where je (x) is the spherical Bessel function of order , and ne(x) is the spherical Neumann function of order . They are defined as follows: je(x) = (-x)² ne(x) = -(-x)² For example, '()' dx 1 d sin x ; X jo (x) = sin x X 1 d j₁(x) = (-x) = x dx ; no (x) sin x x j2(x) = (-x)² (1 d.) ² sin. dx 3 sin x - 3x cos x - x² sin x == = x² l d (¹) dx COS X X sin x x² ² (14+) x dx COS X X COS X x d x cos x sin x (4.46) ||
where je (x) is the spherical Bessel function of order , and ne(x) is the spherical Neumann function of order . They are defined as follows: je(x) = (-x)² ne(x) = -(-x)² For example, '()' dx 1 d sin x ; X jo (x) = sin x X 1 d j₁(x) = (-x) = x dx ; no (x) sin x x j2(x) = (-x)² (1 d.) ² sin. dx 3 sin x - 3x cos x - x² sin x == = x² l d (¹) dx COS X X sin x x² ² (14+) x dx COS X X COS X x d x cos x sin x (4.46) ||
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Introduction to Quantum Mechanics
How do you solve N1 to N2?
For study purposes. Thank you!
Transcribed Image Text:where je (x) is the spherical Bessel function of order , and ne(x) is the spherical
Neumann function of order. They are defined as follows:
ne(x) = -(-x)²
1 d
›(²4) ²
X
dx
je(x) = (-x)²
For example,
sin x
X
jo (x) =
;
sin x
j₁(x) = (-x) =
X
1 d
x dx
;
no (x)
sin x
x
j2(x) = (-x)² (1 d.) ² sin.
dx
3 sin x - 3x cos x - x² sinx
==
= x²
COS Xx
X
d
(¹)
dx
sin x
x²
² (14+)
x dx
COS X
X
COS X
x
d x cos x sin x
(4.46)
||
Transcribed Image Text:Table 4.4 The first few spherical Bessel and Neumann functions, je(x) and ne(x);
asymptotic forms for small x.
jo =
j₁ =
sin x
X
j2 =
sin x
1²
=(-:)sinx−3
2⁰l!
(2l + 1)! x²,
cos x
→
COS X
no ==
n₁ =
cos x
X
ne → -
COS X
+2
sin x
1₂ = -
ne-lo-assins
- ( ²³/² - -/- ) ₁
cos X-
(20)!
1
2²l! xl+I'
for x << 1.
sin x
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