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 = fag = 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,, 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