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" V (x) = 0 x < 0
" V (x) = V x > 0
" x < 0

2 "2¨L
- = E¨L
2m "x2
" x > 0

2 "2¨R
- + V ¨R = E¨R
2m "x2
E V
E > V
E > V
" x < 0
kL - kR x
L L
¨L = eik x + e-ik
kL + kR
" x > 0
2kL x
R
¨R = eik
kL + kR
2m(E) 2m(E - V )
2 2
kL = kR =
2 2
E < V
E < V
" x < 0

"2¨L 2
+ kL¨L = 0
"x2
" x > 0
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"2¨R 2
- kR¨R = 0
"x2
kL - ikR x
L L
¨L = eik x + e-ik
kL + ikR
2kL x
R
¨R = e-k
kL + ikR
kL kR
x > 0
2
2kL x 2kL x 4kL
R R R
à = ¨(x)¨ (x) = e-k e-k = e-2k x
2 2
kL + ikR kL - ikR kL + kR
x
b
b
Ã(b)
T =
Ã(0)
2
4kL Rb
e-2k
2 2
Ã(b) kL+kR
R
T = = = e-2k b
2
4kL R0
Ã(0)
e-2k
2 2
kL+kR
kR

"
2m 8m
-2 V -E b - (V -E) b
2 2
T = e = e
b V - E0
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e2
V (r) = -
4Ä„ 0r

2 "2¨ "2¨ "2¨ e2
- + + - k ¨ = E¨
2m "x2 "y2 "z2 r

r = x2 + y2 + z2
x = r sin ¸ cos Ć
y = r sin ¸ sin Ć
z = r cos ¸
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2 " "¨ 2 1 " "¨ 1 "2¨ e2
- r2 - sin ¸ + - k ¨ = E¨
2mr2 "r "r 2mr2 sin ¸ "¸ "¸ "Ć2 r
sin2 ¸
¨ = ¨(r,¸, Ć)
¨(r,¸, Ć) = R(r) f(¸) g(Ć)
¨(r,¸, Ć) = Rn,l(r) fl,m(¸) eimĆ
l
2r 2r
0
Rn,l(r) = Nnl e-Zr/na L2l+1
n+1
na0 na0
3/2 1/2
2r (n - l - 1)!
Nnl = -
na0 2n[(n + 1)!]3
8Ä„µ0 2
a0 = -
me2
L2l+1
n+1
n

(n + k)!
Lk (x) = (-1)m xm
n
(n - m)!(k + m)!m!
m=0
3/2 2
1 4 r
0
R32(r) = " e-r/3a
a0 81 30 a0
" n = 1, 2, 3 . . .
" l = 0, 1, 2 . . . n - 1
" m = -l,(-l + 1), . . . 0, 1, 2, . . . l
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Ä…1/2
s = Ä…1
2
1
En = const
n2
m = 0 m = 1 m = 2
l = 0
l = 1
l = 2
n = 3
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