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1 2a
mω 2
Shifted Oscillator x’ ω ω h
¯ω
2 mω
2
l(l+1)¯
h
1
mω 2 r2 + 2mr2
Isotropic Oscillator l l+1 2¯ ω
h
2
’ l + 3 ¯ω
(3 dim.) h
2
l(l+1)¯ 2
2
me4 me4
h
’ e + 2mr2 + 2(l+1)2 h2 1
Coulomb l l+1 2¯ 2 (c0 +1)2
r ¯ h
1
’ (c 2
1 +1)
2
B √h
±¯
A2 + c2 ’ c2
Rosen-Morse I + 2B tanh ±x A A’ 0 1
A2 2m
√h
±¯ 1 1
sech2 ±x +B 2
’A A + ’
c2 c2
2m 0 1

√h
A±¯ √h
±¯
A2 + B 2 + A2 + c2 ’ c2
Rosen-Morse II — A A’ 0 1
2m 2m
cosech2 ±r
√h
±¯
’B 2A + —
2m
coth±r cosech±r
Table 5.1 (continued)
© 2001 by Chapman & Hall/CRC




Potential W (x) Energy Levels ψ0 (x)

√ mω
Shifted Oscillator mωx ’ 2a n¯ ω
h exp ’ 2¯
h
2
2a
x’ mω

√ 2
(l+1)¯
h
rl+1 exp ’ mωr
Isotropic Oscillator mωr ’ 2n¯ ω
h


h
mr 2
(3 dim.)
√ 2
me4 2
(l+1)¯
h
e 1 1 me r
rl+1 exp ’ (l+1)¯
Coulomb m (l+1)¯ ’ ’

2¯ 2 (l+1)2 (n+l+1)2
h h
h
2
mr

√ 2mA
B
Rosen-Morse I 2(A tanh ±x + ) (sech±x) —
±¯
h
A

1 1 2mBx
+B 2 ’ exp ’
A2 2 A¯
h
A’ √h±

2m
2m
√ 2
(sin h±r) h± +(B’A)
¯
√h±

A2 ’ A ’
Rosen-Morse II 2(Acoth±r ’ Bcosech±r) √
2m 2m n
(1+cos h±r) h±
¯
A<B
Table 5.1 (continued)
© 2001 by Chapman & Hall/CRC




Potential V+ (x) Shape-invariant Parameters
c0 c1 R(c1 )
B2 √h
±¯
A2 + c2 ’ c2
Eckart-I ’ 2Bcoth±r A A+ 0 1
A2 2m
√h
±¯ 1 1
cosech2 ±r +B 2
+A A ’ ’
c2 c2
2m 0 1

ñ
h
¯ √h
±¯
’A2 + B 2 + A A ’ cosec2 ±x c2 ’ c2
Eckart -II A A+ 1 0
2m 2m

ñ
h
¯
’B 2A ’ cosec±x cot±x (0 ¤ ±x ¤ π, A > B)
2m
2
ñ
h
¯ √h ,
±¯ √h
2±¯
’(A + B)2 + A A ’ sec2 ±x
Poschl-Teller-I (A, B) A+ A+B+
2m 2m 2m

ñ
h
¯ √h
±¯
cosec2 ±x ’(A + B)2
+B B ’ B+
2m 2m

ñ
h
¯ √h ,
±¯
(A ’ B)2 ’ A A + sech2 ±r (A ’ B)2 ’ (A
Poschl-Teller-II (A, B) A’
2m 2m
2
√h
±¯ √h
±¯ 2
cosech2 ±r
+B B ’ B+ ’B ’ ±¯
h
m
2m 2m

¯
A2 + B 2 e’2±x ’ 2Be’±x — c2 ’ c2
Morse - I A A’ √
0 1
2m
√h
±¯
A+
2 2m

√h
±¯ √h
±¯
A2 + B 2 ’ A A + sech2 ±x c2 ’ c2
Hyperbolic A A’ 0 1
2m 2m

√h
±¯
+B 2A + sech±x tanh ±x
2m
B2 ñ
h
¯ √±
h
¯
’A2 + cosec2 ±x c2 ’ c2
Trigonometric +A A+ A A’ 1 0
A2 2m 2m
1 1
+B 2
’2B cot ±x ’
c2 c2
0 1
Table 5.1 (continued)

Potential W (x) Energy Levels ψ0 (x)
© 2001 by Chapman & Hall/CRC





√ 2 2mA
B √h±

A2
Eckart-I ’ 2 Acoth±r ’ ’ A+ (sinh ±r) ±¯
h
A 2m

1 1 2mBr
(B > A2 ) +B 2 ’ exp ’
A2 2 A¯
h
A+ √ h
n±¯
2m

2m (A+B)
√ 2
(sin ±x) h±
¯
√h±

’ A2
Eckart -II 2(’A cot ±x + Bcosecx) A+ √
2m 2m B
(1’cos ±x) h±
¯
(0 ¤ ±x ¤ π, A > B)

√ 2 2m
B
√h
2±¯
Poschl-Teller-I 2(A tan ±x ’ B cot ±x) A+B+ (sin ±x) —
±¯
h
2m

2m

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