10.2 The Euler Equations of Functionals …
631
where, a
2
=
kp
ρ
, a is the local sound velocity of the acoustic field. Equation (3) is
called the vector wave equation or the gas particle equation of motion for the
sound transmission.
Let σ = −∇ · η, σ represents the relative change of the gas density, substituting
it into Eq. (3), and taking the negative value of the divergence, we obtain the scalar
equation
σ −
1
a 2 σ tt = 0
( 4 )
where, = ∇·∇ is the Laplace operator. It follows that Eq. (4) is a three-dimensional
wave equation.
Example 10.2.4 In quantum mechanics, the functional of the time-dependent
particle state wave function is
J [ψ, ψ
∗
] =
t 1
t 0
V
−
2
2m
∇ψ
∗
· ∇ψ +
i
2
(ψ
∗
ψ t − ψ
∗
t ψ) − V ψ
∗
ψ
dV dt
where, = h/2π, with h = 6.626×10
−34 J·s; and h are both Planck(’s) constant,
sometimes is also called the Dirac constant; m is the mass of the free particle;
ψ is the wave function of the particle; ψ
∗ is the complex conjugate of ψ; V is the
potential energy of the particle. Find its Euler equations.
Solution The integrand is F =
−
2
2m
∇ψ
∗
· ∇ψ +
i
2
(ψ
∗
ψ t − ψ
∗
t ψ) − V ψ
∗
ψ, where
ψ
∗ is the complex conjugate wave function of ψ, the various partial derivatives are
∂ F
∂ψ ∗ =
i
2
ψ t − V ψ, ∇ ·
∂ F
∂∇ψ ∗ = −
2
2m
ψ,
∂
∂t
∂ F
∂ψ
∗
t
= −
i
2
ψ t
∂ F
∂ψ
= −
i
2
ψ
∗
t − V ψ
∗
, ∇ ·
∂ F
∂∇ψ
= −
2
2m
ψ
∗
,
∂
∂t
∂ F
∂ψ t
= −
i
2
ψ
∗
t
The Euler equations of the functional are
⎧
⎪ ⎨
⎪ ⎩
i
2
ψ t − V ψ +
2
2m
ψ +
i
2
ψ t = 0
−
i
2
ψ
∗
t − V ψ
∗
+
2
2m
ψ
∗
−
i
2
ψ
∗
t = 0
(1)
or
⎧
⎪ ⎨
⎪ ⎩
−
2
2m
ψ + V ψ = iψ t
−
2
2m
ψ
∗
+ V ψ
∗
= −iψ
∗
t
(2)