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14 Classical Mechanics and the Old Quantum Theory
These derivatives are ordinary derivatives since η and ϕ are functions of one variable.
We divide Eq. (14.3-6) by ϕη to complete the separation of variables:
1
ϕ(x)
d 2 ϕ
dx 2
1
c 2 η(t)
d 2 η
dt 2
(14.3-7)
Each term now depends on only one independent variable.
Since x and t are independent variables, it is possible to keep t fixed while we allow x
to range. The right-hand side of Eq. (14.3-7) is then equal to a constant (the separation
constant), so the term containing x must be a constant function of x. Similarly, the
term containing t must be a constant function of t and must be equal to the separation
constant:
1
ϕ(x)
d 2 ϕ
dx 2 constant −κ
2
(14.3-8)
1
c 2 η(t)
d 2 η
dt 2 −κ
2
(14.3-9)
The separation constant must be negative to give an oscillatory solution. We denote
this constant by −κ 2 so that κ will be a real quantity.
Exercise 14.4
If the separation constant is assumed to be positive, we have the equation
1
ϕ(x)
d 2 ϕ
dx 2 constant a 2
where a is a real constant.
a. Show that the following function is a solution to this equation:
ϕ Le ax + Me −ax
where L and M are constants. This is not a periodic function.
b. Show that
η Pe act + Qe −act
which is not an oscillating function.
Multiplying Eq. (14.3-8) by ϕ and Eq. (14.3-9) by c 2 η gives
d 2 ϕ
dx 2 −κ
2 ϕ(x)
(14.3-10)
d 2 η
dt 2 −κ
2 c
2 η(t)
(14.3-11)
These equations have the same form as Eq. (14.2-14) with different symbols. We
transcribe the solution to that equation with appropriate replacements of symbols:
ϕ(x) Bcos(κx) + Dsin(κx)
(14.3-12)
η(t) F cos(κct) + Gsin(κct)
(14.3-13)
14 Classical Mechanics and the Old Quantum Theory
These derivatives are ordinary derivatives since η and ϕ are functions of one variable.
We divide Eq. (14.3-6) by ϕη to complete the separation of variables:
1
ϕ(x)
d 2 ϕ
dx 2
1
c 2 η(t)
d 2 η
dt 2
(14.3-7)
Each term now depends on only one independent variable.
Since x and t are independent variables, it is possible to keep t fixed while we allow x
to range. The right-hand side of Eq. (14.3-7) is then equal to a constant (the separation
constant), so the term containing x must be a constant function of x. Similarly, the
term containing t must be a constant function of t and must be equal to the separation
constant:
1
ϕ(x)
d 2 ϕ
dx 2 constant −κ
2
(14.3-8)
1
c 2 η(t)
d 2 η
dt 2 −κ
2
(14.3-9)
The separation constant must be negative to give an oscillatory solution. We denote
this constant by −κ 2 so that κ will be a real quantity.
Exercise 14.4
If the separation constant is assumed to be positive, we have the equation
1
ϕ(x)
d 2 ϕ
dx 2 constant a 2
where a is a real constant.
a. Show that the following function is a solution to this equation:
ϕ Le ax + Me −ax
where L and M are constants. This is not a periodic function.
b. Show that
η Pe act + Qe −act
which is not an oscillating function.
Multiplying Eq. (14.3-8) by ϕ and Eq. (14.3-9) by c 2 η gives
d 2 ϕ
dx 2 −κ
2 ϕ(x)
(14.3-10)
d 2 η
dt 2 −κ
2 c
2 η(t)
(14.3-11)
These equations have the same form as Eq. (14.2-14) with different symbols. We
transcribe the solution to that equation with appropriate replacements of symbols:
ϕ(x) Bcos(κx) + Dsin(κx)
(14.3-12)
η(t) F cos(κct) + Gsin(κct)
(14.3-13)