Masujima M. Applied Mathematical Methods in Theoretical Physics
Подождите немного. Документ загружается.
358 10 Calculus of Variations: Applications
equation was first deduced as
H
q,
i
∇
q
φ(
q) = Eφ(
q), (10.1.24)
and then, from the observation that the time-dependence of the optical scalar
wave is given by exp[−iωt] and with the assumption that the time-dependence
of the wavefunction for the stationary state must be given by exp[−iEt], the
time-dependent Schr
¨
odinger equation is deduced as
i
∂
∂t
ψ(
q, t ) = H
q,
i
∇
q
, t
ψ(
q, t). (10.1.25)
The ‘‘derivation’’ of Eq. (10.1.24) is possible in the following manner. We assume
that the Hamiltonian H(
q(t),
p(t), t) is independent of time t and is equal to the total
energy E. We separate the generator S(
q(t),
P(t), t )as
S(
q(t),
P(t), t) = W(
q(t),
P(t)) − Et. (10.1.26)
We observe that the Hamilton–Jacobi equation becomes
H (
q,
∇
q
W(
q)) =
1
2m
{
∇
q
W(
q)}
2
+ V(
q) = E. (10.1.27)
Namely, we have
{
∇
q
W(
q)}
2
=
p
2
= 2m{E − V(
q)}, (10.1.28)
which is nothing but the mechanical analog of the Eikonal equation for geometrical
optics,
{
∇
q
L(
q)}
2
= n
2
(
q), (10.1.29)
where L(
q) is the Eikonal of the optical scalar wave φ(
q, t)definedby
φ(
q, t ) = A(
q, t )exp[i(L(
q) −ωt)]. (10.1.30)
From Eqs. (10.1.28) and (10.1.29), we find that the magnitude of
p in classical
mechanics corresponds to the index of refraction n(
q) in geometrical optics. We
recall that the time-independent optical scalar wave equation is given by
∇
2
q
φ(
q) +
4π
2
λ
2
φ(
q) = 0. (10.1.31)
According to de Broglie’s matter wave hypothesis, we have
λ
2π
=
p
, (10.1.32)
10.1 Hamilton–Jacobi Equation and Quantum Mechanics 359
and, substituting Eq. (10.1.32) into Eq. (10.1.31) and using Eq. (10.1.28), we obtain
the time-independent Schr
¨
odinger equation,
∇
2
q
φ(
q) +
2m{E − V(
q)}
2
φ(
q) = 0,
or
H
q,
i
∇
q
φ(
q) = Eφ(
q). (10.1.33)
We invoked de Broglie’s matter wave hypothesis instead of the inverse of Eikonal
approximation to derive the time-independent Schr
¨
odinger equation.
We point out that the optical–mechanical analogy is not complete. We observe
that the time-dependent optical scalar wave equation is given by
∇
2
q
φ(
q, t) −
1
v
2
∂
2
∂t
2
φ(
q, t) = 0. (10.1.34)
This is second order in partial time derivative. As the initial value problem, we
need to specify the value of φ(
q, t)and∂φ(
q, t)∂t at some time for the complete
specification of φ(
q, t) at a later time. On the other hand, we observe that the
time-dependent Schr
¨
odinger equation, Eq. (10.1.25), is first order in the partial
time derivative. As the initial value problem, we need to specify only the value of
ψ(
q, t) at some time for the complete specification of ψ(
q, t)atalatertime.We
remark that this is Huygens’ principle in wave mechanics.
We note that Hamilton was one step short of arriving at wave mechanics as early
as 1834, although he did not have any experimentally compelling reason to take
such a step. On the other hand, by 1924, L. de Broglie and E. Schr
¨
odinger had
sufficient experimentally compelling reasons to take such a step.
In the summer of 1925, M. Born, W. Heisenberg, and P. Jordan deduced
matrix mechanics from the consistency of the Ritz combination principle, the
Bohr quantization condition, the Fourier analysis of physical quantity in classical
physics, and Hamilton’s canonical equation of motion . We will state the basic
principles of matrix mechanics below.
Assumption 1. All physical quantities are represented by matrices. If the physical
quantities are real, the corresponding matrices are Hermitian.
Assumption 2. The time dependence of the (a, b) element of the physical quantity
is of the form given by exp[2πiν
a,b
t].
Assumption 3. The frequency ν
a,b
follows the Ritz combination principle,
ν
a,b
+ ν
b,c
= ν
a,c
, ν
a,a
= 0, ν
a,b
=−ν
b,a
. (10.1.35)
Assumption 4. The time derivative of the physical quantity is represented by the
time derivative of the corresponding matrix.
Assumption 5. The sum A + B of the two physical quantities, A and B, is represented
by the sum of the corresponding two matrices.
360 10 Calculus of Variations: Applications
Assumption 6. The product AB of the two physical quantities, A and B,is
represented by the product of the corresponding two matrices.
Assumption 7. The coordinate q satisfies the equation of the motion of the
mechanical system under consideration.
Assumption 8. The canonical momentum p conjugate to the coordinate q is defined
for the system with one degree of freedom and they satisfy
(pq)
a,b
− (qp)
a,b
=
i
δ
a,b
. (10.1.36)
Assumption 9. The canonical momenta {p
r
}
f
r=1
conjugate to the coordinates {q
s
}
f
s=1
aredefinedforthesystemwithf degrees of freedom and they satisfy
q
r
q
s
− q
s
q
r
= 0,
p
r
p
s
− p
s
p
r
= 0,
(p
r
q
s
)
a,b
− (q
s
p
r
)
a,b
= (i)δ
r,s
δ
a,b
.
(10.1.37)
Assumption 10. The equation of motion of the system is given by Hamilton’s
canonical equation of motion,
d
dt
q
r
=
∂H
∂p
r
,
d
dt
p
r
=−
∂H
∂q
r
, (10.1.38)
where the mechanical energy of the system is given by H({q
r
}
f
r=1
, {p
r
}
f
r=1
).
With these assumptions, we can prove the following theorem with some mathe-
matical preliminary.
Theorem. Given the function F({q
r
}
f
r=1
, {p
r
}
f
r=1
) of {q
r
}
f
r=1
and {p
r
}
f
r=1
,wehave
∂F
∂q
r
=
i
(p
r
F − Fp
r
),
∂F
∂p
r
=−
i
(q
r
F − Fq
r
). (10.1.39)
Hence it follows that
i
d
dt
q
r
= q
r
H − Hq
r
, i
d
dt
p
r
= p
r
H − Hp
r
. (10.1.40)
P.A.M. Dirac developed the transformation theory of quantum mechanics with the
abstract notion of the bra-vector and the ket-vector in 1926. With the transformation
theory, the distinction between the Schr
¨
odinger approach and the Heisenberg
approach is that the state vector is time dependent and the operators are time-
independent for the former, and the state vector is time-independent and the
operators are time dependent for the latter.
Through 1940s and 1950s, two radically different approaches to quantum theory
were developed. One is Feynman’s action principle and the other is Schwinger’s
action principle. Schwinger’s action principle proposed in the early 1950s is the
differential formalism of quantum action principle to be compared with Feynman’s
10.2 Feynman’s Action Principle in Quantum Theory 361
action principle proposed in the early 1940s, which is the integral formalism of
quantum action principle. These two quantum action principles are equivalent to
each other and are essential in carrying out the computation of electrodynamic
level shifts of the atomic energy level.
10.2
Feynman’s Action Principle in Quantum Theory
Feynman’s Action Principle
We note that the operator
ˆ
q(t)atalltimet (the operator
ˆ
φ(x) at all space–time
indices x on space-like hypersurface σ ) forms a complete set of the operators. In
other words, the quantum theoretical state vector can be expressed by the complete
set of eigenket
q, t
of the commuting operators
ˆ
q(t) (the complete set of eigenket
φ, σ
of the commuting operators
ˆ
φ(x)). Feynman’s action principle states that the
transformation function
q
, t
q
, t
(
φ
, σ
φ
, σ
)isgivenby
q
, t
q
, t
=
1
N()
q(t
)=q
q(t
)=q
D [q]exp
i
t
t
dtL(q(t),
˙
q(t))
,
(10.2.1M)
φ
, σ
φ
, σ
=
1
N()
φ
,σ
φ
,σ
D [φ]exp
i
(σ
,σ
)
d
4
x L(φ(x), ∂
µ
φ(x))
.
(10.2.1F)
The space–time region is given by that of space–time region sandwiched
between t
and t
(σ
and σ
). We state here three assumptions involved:
(A-1) The principle of superposition.
q(t
)=q
q(t
)=q
D [q] =
t=t
dq
q
II
(t
)=q
q
II
(t
)=q
D [q
II
]
q
I
(t
)=q
q
I
(t
)=q
D [q
I
], (10.2.2M)
φ
,σ
φ
,σ
D [φ] =
σ
dφ
φ
,σ
φ
,σ
D [φ
II
]
φ
,σ
φ
,σ
D [φ
I
]. (10.2.2F)
(A-2) Functional integral by parts is allowed.
(A-3) Resolution of identity.
dq
q
, t
q
, t
= 1,
dφ
φ
, σ
φ
, σ
= 1. (10.2.3M,F)
From the consistency of the three assumptions, the normalization constant N()
must satisfy
N(
1
+
2
) = N(
1
)N(
2
), (10.2.4M,F)
362 10 Calculus of Variations: Applications
which also originates from the additivity of the action functional,
I[q;t
, t
] = I [q;t
, t
] +I[q;t
, t
], I[φ;σ
, σ
] = I [φ;σ
, σ
] + I[φ;σ
, σ
].
(10.2.5M,F)
The action functional is defined by
I[q;t
, t
] =
t
t
dtL(q(t),
˙
q(t)), I[φ;σ
, σ
] =
(σ
,σ
)
d
4
x L(φ(x), ∂
µ
φ(x)).
(10.2.6M,F)
Operator, Equation of Motion, and Time-Ordered Product:WenotethatinFeyn-
man’s action principle, the operator is defined by its matrix elements.
q
, t
ˆ
q(t)
q
, t
=
1
N()
q(t
)=q
q(t
)=q
D [q]q(t)exp
i
I[q;t
, t
]
,
(10.2.7M)
φ
, σ
ˆ
φ(x)
φ
, σ
=
1
N()
φ
,σ
φ
,σ
D [φ]φ(x)exp
i
I[φ;σ
, σ
]
. (10.2.7F)
If t lies on the t
-surface (x lies on the σ
-surface), Eqs. (10.2.7M) and (10.2.7F) can
be rewritten on the basis of (10.2.1M) and (10.2.1F) as
q
, t
ˆ
q(t
)
q
, t
= q
q
, t
q
, t
, (10.2.8M)
φ
, σ
ˆ
φ(x
)
φ
, σ
= φ
(x
)
φ
, σ
φ
, σ
, (10.2.8F)
for all
q
, t
(for all
φ
, σ
) which form a complete set.
Then we have
q
, t
ˆ
q(t
) = q
q
, t
, (10.2.9M)
φ
, σ
ˆ
φ(x
) = φ
(x
)
φ
, σ
, (10.2.9F)
which are the defining equations of the eigenbras,
q
, t
and
φ
, σ
.
Next we consider the variation of the action functional,
δI [q;t
, t
] =
t
t
dt
∂L(q(t),
˙
q(t))
∂q(t)
−
d
dt
∂L(q(t),
˙
q(t))
∂
˙
q(t)
δq(t)
+
t
t
dt
d
dt
∂L(q(t),
˙
q(t))
∂
˙
q(t)
δq(t)
, (10.2.10M)
10.2 Feynman’s Action Principle in Quantum Theory 363
δI [φ;σ
, σ
] =
d
4
x
∂L(φ(x), ∂
µ
φ(x))
∂φ(x)
− ∂
µ
∂L(φ(x), ∂
µ
φ(x))
∂(∂
µ
φ(x))
δφ(x)
+
d
4
x ∂
µ
∂L(φ(x), ∂
µ
φ(x))
∂(∂
µ
φ(x))
δφ(x)
. (10.2.10F)
We consider a particular variations in which the end points are fixed,
δq(t
) = δq(t
) = 0; δφ(x
on σ
) = δφ(x
on σ
) = 0. (10.2.11M,F)
Then the second terms in (10.2.10M) and (10.2.10F) vanish. We obtain the Euler
derivatives,
δI[q;t
, t
]
δq(t)
=
∂L(q(t),
˙
q(t))
∂q(t)
−
d
dt
∂L(q(t),
˙
q(t))
∂
˙
q(t)
,
δI[φ;σ
, σ
]
δφ(x)
=
∂L(φ(x), ∂
µ
φ(x))
∂φ(x)
−∂
µ
∂L(φ(x), ∂
µ
φ(x))
∂(∂
µ
φ(x))
.
The (Euler-) Lagrange equation of motion follows from the identities:
D [q + ε] = D [q]; and D [φ + ε] = D [φ].
Under these identities, we have the following equations:
1
N()
q(t
)=q
q(t
)=q
D [q]exp
i
I[q;t
, t
]
=
1
N()
q(t
)=q
q(t
)=q
D [q + ε]exp
i
I[q + ε;t
, t
]
=
1
N()
q(t
)=q
q(t
)=q
D [q]exp
i
I[q + ε;t
, t
]
,
1
N()
φ
,σ
φ
,σ
D [φ]exp
i
I[φ;σ
, σ
]
=
1
N()
φ
,σ
φ
,σ
D [φ + ε]exp
i
I[φ + ε;σ
, σ
]
=
1
N()
φ
,σ
φ
,σ
D [φ]exp
i
I[φ + ε;σ
, σ
]
.
Expanding the above equations in the powers of ε(t)andε(x), and picking up the
lowest order terms, we obtain
1
N()
q(t
)=q
q(t
)=q
D [q]
i
δ
δq(t)
exp
i
I[q;t
, t
]
= 0,
1
N()
φ
,σ
φ
,σ
D [φ]
i
δ
δφ(x)
exp
i
I[φ;σ
, σ
]
= 0.
364 10 Calculus of Variations: Applications
We thus have, in accordance with Feynman’s action principle,
q
, t
δI[
ˆ
q;t
, t
]
δ
ˆ
q(t)
q
, t
=
1
N()
q(t
)=q
q(t
)=q
D [q]
δI
δq(t)
exp
i
I[q;t
, t
]
=
1
N()
q(t
)=q
q(t
)=q
D [q]
i
δ
δq(t)
exp
i
I[q;t
, t
]
= 0, (10.2.12M)
φ
, σ
δI[
ˆ
φ;σ
, σ
]
δ
ˆ
φ(x)
φ
, σ
=
1
N()
φ
,σ
φ
,σ
D [φ]
δI
δφ(x)
exp
i
I[φ;σ
, σ
]
=
1
N()
φ
,σ
φ
,σ
D [φ]
i
δ
δφ(x)
exp
i
I[φ;σ
, σ
]
= 0. (10.2.12F)
By the assumption that the eigenkets
q, t
(
|
φ, σ
) form the complete set, we obtain
the (Euler-) Lagrange equation of motion at the operator level from (10.2.12M) and
(10.2.12F) as
−
δI[
ˆ
q]
δ
ˆ
q(t)
=
d
dt
∂L(
ˆ
q(t),
·
ˆ
q(t))
∂
·
ˆ
q(t)
−
∂L(
ˆ
q(t),
·
ˆ
q(t))
∂
ˆ
q(t)
= 0, (10.2.13M)
−
δI[
ˆ
φ]
δ
ˆ
φ(x)
= ∂
µ
∂L(
ˆ
φ(x), ∂
µ
ˆ
φ(x))
∂(∂
µ
ˆ
φ(x))
−
∂L(
ˆ
φ(x), ∂
µ
ˆ
φ(x))
∂
ˆ
φ(x)
= 0. (10.2.13F)
As for the time-ordered product, we have
1
N()
q(t
)=q
q(t
)=q
D [q]q(t
1
) ···q(t
n
)exp
i
I
=
q
, t
T(
ˆ
q(t
1
) ···
ˆ
q(t
n
))
q
, t
,
(10.2.14M)
1
N()
φ
,σ
φ
,σ
D [φ]φ(x
1
) ···φ(x
n
)exp
i
I
=
φ
, σ
T(
ˆ
φ(x
1
) ···
ˆ
φ(x
n
))
φ
, σ
. (10.2.14F)
We can prove the above by mathematical induction starting from n = 2. The left-
hand sides of (10.2.14M) and (10.2.14F) are the matrix elements of the canonical
T
∗
-product,
T
∗
(∂
x
1
µ
ˆ
O
1
(x
1
) ···
ˆ
O
n
(x
n
)) ≡ ∂
x
1
µ
T
∗
(
ˆ
O
1
(x
1
) ···
ˆ
O
n
(x
n
)),
with
T
∗
(
ˆ
φ
r
1
(x
1
) ···
ˆ
φ
r
n
(x
n
)) ≡ T(
ˆ
φ
r
1
(x
1
) ···
ˆ
φ
r
n
(x
n
)).
10.2 Feynman’s Action Principle in Quantum Theory 365
Canonical Momentum and Equal-Time Canonical Commutators. We define the
momentum operator as the displacement operator,
q
, t
ˆ
p(t
)
q
, t
=
i
∂
∂q
q
, t
q
, t
, (10.2.15M)
φ
, σ
ˆπ (x
)
φ
, σ
=
i
δ
δφ
φ
, σ
φ
, σ
. (10.2.15F)
In order to express the right-hand sides of (10.2.15M) and (10.2.15F) in the form in
which we can use Feynman’s action principle, we consider the following variation
of the action functional;
(1) Inside , we consider the infinitesimal variations of q(t)andφ(x),
q(t) −→ q(t) +δq(t)andφ(x) −→ φ(x) +δφ(x), (10.2.16M,F)
where, as δq(t)andδφ(x ), we take
δq(t
) = 0, δq(t) = ξ (t), δq(t
) = ξ
; δφ(x
) = 0, δφ(x) = ξ (x ), δφ(x
) = ξ
.
(10.2.17M,F)
(2) Inside , the physical system evolves in time in accordance with the (Euler-)
Lagrange equation of motion.
As the response of the action functional to a particular variations, (1) and (2), we
have
δI[q;t
, t
] =
∂L(q(t
),
˙
q(t
))
∂
˙
q(t
)
ξ
, (10.2.18M)
δI[φ;σ
, σ
] =
σ
dσ
µ
∂L(φ(x
), ∂
µ
φ(x
))
∂(∂
µ
φ(x
))
ξ
. (10.2.18F)
Thus we obtain
δI[q;t
, t
]
δq(t
)
=
∂L(q(t
),
˙
q(t
))
∂
˙
q(t
)
,
δI[φ;σ
, σ
]
δφ(x
)
= n
µ
(x
)
∂L(φ(x
), ∂
µ
φ(x
))
∂(∂
µ
φ(x
))
,
(10.2.19M,F)
where n
µ
(x
) is the unit normal vector at point x
on the space-like surface σ
.
With this, we obtain
i
∂
∂q(t
)
q
, t
q
, t
=
i
lim
ξ
→0
q
+ ξ
, t
q
, t
−
q
, t
q
, t
ξ
=
i
lim
ξ
→0
1
N()
q(t
)=q
q(t
)=q
D [q]exp
i
I[q;t
, t
]
366 10 Calculus of Variations: Applications
×
1
ξ
exp
i
(I[q + ξ ;t
, t
] −I[q;t
, t
])
− 1
= lim
ξ
→0
1
N()
q(t
)=q
q(t
)=q
D [q]exp
i
I[q;t
, t
]
×
1
ξ
{I[q + ξ ;t
, t
] −I[q;t
, t
] +0(ξ
2)}
=
1
N()
q(t
)=q
q(t
)=q
D [q]exp
i
I[q;t
, t
]
δI[q;t
, t
]
δq(t
)
=
1
N()
q(t
)=q
q(t
)=q
D [q]exp
i
I[q;t
, t
]
∂L(q(t
),
˙
q(t
))
∂
˙
q(t
)
=
q
, t
∂L(
ˆ
q(t
),
·
ˆ
q(t
))
∂
·
ˆ
q(t
)
q
, t
. (10.2.20M)
Likewise, we obtain
i
δ
δφ
(x
)
φ
, σ
φ
, σ
=
1
N()
φ
,σ
φ
,σ
D [φ]exp
i
I[φ;σ
, σ
]
×
δI[φ;σ
, σ
]
δφ(x
)
=
1
N()
φ
,σ
φ
,σ
D [φ]exp
i
I[φ;σ
, σ
]
n
µ
(x
)
∂L(φ(x
), ∂
µ
φ(x
))
∂(∂
µ
φ(x
))
=
φ
, σ
n
µ
(x
)
∂L(
ˆ
φ(x
), ∂
µ
ˆ
φ(x
))
∂(∂
µ
ˆ
φ(x
))
φ
, σ
. (10.2.20F)
From (10.2.15M) and (10.2.15F), and (10.2.20M) and (10.2.20F), we obtain the
identities
q
, t
ˆ
p(t
)
q
, t
=
q
, t
∂L(
ˆ
q(t
),
·
ˆ
q(t
))
∂
·
ˆ
q(t
)
q
, t
,
φ
, σ
ˆπ (x
)
φ
, σ
=
φ
, σ
n
µ
(x
)
∂L(
ˆ
φ(x
), ∂
µ
ˆ
φ(x
))
∂(∂
µ
ˆ
φ(x
))
φ
, σ
,
or
ˆ
p(t) =
∂L(
ˆ
q(t),
·
ˆ
q(t))
∂
·
ˆ
q(t)
and ˆπ (x) = n
µ
(x)
∂L(
ˆ
φ(x), ∂
µ
ˆ
φ(x))
∂(∂
µ
ˆ
φ(x))
. (10.2.21M, F)
Equation (10.2.21M) is the definition of the canonical momentum
ˆ
p(t). With a
choice of the unit normal vector as n
µ
(x) = (1, 0, 0, 0), Eq. (10.2.21F) is also the
10.2 Feynman’s Action Principle in Quantum Theory 367
definition of the canonical momentum ˆπ(x). A noteworthy point is that ˆπ (x)isa
normal dependent quantity.
Lastly, for quantum mechanics of the Bose particle system, from
q
, t
ˆ
q
B
(t
)
q
, t
= q
B
q
, t
q
, t
(10.2.22M)
and from (10.2.15M), we have
q
, t
ˆ
p
B
(t
)
ˆ
q
B
(t
)
q
, t
=
i
∂
∂q
B
q
B
q
, t
q
, t
=
i
q
, t
q
, t
+q
B
q
, t
ˆ
p
B
(t
)
q
, t
=
i
q
, t
q
, t
+
q
, t
ˆ
q
B
(t
)
ˆ
p
B
(t
)
q
, t
, (10.2.23M)
i.e., we have
[
ˆ
p
B
(t),
ˆ
q
B
(t)] =
i
, (10.2.24M)
where the commutator, [A, B], is given by
[A, B] = AB − BA.
For quantum mechanics of the Fermi particle system, we have a minus sign in
front of the second terms of the second line and the third line of Eq. (10.2.23M)
which originates from the anticommuting Fermion number, so that we obtain
{
ˆ
p
F
(t),
ˆ
q
F
(t)}=
i
, (10.2.25M)
where the anticommutator, {A, B},isgivenby
{A, B}=AB + BA.
Equations (10.2.24M) and (10.2.25M) are the natural consequences of the choice of
the momentum operator as the displacement operator, (10.2.15M).
For quantum field theory of the Bose field, from
φ
, σ
ˆ
φ
B
(x
)
φ
, σ
= φ
B
(x
)
φ
, σ
φ
, σ
(10.2.22F)
and from (10.2.15F), we have
φ
, σ
ˆπ
B
(x
1
)
ˆ
φ
B
(x
2
)
φ
, σ
=
i
δ
δφ
B
(x
1
)
φ
B
(x
2
)
φ
, σ
φ
, σ
=
i
δ
σ
(x
1
− x
2
)
φ
, σ
φ
, σ
+ φ
B
(x
2
)
φ
, σ
ˆπ
B
(x
1
)
φ
, σ
=
i
δ
σ
(x
1
− x
2
)
φ
, σ
φ
, σ
+
φ
, σ
ˆ
φ
B
(x
2
)ˆπ
B
(x
1
)
φ
, σ
. (10.2.23F)