees explicitly how the force-slope operator (dF/dx) determines the quantum correction to the classical power (Fv). By integrating the quantum equation, K̇=Fv−(iℏ/2m)dF/dx, over time, it can be shown that the net quantum correction (terms involving iℏ) exactly vanishes leaving an expression that is identical in form to the classical expression, ΔK=∫Fdx.
32.
Reference 2, p. 242.
33.
Reference 2, pp. 242–244.
34.
For any force function F(x), it is easy to show that the wave-packet energy equation (39) assumes the general form: dH(〈x〉,〈p〉)/dt=(〈p〉/m)[〈F(x)〉−F(〈x〉)].
35.
Reference 2, pp. 487–499.
36.
Reference 3, p. 117.
37.
As an alternative proof of the time independence of K(t)+V(t) for a harmonic oscillator, we can solve the Heisenberg operator equations directly. The general solutions of ẋ=p/m and ṗ=−mω2x are x(t)=x(0)cos ωt+(p(0)/mω)sin ωt and p(t)=p(0)cos ωt−mωx(0)sin ωt. By squaring these expressions for x(t) and p(t), we obtain the relation, p2(t)+m2ω2x2(t)=p2(0)+m2ω2x2(0). This operator relation establishes the energy-conservation relation, K(t)+V(t)=K(0)+V(0). Whereas the time independence of K(t)+V(t) is due to the general form of the canonical operators x(t) and p(t), the value of the conserved energy En=〈n|K+V|n〉 is determined by the elements, 〈n|x2(0)|n〉 and 〈n|p2(0)|n〉, of the canonical matrices.
38.
The general solutions to the Ehrenfest oscillator equations, d〈x〉/dt=〈p〉/m and d〈p〉/dt=−mω2〈x〉, are 〈x〉(t)=〈x〉0cos ωt+(〈p〉0/mω)sin ωt and 〈p〉(t)=〈p〉0cos ωt−mω〈x〉0sin ωt. For an oscillator in the initial state |Ψ(0)〉=(|0〉+i|1〉+i|2〉)/∛, we find that the initial averages, 〈x〉0=〈Ψ(0)|x|Ψ(0)〉 and 〈p〉0=〈Ψ(0)|p|Ψ(0)〉, take the values 〈x〉0=4ℏ/9mω and 〈p〉0=2ℏmω/9. These results agree with those in Eqs. (52) and (53).
39.
Reference 21, Van Der Waerden, p. 296. Born and Jordan say “Whereas in classical mechanics, energy conservation (Ḣ=0) is directly apparent from the canonical equations, the same law of energy conservation in quantum mechanics, Ḣ=0, lies, as one can see, more deeply hidden beneath the surface. That its demonstrability from the assumed postulates is far from being trivial will be appreciated if, following more closely the classical method of proof, one sets out to prove H to be constant simply by evaluating Ḣ.” In a footnote, they say “For the case H=(1/2m)p2+U(q), it can immediately be carried out with the aid of the equation qnp=pqn−n(h/2πi)qn−1.”
40.
R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (Wiley, New York, 1985), pp. 140–144, 157–158.
41.
Reference 2, pp. 240–241.
42.
Reference 2, p. 249.
43.
Reference 14, pp. 54–63.
44.
Reference 11, Chap. 17.
45.
For a general discussion of the time evolution operator, see Ref. 2, pp. 308–310.
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