V THE EQUATIONS OF MOTION
運動方程式

27. Schrödinger’s form for the equations of motion

Our work from § 5 onwards has all been concerned with one instant of time. It gave the general scheme of relations between states and dynamical variables for a dynamical system at one instant of time. To get a complete theory of dynamics we must consider also the connexion between different instants of time. When one makes an observation on the dynamical system, the state of the system gets changed in an unpredictable way, but in between observations causality applies, in quantum mechanics as in classical mechanics, and the system is governed by equations of motion which make the state at one time determine the state at a later time. These equations of motion we now proceed to study. They will apply so long as the dynamical system is left undisturbed by any observation or similar process.t Their general form can be deduced from the principle of superposition of Chapter I. Let us consider a particular state of motion throughout the time during which the system is left undisturbed. We shall have the state at any time t corresponding to a certain ket which depends on ¢ and which may be written |f>. If we deal with several of these states of motion we distinguish them by giving them labels such as A, and we then write the ket which corresponds to the state at time ¢ for one of them |At>. The requirement that the state at one time determines the state at another time means that |At,> determines |At> except for a numerical factor. The principle of superposition applies to these states of motion throughout the time during which the system is undisturbed, and means that if we take a superposition relation holding for certain states at time f, and giving rise to a linear equation between the corresponding kets, e.g. the equation | Rty> = ¢y|Aty>+¢q|Bto>, the same superposition relation must hold between the states of motion throughout the time during which the system is undisturbed and must lead to the same equation between the kets corresponding + The preparation of a state is a process of this kind. It often takes the form of making an observation and selecting the system when the result of the observation turns out to be a cortain pre-assigned number. $27 SCHRODINGER’S FORM FOR THE EQUATIONS OF MOTION 109 to these states at any time ¢ (in the undisturbed time interval), ie. he equation | Re = ¢y|Aty-+e4| Bt), provided the arbitrary numerical factors by which these kets may be multiplied are suitably chosen. It follows that the |Pt>’s are linear functions of the |Pé,>’s and each [Pf is the result of some linear operator applied to |Pt)>. In symbols |Pi> = T|Pig), (1) where 7 is a linear operator independent of P and depending only on é (and fp). We now assume that each | Pé> has the same length as the corre- sponding |Pt >. It is not necessarily possible to choose the arbitrary numerical factors by which the |Pt>’s may be multiplied so as to make this so without destroying the linear dependence of the [P#>’s on the | Pt,>’s, so the new assumption is a physical one and not just a question of notation. It involves a kind of sharpening of the principle of superposition. The arbitrariness in |Pt>) now becomes merely a phase factor, which must be independent of P in order that the linear dependence of the |Pt>’s on the |Pt)>’s may be preserved. From the condition that the length of c,|Pi>-++c,|Qi equals that of ¢,|Ptp>+¢,|Qéy> for any complex numbers ¢,, c,, we can deduce that (QE Pt> = (Qty) Pty. . (2) The connexion between the |Pé>’s and |Pt,>’s is formally similar to the connexion we had in § 25 between the displaced and undisplaced kets, with a process of time displacement instead of the space displace- ment of § 25. Equations (1) and (2) play the part of equations (59) and (60) of § 25. We can develop the consequences of these equations as in § 25 and can deduce that 7 contains an arbitrary numerical factor of modulus unity and satisfies TT = 1, (3) corresponding to (62) of § 25, so T ts unitary. We pass to the infinitesi- mal case by making ¢ > é, and assure from physical continuity that the limit im PO—IP 4) tot, i—ty exists. This limit is just the derivative of | Pt,> with respect to fy. From (1) it equals d| Pty» . £1 —) ii S. 4 dty rs t—ty Plo (#) 110 THE EQUATIONS OF MOTION § 27 The limit operator occurring here is, like (64) of § 25, a pure imaginary linear operator and is undetermined to the extent of an arbitrary additive pure imaginary number. Putting this limit operator multi- plied by 1% equal to H, or rather H(f)) since it may depend on ft), equation (4) becomes, when written for a general f, d|Pty dt Equation (5) gives the general law for the variation with time of the ket corresponding to the state at any time. It is Schrédinger’s form for the equations of motion. It involves just one real linear operator H(¢), which must be characteristic of the dynamical system under consideration. We assume that H(t) is the total energy of the system. There are two justifications for this assumption, (i) the analogy with classical mechanics, which will be developed in the next section, and (ii) we have H(t) appearing as i% times an operator of displacement in time similar to the operators of displacement in the x, y, and z directions of § 25, so corresponding to (69) of § 25 we should have H(t) equal to the total energy, since the theory of relativity puts energy in the same relation to time as momentum to distance. We assume on physical grounds that the total energy of a system is always an observable. For an isolated system it is a constant, and may then be written H. Even when it is not a constant we shall often write it simply H, leaving its dependence on ¢ understood. If the energy depends on #, it means the system is acted on by external forces. An action of this kind is to be distinguished from a distur- bance caused by a process of observation, as the former is compatible with causality and equations of motion while the latter is not. We can get a connexion between H(t) and the T of equation (1) by substituting for |Pé> in (5) its value given by equation (1). This gives ine |Pig> = HQ)T| Pty. ih H(t)|Pty. . (5) Since |Pt,> may be any ket, we have aT ih = HOT. (6) Equation (5) is very important for practical problems, where it is usually used in conjunction with a representation. Introducing a § 27 SCHRODINGER’S FORM FOR THE EQUATIONS OF MOTION 111 representation with a complete set of commuting observables diagonal and putting <é’| Pt> equal to (€'t), we have, passing to the | standard ket notation, [Pty = (éi)). Equation (5) now becomes nS et) = Hye). (7 Equation (7) is known as Schrédinger’s wave equation and its solutions (Et) are time-dependent wave functions. Each solution corresponds to a state of motion of the system and the square of its modulus gives ‘ the probability of the é’s having specified values at any time ¢. For a system describable in terms of canonical coordinates and momenta we may use Schrédinger’s representation and can then take H to be an operator of differentiation in accordance with (42) of § 22. 28. Heisenbers’s form for the equations of motion In the preceding section we set up a picture of the states of undisturbed motion by making each of them correspond to a moving ket, the state at any time corresponding to the ket at that time. We shall call this the Schrodinger picture. Let us apply to our kets the unitary transformation which makes each ket |a> go over into ja*> = Ta). (8) This transformation is of the form given by (75) of § 26 with 7'-! for U, but it depends on the tume t since T depends on t. It is thus to be pictured as the application of a continuous motion (consisting of rotations and uniform deformations) to the whole ket vector space. A ket which is originally fixed becomes a moving one, its motion being given by (8) with |a> independent of #. On the other hand, a ket which is originally moving to correspond to a state of undisturbed motion, i.e. in accordance with equation (1), becomes fixed, since on substituting |Pt for |a> in (8) we get ja*> independent of t. Thus the transformation brings the kets corresponding to states of undisturbed motion to rest. The unitary transformation must be applied also to bras and linear operators, in order that equations between the various quantities may remain invariant. The transformation applied to bras is given by the conjugate imaginary of (8) and applied to linear operators it is given by (70) of § 26 with 7’! for U, i.e. a*¥ = TolyT, (9) 112 THE EQUATIONS OF MOTION § 28 A linear operator which is originally fixed transforms into a moving linear operator in general. Now a dynamical variable corresponds to a linear operator which is originally fixed (because it does not refer to t at all), so after the transformation it corresponds to a moving linear operator. The transformation thus leads us to a new picture of the motion, in which the states correspond to fixed vectors and the dynamical variables to moving linear operators. We shall call this the Heisenberg picture. The physical condition of the dynamical system at any time inyolves the relation of the dynamical variables to the state, and the change of the physical condition with time may be ascribed either to a change in the state, with the dynamical variables kept fixed, which gives us the Schrédinger picture, or to a change in the dynamical variables, with the state kept fixed, which gives us the Heisenberg picture. In the Heisenberg picture there are equations of motion for the dynamical variables. Take a dynamical variable corresponding to the fixed linear operator v in the Schrédinger picture. In the Heisen- berg picture it corresponds to a moving linear operator, which we write as v, instead of v*, to bring out its dependence on t, and which is given by v= TT (10) or Tou, = vt. Differentiating with respect to ¢, we get dT du, aT | = det ae With the help of (6), this gives HT ob ine = HT . AY; _ or he = TwHT—TOHHAT», = v,H— Hv, (11) where A,= TAT. (12) Equation (11) may be written in P.B. notation dw §28 HEISENBERG’S FORM FOR THE EQUATIONS OF MOTION 113 Equation (11) or (13) shows how any dynamical variable varies with time in the Heisenberg picture and gives us Hetsenberg’s form for the equations of motion. These equations of motion are determined by the one linear operator H,, which is just the transform of the linear operator H occurring in Schrédinger’s form for the equations of motion and corresponds to the energy in the Heisenberg picture. We shall call the dynamical variables in the Heisenberg picture, where they vary with the time, Heisenberg dynamical variables, to distinguish them from the fixed dynamical variables of the Schrédinger picture, which we shall call Schrédinger dynamical variables. Each Heisenberg dynamical variable is connected with the corresponding Schrédinger dynamical variable by equation (10). Since this connexion isa unitary transformation, all algebraic and functional relationships are the same for both kinds of dynamical variable. We have 7 = 1 for t = fy, so that v, = v and any Heisenberg dynamical variable at time tf, equals the corresponding Schrédinger dynamical variable. Equation (13) can be compared with classical mechanics, where we also have dynamical variables varying with the time. The equations of motion of classical mechanics can be written in the Hamiltonian dq, oH dp, oH dt — ap,’ dt sg,” form (14) where the qg’s and p’s are a set of canonical coordinates and momenta and H is the energy expressed as a function of them and possibly also of t. The energy expressed in this way is called the Hamiltonian. Equations (14) give, for v any function of the q’s and p’s that does not contain the time ¢ explicitly, dy du dq, , ov dp, aD = [», A], (15) with the classical definition of a P.B., equation (1) of § 21. This is of the same form as equation (13) in the quantum theory. We thus get an analogy between the classical equations of motion in the Hamiltonian form and the quantum equations of motion in Heisen- berg’s form. This analogy provides a justification for the assumption ll4 THE EQUATIONS OF MOTION § 28 that the linear operator H introduced in the preceding section is the energy of the system in quantum mechanics. In classical mechanics a dynamical system is defined mathemati- cally when the Hamiltonian is given, ie. when the energy is given in terms of a set of canonical coordinates and momenta, as this is sufficient to fix the equations of motion. In quantum mechanics a . dynamical system is defined mathematically when the energy is given in terms of dynamical variables whose commutation relations are known, as this is then sufficient to fix the equations of motion, in both Schrédinger’s and Heisenberg’s form. We need to have either H expressed in terms of the Schrédinger dynamical variables . or H, expressed in terms of the corresponding Heisenberg dynamical variables, the functional relationship being, of course, the same in both cases. We call the energy expressed in this way the Hamiltonian of the dynamical system in quantum mechanics, to keep up the analogy with the classical theory. Asystem in quantum mechanics always hasa Hamiltonian, whether the system is one that has a classical analogue and is describable in terms of canonical coordinates and momenta or not. However, if the system does have a classical analogue, its connexion with classical mechanics is specially close and one can usually assume that the Hamiltonian is the same function of the canonical coordinates and. momenta in the quantum theory as in the classical theory.t There would be a difficulty in this, of course, if the classical Hamiltonian involved a product of factors whose quantum analogues do not com- mute, as one would not know in which order to put these factors in the quantum Hamiltonian, but this does not happen for most of the elementary dynamical systems whose study is important for atomic physics. In consequence we are able also largely to use the same language for describing dynamical systems in the quantum theory as in the classical theory (e.g. to talk about particles with given masses moving through given fields of force), and when given a system in classical mechanics, can usually give a meaning to ‘the same’ system in quantum mechanics. Equation (13) holds for v, any function of the Heisenberg dynamical variables not involving the time explicitly, ie. for v any constant + This assumption is found in practice to be successful only when applied with the dynamical coordinates and momenta referring to a Cartesian system of axes and not to more general curvilinear coordinates. §28 HEISENBERG’S FORM FOR THE EQUATIONS OF MOTION 115 linear operator in the Schrédinger picture. It shows that such a function v,is constant if it commutes with H, or if v commutes with 7. We then have y= %, =», and we call v, or v a constant of the motion. It is necessary that v shall commute with # at all times, which is usually possible only if H is constant. In this case we can substitute H for v in (13) and deduce that H, is constant, showing that # itself is then a constant of the motion. Thus if the Hamiltonian is constant in the Schrédinger picture, it is also constant in the Heisenberg picture. For an isolated system, a system not acted on by any external forces, there are always certain constants of the motion. One of these is the total energy or Hamiltonian. Others are provided by the displacement theory of § 25. It is evident physically that the total energy must remain unchanged if all the dynamical variables are displaced in a certain way, so equation (63) of § 25 must hold with vg v= H. Thus D commutes with H and is a constant of the motion. Passing to the case of an infinitesimal displacement, we see that the displacement operators d,, d,, and d, are constants of the motion and hence, from (69) of § 25, the total momentum is a constant of the motion. Again, the total energy must remain unchanged if all the dynamical variables are subjected to a certain rotation. This leads, as will be shown in § 35, to the result that the total angular momentum is a constant of the motion. The laws of conservation of energy, momentum, and angular momentum hold for an isolated system in the Heisenberg picture in quantum mechanics, as they hold in classical mechanics. Two forms for the equations of motion of quantum mechanics have now been given. Of these, the Schrédinger form is the more useful one for practical problems, as it provides the simpler equations. The unknowns in Schrédinger’s wave equation are the numbers which form the representative of a ket vector, while Heisenberg’s equation of motion for a dynamical variable, if expressed in terms of a repre- sentation, would involve as unknowns the numbers forming the representative of the dynamical variable. The latter are far more numerous and therefore more difficult to evaluate than the Schri- dinger unknowns. Heisenberg’s form for the equations of motion is of value in providing an immediate analogy with classical mechanics and enabling one to see how various features of classical theory, such 116 THE EQUATIONS OF MOTION § 28 as the conservation laws referred to above, are translated into quan- tum theory. 29. Stationary states We shall here deal with a dynamical system whose energy is con- stant. Certain specially simple relations hold for this case. Equation (6) can be integrated{ to give : T = eit), with the help of the initial condition that 7 = 1 for i=). This result substituted into (1) gives [Pi = ett) Pty), (18) which is the integral of Schrédinger’s equation of motion (5), and substituted into (10) it gives UY, = ChHt-l0lRye—tHl—toVh (17) which is the integral of Heisenberg’s equation of motion (11), H, being now equal to H. Thus we have solutions of the equations of motion in a simple form. However, these solutions are not of much practical value, because of the difficulty involved in evaluating the operator etH-o% unless H is particularly simple, and for practical purposes one usually has to fall back on Schrédinger’s wave equation. Let us consider a state of motion such that at time ¢, it is an eigen- state of the energy. The ket |Pi,> corresponding to it at this time must be an eigenket of H. If H’ is the eigenvalue to which it belongs, equation (16) gives |Pt) = e~tE' toh | Pty), showing that |P¢> differs from [Pi)> only by a phase factor. Thus the state always remains an eigenstate of the energy, and further, it does not vary with the time at all, since the direction of the ket | Pt» does not vary with the time. Such a state is called a stationary state. The probability for any particular result of an observation on it is independent of the time when the observation is made. From our assumption that the energy is an observable, there are sufficient stationary states for an arbitrary state to be dependent on them. The time-dependent wave function (Et) representing a stationary state of energy H’ will vary with time according to the law PEt) = Yoo(E)e*, (18) f The integration can be carried out as though H were an ordinary algebraic variable instead of a linear operator, because there is no quantity that does not commute with H in the work. § 29 STATIONARY STATES 117 and Schrédinger’s wave equation (7) for it reduces to A'by) = Hf. (19) This equation merely asserts that the state represented by yy is an eigenstate-of H. We call a function yy satisfying (19) an eagenfunction of H, belonging to the eigenvalue 2’. In the Heisenberg picture the stationary states correspond to fixed eigenvectors of the energy. We can set up a representation in which all the basic vectors are eigenvectors of the energy and so correspond to stationary states in the Heisenberg picture. We call such a repre- sentation a Heisenberg representation. The first form of quantum mechanics, discovered by Heisenberg in 1925, was in terms of a representation of this kind. The energy is diagonal in the representa- tion. Any other diagonal dynamical variable must commute with the energy and is therefore a constant of the motion. The problem of setting up a Heisenberg representation thus reduces to the problem of finding a complete set of commuting observables, each of which is a constant of the motion, and then making these observables diagonal. The energy must be a function of these observables, from Theorem 2 of §19. It is sometimes convenient to take the energy itself as one of them. Let « denote the complete set of commuting observables in a Heisenberg representation, so that the basic vectors are written == Cox! [ett tol iyye tH -t0)f [> = eH’ -H MI bela"), (20) where H’ = H(«’) and H” = H(«"). The factor on the right- hand side here is independent of ¢, being an element of the matrix representing the fixed linear operator v. Formula (20) shows how the Heisenberg matrix elements of any Heisenberg dynamical variable vary with time, and it makes », satisfy the equation of motion (11), as is easily verified. The variation given by (20) is simply periodic with the frequency depending only on the energy difference of the two stationary states to which the matrix element refers. This result is closely connected with the Combination Law of Spectroscopy and Bohr’s Frequency 3595.57 T 8 THE EQUATIONS OF MOTION § 29 Condition, according to which (21) is the frequency of the electro- magnetic radiation emitted or absorbed when the system makes a transition under the influence of radiation between the stationary states «’ and «”, the eigenvalues of H being Bohr’s energy levels. These matters will be dealt with in § 45. 30. The free particle The most fundamental and elementary application of quantum mechanics is to the system consisting merely of a free particle, or particle not acted on by any forces. For dealing with it we use as dynamical variables the three Cartesian coordinates z, y, 2 and their conjugate momenta p,, p,, Pp, The Hamiltonian is equal to the kinetic energy of the particle, namely 1 H = <—(pi-+pi+92) (22) according to Newtonian mechanics, m being the mass. This formula is valid only if the velocity of the particle is small compared with c, the velocity of light. For a rapidly moving particle, such as we often have to deal with in atomic theory, (22) must be replaced by the relativistic formula H = o(mic?+ p+ py +p). (23) For small values of p,, p,, and p, (23) goes over into (22), except for the constant term mc? which corresponds to the rest-energy of the particle in the theory of relativity and which has no influence on the equations of motion. Formulas (22) and (23) can be taken over directly into the quantum theory, the square root in (23) being now understood as the positive square root defined at the end of § 11. The constant term mc? by which (23) differs from (22) for small values of p,, py, and p, can still have no physical effects, since the Hamil- tonian in the quantum theory, as introduced in § 27, is undefined to the extent of an arbitrary additive real constant. We shall here work with the more accurate formula (23). We shall first solve the Heisenberg equations of motion. From the quantum conditions (9) of § 21, p, commutes with p, and p,, and hence, from Theorem 1 of § 19 extended to a set of commuting observables, p, commutes with any function of p,, p,, and p, and therefore with H. It follows that p,, is a constant of the motion. Similarly p, and p, are constants of the motion. These results are the same as in the classical § 30 THE FREE PARTICLE 119 theory. Again, the equation of motion for a coordinate, 2, say, is, according to (11), wa, = th = x,c(mc? +p -+-pyt- pz)! —e( mc? + pe + pat De) hp The right-hand side here can be evaluated by means of formula (31) of § 22 with the roles of coordinates and momenta interchanged, so that it reads q,f—fa, = ih of|ap,, (24) f now being any function of the p’s. This gives . é 22 2 tL p2tp2)t an © Pa yO (25) . , _@ , ep, Similarly, Y= “ee y= + . The magnitude of the velocity is v= (+ 9+ a)? = (pit py t+ pz) / A. (26) Equations (25) and (26) are just the same as in the classical theory. Let us consider a state that is an eigenstate of the momenta, belonging to the eigenvalues p,, p;,, p;. This state must be an eigen- state of the Hamiltonian, belonging to the eigenvalue = e(m?-+p? +p, +pe)h, (27) and must therefore be a stationary state. The possible values for H’ are all numbers from mc? to oo, as in the classical theory. The wave function (xyz) representing this state at any time in Schrédinger’s representation must satisfy peahorye)> = peislnye)> = —i6 HEV)», with similar equations for p, and p,. These equations show that (xyz) is of the form . b(ayz) = aetPrtt Py Par, (28) where a@ is independent of x, y, and z. From (18) we see now that the time-dependent wave function ¥(xyzt) is of the form b(xyzt) == dy elPsttPyytP,2- HD, (29) where a, is independent of x, y, z, and ¢. The function (29) of x, y, z, and ¢ describes plane waves in space- time. We see from this example the suitability of the terms ‘wave function’ and ‘wave equation’. The frequency of the waves is v= H'/h, (30) 120 THE EQUATIONS OF MOTION § 30 their wavelength is A= hl(p2+pP +p) = h/P’, (31) P’ being the length of the vector (pi, },p,), and their motion is in the direction specified by the vector (p;, 9, p;) with the velocity My = H'/P! = fv", (32) v’ being the velocity of the particle corresponding to the momentum (Des Di P;) a8 given by formula (26). Equations (30), (31), and (32) are easily seen to hold in all Lorentz frames of reference, the expres- sion on the right-hand side of (29) being, in fact, relativistically invariant with p;,,p,,p, and H’ as the components of a 4-vector. These properties of relativistic invariance led de Broglie, before the discovery of quantum mechanics, to postulate the existence of waves of the form (29) associated with the motion of any particle. They are therefore known as de Broglie waves. In the limiting case when the mass m is made to tend to zero, the classical velocity of the particle v becomes equal to c and hence, from (32), the wave velocity also becomes c. The waves are then like the light-waves associated with a photon, with the difference that they contain no reference to the polarization and involve a complex ex- ponential instead of sines and cosines. Formulas (30) and (31) are still valid, connecting the frequency of the light-waves with the energy of the photon and the wavelength of the light-waves with the momentum of the photon. For the state represented by (29), the probability of the particle being found in any specified small volume when an observation of its position is made is independent of where the volume is. This provides an example of Heisenberg’s principle of uncertainty, the state being one for which the momentum is accurately given and for which, in consequence, the position is completely unknown. Such a state is, of course, a limiting case which never occurs in practice. The states usually met with in practice are those represented by wave packets, which may be formed by superposing a number of waves of the type (29) belonging to slightly different values of (9%, p;,, p.), a8 discussed in § 24. The ordinary formula in hydrodynamics for the velocity of such a wave packet, i.e. the group velocity of the waves, is dv d(1/d) (33) § 30 THE FREE PARTICLE 121 which gives, from (30) and (31) aH’ qo oy ping COP’ , = a =U. 4 This is just the velocity-of the particle. The wave packet moves in the same direction and with the same velocity as the particle moves in classical mechanics. 31. The motion of wave packets The result just deduced for a free particle is an example of a general principle. For any dynamical system with a classical analogue, a state for which the classical description is valid as an approximation is represented in quantum mechanics by a wave packet, all the co- ordinates and momenta having approximate numerical values, whose accuracy is limited by Heisenberg’s principle of uncertainty. Now Schrédinger’s wave equation fixes how such a wave packet varies with time, so in order that the classical description may remain valid, the wave packet should remain a wave packet and should move according to the laws of classical dynamics. We shall verify that this is so. We take a dynamical system having a classical analogue and let its Hamiltonian be H(q,,p,) (r = 1, 2,...,n). The corresponding classi- cal dynamical system will have as Hamiltonian H,(q,, p,) say, obtained by putting ordinary algebraic variables for the g, and p, in A(g,, 7,) and making # — 0 if it occurs in H(q,,p,). The classical Hamiltonian Hf, is, of course, a real function of its variables. It is usually a quadratic function of the momenta p,, but not always so, the relativistic theory of a free particle being an example where it is not. The following argument is valid for H, any algebraic function of the p’s. We suppose that the time-dependent wave function in Schré- dinger’s representation is of the form (gt) = AekSi, (35) where A and S are real functions of the q’s and ¢ which do not vary very rapidly with their arguments. The wave function is then of the form of waves, with A and S determining the amplitude and phase respectively. Schrédinger’s wave equation (7) gives RS Aci = H(y,,p,)Aei™) or aZ _A =) — ¢-tSI(q,, p,) ActSIAy, (36) 122 THE EQUATIONS OF MOTION § 3] Now e~*5# is evidently a unitary linear operator and may be used for U in equation (70) of § 26 to give us a unitary transformation. The q’s remain unchanged by this transformation, each p, goes over into “e-iSihi esi ~— Dp, +é8/éq,5 with the help of (81) of § 22, and H goes over into e Sh (g,, pest = (Gy, Pp +88 /09,); since algebraic relations are preserved by the transformation. Thus (86) becomes aA {aZ—aS) = 8( an. +53 54. (37) Let us now suppose that # can be counted as small and let us neglect terms involving # in (37). This involves neglecting the p,’s that occur in H in (37), since each p, is equivalent to the operator — ii d/éq, operating on the functions of the q’s to the right of it. The surviving terms give ag as Yr, This is a differential equation which the phase function S has to satisfy. The equation is determined by the classical Hamiltonian function H, and is known as the Hamilton-Jacobi equation in classical dynamics. It allows S to be real and so shows that the assumption of the wave form (35) does not lead to an inconsistency. To obtain an equation for A, we must retain the terms in (37) which are linear in % and see what they give. A direct evaluation of these terms is rather awkward in the case of a general function H, and we can get the result we require more easily by first multiplying both sides of (37) by the bra vector (Af, where f is an arbitrary real function of the g’s. This gives cafin aS} — ApH (an.p.-+ 5} Y. (38) The conjugate complex equation is 04 (as as cas{ 8 FAG) = (aH (gn2 +S) sA». Subtracting and ee out by 2, we obtain ApS = (ALLE (tn, +) ] >. (39) § 31 THE MOTION OF WAVE PACKETS 123 We now have to evaluate the P.B. Lf, 2 (G,.p,+8S/6q,)].- Our assumption that % can be counted as small enables us to expand Al(q,, P,+ 08/éq,) a8 a power series in the p’s. The terms of zero degree will contribute nothing to the P.B. The terms of the first degree in the p’s give a contribution to the P.B. which can be evaluated most easily with the help of the classical formula (1) of § 21 (this formula being valid also in the quantum theory if u is independent of the »’s and v is linear in the y’s). The amount of this contribution is oF |= fetal] 3 ag, aps Pr=OS/8ar the notation meaning that we must substitute 0S/éq, for each p, in the function [ ] of the g’s and y’s, so as to obtain a function of the q’s only. The terms of higher degree in the p’s give contributions to the P.B. which vanish when i > 0. Thus (39) becomes, with neglect of terms involving #, which is equivalent to the neglect of 4? in (37), GZ of Ste Ie PD ] mer . 40 Now if a(g) and 6(q) are any two functions of the q’s, formula > and so (al) > = —< oan, (41) provided a(g) and 4(g) satisfy suitable boundary conditions, as dis- cussed in §§ 22 and 23. Hence (40) may be written Since this holds for an arbitrary real function f, we must have at : a, oD, pr=aSlegr This is the equation for the amplitude A of the wave function. To get an understanding of its significance, let us suppose we have a fluid moving in the space of the variables qg, the density of the fluid at any point and time being A? and its velocity being dq, _ [ Geel d OP, | p,=aSlagy (43) 124 THE EQUATIONS OF MOTION § 31 Equation (42) is then just the equation of conservation for such a fluid. The motion of the fluid is determined by the function S satisfying (38), there being one possible motion for each solution of (38). For a given S, let us take a solution of (42) for which at some definite time the density A? vanishes everywhere outside a certain small region. We may suppose this region to move with the fluid, its velocity at each point being given by (43), and then the equation of conservation (42) will require the density always to vanish outside the region. There is a limit to how small the region may be, imposed by the approximation we made in neglecting # in (39). This approxi- mation is valid only provided ej as or —— <—- — which requires that A shall vary by an appreciable fraction of itself only through a range of the qg’s in which S varies by many times #, i.e. arange consisting of many wavelengths of the wave function (35). Our solution is then a wave packet of the type discussed in § 24 and remains so for all time. We thus get a wave function representing a state of motion for which the coordinates and momenta have approximate numerical values throughout all time. Such a state of motion in quantum theory corresponds to the states with which classical theory deals. The motion of our wave packet is determined by equations (38) and (43). From these we get, defining p, as 0S/ég,, dp, d2S 88 , es dq, dt — dt aq, btaq, ' < aq, aq, dt gs 8q,] <1 89,09, Dy 0. aq where in the last line the y’s are counted as independent of the q’s before the partial differentiation. Equations (43) and (44) are just the classical equations of motion in Hamiltonian form and show that the wave packet moves according to the laws of classical mechanics. § 31 THE MOTION OF WAVE PACKETS 125 We see in this way how the classical equations of motion are derivable from the quantum theory as a limiting case. .By a more accurate solution of the wave equation one can show that the accuracy with which the coordinates and momenta simul- taneously have numerical values cannot remain permanently as favourable as the limit allowed by Heisenberg’s principle of un- certainty, equation (56) of § 24, but if it is initially so it will become less favourable, the wave packet undergoing a spreading. 32. The action principlet Equation (10) shows that the Heisenberg dynamical variables at time ¢, v, are connected with their values at time f, v,, or v, by a unitary transformation. The Heisenberg variables at time ¢+-8¢ are connected with their values at time ¢ by an infinitesimal unitary transformation, as is shown by the equation of motion (11) or (13), which gives the connexion between v,,5, and v, of the form of (79) or (80) of § 26 with HA, for F and 8¢/# for «. The variation with time of the Heisenberg dynamical variables may thus be looked upon as the continuous unfolding of a unitary transformation. In classical mechanics the dynamical variables at time ¢+6¢ are connected with their values at time ¢ by an infinitesimal contact transformation and the whole motion may be looked upon as the continuous unfolding of a contact transformation. We have here the mathematical foundation of the analogy between the classical and quantum equations of motion, and can develop it to bring out the quantum analogue of all the main features of the classical theory of dynamics. Suppose we have a representation in which the complete set of commuting observables € are diagonal, so that a basic bra is <é’|. We can introduce a second representation in which the basic bras are cel = EIT. (45) The new basic bras depend on the time ¢ and give us a moving representation, like a moving system of axes in an ordinary vector space. Comparing (45) with the conjugate imaginary of (8), we see that the new basic vectors are just the transforms in the Heisenberg picture of the original basic vectors in the Schrédinger picture, and hence they must be connected with the Heisenberg dynamical f See Kennard, Z. f. Physik, 44 (1927), 344; Darwin, Proc. Roy. Soc. A, 117 (1927), 258. { This section may be omitted by the student who is not specially concerned with higher dynamics. 126 THE EQUATIONS OF MOTION § 82 variables v, in the same way in which the original basic vectors are connected with the Schrédinger dynamical variables v. In particular, each <é’*| must be an eigenvector of the &’s belonging to the eigen- values é’. It may therefore be written <;|, with the understanding that the numbers &; are the same eigenvalues of the é,’s that the €”s are of the &’s. From (45) we get CEE = EIT IE", (46) showing that the transformation function is just the representative of T in the original representation. Differentiating (45) with respect to ¢ and using (6), we get ey @ ae wy per AL napm ‘ the Eel = KE |e OID = oi with the help of (12). Multiplying on the right by any ket |a> independent of t, we get iS is the same function of the variables & and & that CEH \é") is of &’ and &’, equations (47) and (48) are of precisely the same form, with the variables &;,€ in (47) playing the role of the variables €’ and &" in (48) and the function <£|@> playing the role of the function <é"|Pt>. We can thus look upon (47) as a form of Schrédinger’s wave equation, with the function (&|a> of the variables €, as the wave function. In this way Schrédinger’s wave equation appears in a new light, as the condition on the representative, in the moving representation with the Heisenberg variables &, diagonal, of the jived ket corresponding to a state in the Heisenberg picture. The function <&la> owes its variation with time to its left factor <&|, in contra- distinction to the function <&’| Pt), which owes its variation with time to its right factor |.P>. lf we put |a> = [é"5 in (47), we get HSCEI = | CEIIELIEED By CAPE"), (49) § 32 THE ACTION PRINCIPLE 127 showing that the transformation function <é|&"> satisfies Schré- dinger’s wave equation. Now & = &, so we must have CE4,18"> = (8, —€"), (50) the 6 function here being understood as the product of a number of factors, one for each é-variable, such as occurs for the variables evap €, on the right-hand side of equation (34) of §16. Thus the transformation function <&;|é"> is that solution of Schrédinger’s wave equation for which the é’s certainly have the values é” at time ft, The square of its modulus, |<(&|é"]*, is the relative probability of the é’s having the values & at time t > t, if they certainly have the values &" at time t). We may write ¢&|é"> as <&/&,> and consider it as depending on f) as well as on é. To get its dependence on {, we take the conjugate complex of equation (49), interchange ¢ and f, and also interchange single primes and double primes. This gives — ih Eilg> = [CEE aE ERED. (51) 0 The foregoing discussion of the transformation function <£|é”> is valid with the é’s any complete set of commuting observables. The equations were written down for the case of the é’s having continuous eigenvalues, but they would still be valid if any of the é’s have discrete eigenvalues, provided the necessary formal changes are made in them. Let us now take a dynamical system having a classical analogue and let us take the ¢’s to be the coordinates g. Put = es (52) and so define the function S of the variables gj, q”. This function also depends explicitly on ¢. (52) is a solution of Schrédinger’s wave equation and, if % can be counted as small, it can be handled in the same way as (35) was. The S of (52) differs from the S of (35) on account of there being no A in (52), which makes the S of (52) com- plex, but the real part of this S equals the S of (35) and its pure imaginary part is of the order #. Thus, in the limit i > 0, the S of (52) will equal that of (35) and will therefore satisfy, corresponding to (38), —aS/at = Hqy D4), (53) where Dy = O8/0qy, (54) and H, is the Hamiltonian of the classical analogue of our quantum dynamical system. But (52) is also a solution of (51) with q’s for é’s, 128 THE EQUATIONS OF MOTION § 32 which is the conjugate complex of Schrédinger’s wave equation in the variables q” or gj,. This causes S to satisfy alsot where p, = —08/aq,. (56) The solution of the Hamilton-Jacobi equations (53), (55) is the action function of classical mechanics for the time interval tg to ¢, i.e. it is the time integral of the Lagrangian L, t S= { Le’) de’. (57) ! Thus the S defined by (52) is the quantum analogue of the classical action function and equals it in the limit i > 0. To get the quantum analogue of the classical Lagrangian, we pass to the case of an infinitesimal ° time interval by putting ¢ = t)+5¢ and we then have <@,,5/|¢;,> 8 the analogue of e*o5i%, For the sake of the analogy, one should consider E(t )) as a function of the coordinates gq’ at time é,4-6¢ and the co- ordinates q” at time ¢), rather than as a function of the coordinates and velocities at time t), as one usually does. The principle of least action in classical mechanics says that the action function (57) remains stationary for small variations of the tra- jectory of the system which do not alter the end points, i.e. for small variations of the q’s at all intermediate times between f, and ¢ with q,, and q, fixed. Let us see what it corresponds to in the quantum theory. ty Put exp [ Lt) aj) = exp{iS(t,, t,)/f} = Blty, ta), (58) te so that B(t,,t,) corresponds to in the quantum theory. (We here allow qj, and ¢/, to denote different eigenvalues of q,, and q;,, to save having to introduce a large number of primes into the analysis.) Now suppose the time interval 4 >¢ to be divided up into a large number of small time intervals ty > t,, t, > tos.) bm > bm bm > t, bY the introduction of a sequence of intermediate times t,, fg,..., t,. Then Bt, to) = Bii, tm) Btn, bm1)++-B (ty, t) Bt, to). (59) The corresponding quantum equation, which follows from the pro- perty of basic vectors (35) of § 16, is = fff 4am dia ldin-a> Uin-1-+- 414i |G?» (60) } For a more accurate comparison of transformation functions with classical theory, see Van Vleck, Proc. Nat. Acad. 14, 178. § 32 THE ACTION PRINCIPLE 129 g;, being written for g;, for brevity. At first sight there does not seem to be any close correspondence between (59) and (60). We must, however, analyse the meaning of (59) rather more carefully. We must regard each factor B as a function of the q’s at the two ends of the time interval to which it refers. This makes the right-hand side of (59) a function, not only of g, and q,, but also of all the intermediate qs. Equation (59) is valid only when we substitute for the inter- mediate q’s in its right-hand side their values for the real trajectory, small variations in which values leave S stationary and therefore also, from (58), leave B(t,t,) stationary. It is the process of substituting these values for the intermediate q’s which corresponds to the inte- grations over all values for the intermediate ¢’’s in (60). The quantum analogue of the action principle is thus absorbed in the composition law (60) and the classical requirement that the values of the inter- mediate q’s shall make S stationary corresponds to the condition in quantum mechanics that all values of the intermediate q’’s are important in proportion to their contribution to the integral in (60). Let us see how (59) can be a limiting case of (60) for A small. We must suppose the integrand in (60) to be of the form e”*, where F is a function of 95,491, 2.---» Gm: Which remains continuous as % tends to zero, so that the integrand is a rapidly oscillating function when his small. The integral of such a rapidly oscillating function will be extremely small, except for the contribution arising from a region in the domain of integration where comparatively large variations in the g;, produce only very small variations in #. Such a region must be the neighbourhood ofa point where F is stationary for small varia- tions of the g;. Thus the integral in (60) is determined essentially by the value of the integrand at a point where the integrand is stationary for small variations of the intermediate q’’s, and so (60) goes over into (59). Equations (54) and (56) express that the variables qj, p; are con- nected with the variables g”",p” by a contact transformation and are one of the standard forms of writing the equations of a contact trans- formation. There is an analogous form for writing the equations of a unitary transformation in quantum mechanics. We get from (52), with the help of (45) of § 22, OS(% 9") a . (gj , (GlPuld’> = th Cailg"’> = 4 Ge|PrelZ qi, CAL agi, . (61) “130 . . THE EQUATIONS OF MOTION § 32 Similarly, with the help of (46) of § 22, . f i . a J a as tb . , a Cailprla”> = 2, =

= fag , (63) where f(q,) and g(q) are functions of the g,'s and q’s respectively. Let G(q,q@) be any function of the q/s and q’s consisting of a sum or integral of terms each of the form f(q,)g(qg), so that all the qs in G occur to the left of all the g’s. Such a function we call well ordered. Applying (63) to each of the terms in G and adding or integrating, t ql a , a AP meee GGG Dlg"> = Gana" aila">. Now let us suppose each p,, and p, can be expressed as a well-ordered function of the q¢,’s and g’s and write these functions 9,(q;, 9), Pr(Q Q)- Putting these functions for G, we get = Pale 1 )alg">, G1P.1e> = Pr(Ge 9")- Comparing these equations with (61) and (62) respectively, we see that , rom OS(G,9") rom — SG, 9") PrlGs ') = age : PAG Y') = ~~ agh This means that aS(G, 9g) OS (Gp 2) = = 4, Prt Ody 3 Pr aq, F (6 ) provided the right-hand sides of (64) are written as well-ordered functions. These equations are of the same form as (54) and (56), but refer to the non-commuting quantum variables ¢,,¢ instead of the ordinary algebraic variables g;,¢q". They show how the conditions for a unitary transformation between quantum variables are analogous to the condi- tions for a contact transformation between classical variables. The analogy is not complete, however, because the classical S must be real and there is no simple condition corresponding to this for the S of (64). 33. The Gibbs ensemble In our work up to the present we have been assuming all along that our dynamical system at each instant of time is in a definite state, that is to say, its motion is specified as completely and accurately as is possible without conflicting with the general principles of the theory § 33 THE GIBBS ENSEMBLE 131 In the classical theory this would mean, of course, that all the coordi- nates and momenta have specified values. Now we may be interested in a motion which is specified to a lesser extent than this maximum possible. The present section will be devoted to the methods to be used in such a case. The procedure in classical mechanics is to introduce what is called a Gibbs ensemble, the idea of which is as follows. We consider all the dynamical coordinates and momenta as Cartesian coordinates in a certain space, the phase space, whose number of dimensions is twice the number of degrees of freedom of the system. Any state of the system can then be represented by a point in this space. This point will move according to the classical equations of motion (14). Sup- pose, now, that we are not given that the system is in a definite state at any time, but only that it is in one or other of a number of possible states according to a definite probability law. We should then be able to represent it by a fluid in the phase space, the mass of fluid in any volume of the phase space heing the total probability of the system being in any state whose representative point lies in that volume. Each particle of the fluid will be moving according to the equations of motion (14). If we introduce the density p of the fluid at any point, equal to the probability per unit volume of phase space of the system being in the neighbourhood of the corresponding state, we shall have the equation of conservation ap a( dq\ , 2/ dp, a> ~ Dd lagloai) tala] This may be considered as the equation of motion for the fluid, since it determines the density p for all time if p is given initially as a function of the g’sand p’s. It is, apart from the minus sign, of the same form as the ordinary equation of motion (15) for a dynamical variable. The requirement that the total probability of the system being in any state shall be unity gives us a normalizing condition for p ff pdgdp = 1, (66) the integration being over the whole of phase space and the single 132 THE EQUATIONS OF MOTION § 33 differential dg or dp being written to denote the product of all the dq’s or dp’s. If 8 denotes any function of the dynamical variables, the average value of B will be [ f Be dade. (67) It makes only a trivial alteration in the theory, but often facilitates discussion, if we work with a density p differing from the above one by a positive constant factor, k say, so that we have instead of (66) [f edadp = &. With this density we can picture the fluid as representing a number k of similar dynamical systems, all following through their motions independently in the same place, without any mutual disturbance or interaction. The density at any point would then be the probable or average number of systems in the neighbourhood of any state per unit volume of phase space, and expression (67) would give the average total value of 8 for all the systems. Such a set of dynamical systems, which is the ensemble introduced by Gibbs, is usually not realizable in practice, except as a rough approximation, but it forms all the same a useful theoretical abstraction. We shall now see that there exists a corresponding density p in quantum mechanics, having properties analogous to the above. It was first introduced by von Neumann. Its existence is rather surprising in view of the fact that phase space has no meaning in quantum mechanics, there being no possibility of assigning numerical values simultaneously to the g’s and p’s. We consider a dynamical system which is at a certain time in one or other of a number of possible states according to some given probability law. These states may be either a discrete set or a con- tinuous range, or both together. We shall here take for definiteness the case of a discrete set and suppose them labelled by a parameter m. Let the normalized ket vectors corresponding to them be |m)> and let the probability of the system being in the mth state be P,. We then define the quantum density p by p= > Im>P, cin. (68) Let p’ be any eigenvalue of p and |p’> an eigenket belonging to this eigenvalue. Then > |m>P, = plp’> = p'lp’> m § 33 THE GIBBS ENSEMBLE 133 so that > = p' or & Falomip >? = p’. Now P,,, being a probability, can never be negative. It follows that p’ cannot be negative. Thus p has no negative eigenvalues, in analogy with the fact that the classical density p is never negative. Let us now obtain the equation of motion for our quantum p. In Schrédinger’s picture the kets and bras in (68) will vary with the time in accordance with Schrédinger’s equation (5) and the conjugate imaginary of this equation, while the P,,’s will remain constant, since the system, so long as it is left undisturbed, cannot change over from a state corresponding to one ket satisfying Schrédinger’s equation to a state corresponding to another. We thus have dp _ .,{a|m> ; dim| m = ¥ (Alm>P,,P,cm lH} = Hp—pH. (69) This is the quantum analogue of the classical equation of motion (65). Our quantum p, like the classical one, is determined for all time if it is given initially. From the assumption of § 12, the average value of any observable 8 when the system is in the state m is . Hence if the system is distributed over the various states m according to the probability law P,,, the average value of B will be ¥ P,. If we introduce m & representation with a discrete set of basic ket vectors |¢’> say, this equals , PaCimie' = & Pacone> = § ioe) = Fe UeBle>, (70) the last step being easily verified with the law of matrix multiplica- tion, equation (44) of §17. The expressions (70) are the analogue of the expression (67) of the classical theory. Whereas in the classical theory we have to multiply 8 by p and take the integral of the product over all phase space, in the quantum theory we have to multiply 8 by p, with the factors in either order, and take the 3598.57 K 134 THE EQUATIONS OF MOTION § 33 diagonal sum of the product in a representation. If the representa- tion involves a continuous range of basic vectors |é>, we get instead f (70 “em [ de” = f ce" ppie’> ae", (71) so that we must carry through a process of ‘integrating along the diagonal’ instead of summing the diagonal elements. We'shall define (71) to be the diagonal sum of Bp in the continuous case. It can easily be verified, from the properties of transformation functions (56) of § 18, that the diagonal sum is the same for all representations. From the condition that the |m)>’s are normalized we get, with discrete £’’s since the total probability of the system being in any state is unity. This is the analogue of equation (66). The probability of the system being in the state £', or the probability of the observables € which are diagonal in the representation having the values é’, is, according to the rule for interpreting representatives of kets (51) of § 18, S Ie’ Im>2Py = , (73) which gives us a meaning for each term in the sum on the left-hand side of (72). For continuous é’’s, the right-hand side of (73) gives the probability of the és having values in the neighbourhood of €’ per unit range of variation of the values £’. As in the classical theory, we may take a density equal to & times the above p and consider it as representing a Gibbs ensemble of & similar dynamical systems, between which there is no mutual dis- turbance or interaction. We shall then have & on the right-hand side of (72), and (70) or (71) will give the total average @ for all the members of the ensemble, while (73) will give the total probability of a member of the ensemble having values for its &’s equal to £' or in the neighbourhood of £’ per unit range of variation of the values €’. , An important application of the Gibbs ensemble is to a dynamical system in thermodynamic equilibrium with its surroundings at a given temperature 7’. Gibbs showed that such a system is repre- sented in classical mechanics by the density p == ce Hkh, (74) § 33 THE GIBBS ENSEMBLE 135 H being the Hamiltonian, which is now independent of the time, k being Boltzmann’s constant, and ¢ being a number chosen to make the normalizing condition (66) hold. This formula may be taken over unchanged into the quantum theory. At high temperatures, (74) becomes p = ¢, which gives, on being substituted into the right-hand side of (73), c<é’|é’ == c in the case of discrete €”s. This shows that at high temperatures all discrete states aré equaily probable,