Dong S.-H. Wave Equations in Higher Dimensions
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294 Index
Similarity transformation, 27, 29, 30, 32, 90,
247
unitary, 31
Simple roots, 19, 21, 22, 53
Simple weight, 53
Single-particle density, 78
Single-valued representation, 21, 23, 32
Singular potentials, 6
SJWKB
quantization condition, 147
quantization rule, 148
Skew-symmetric matrix, 258
SO(1, 3), 232
SO(1, 3 +D), 232
SO(2) group, 204
SO(2l), 19, 21–23, 26, 34, 37, 48
SO(2l +1), 19–22, 26, 33, 34, 36, 37, 48
SO(2N), 52, 55, 232
SO(2N +1),52
SO(3) group, 20, 35, 36
SO(4) group, 22, 204
SO(D), 44, 48, 50, 52, 61, 203, 231, 232
SO(D), 233
SO(d +1), 5
SO(D +1), 230
SO(D +1), 233
Solvable
models, 147
potentials, 129–134, 146, 147
systems, 130
SO(N ), vii, 5, 8
SO(N), 16–19, 24–26, 30–36, 38, 237
Spatial dimensions, 3, 4, 229, 234
Spherical coordinates, 138, 139, 261
Spherical harmonic basis
tensor, 22
vectors, 20, 22
Spherical harmonic basis tensor, 20
Spherical harmonics, 14, 48, 49
Spherical symmetry, 56, 149
Spherical well, 5
Spherically symmetric potential, 3, 52, 61, 82,
91, 109, 134, 142, 181, 203, 204
Spin algebra, 7
Spin-orbit coupling, 5
Spinor operators, 52, 53
Spinor representation, 27, 29, 32–38, 53, 55,
231, 232
Spinor space, 34
Squeezed states, 14
String theory, viii, 3, 7, 226
Structure constants, 16, 230, 257, 258
Sturm basis, 88, 96
Sturm-Liouville theorem, 57, 109, 111–113,
117, 171, 190, 203, 204, 206, 218
Sturmian bases, 81
Sturmian functions, 89
SU(1, 1
), 72, 77, 88
su(1, 1) algebra, 5
su(1, 1) algebra, 72, 77, 89
SU(2), 204, 228, 263
Subgroup, 15, 239–242, 244, 245
improper, 240
invariant, 26, 31, 244, 245
proper, 240–242
Subset, 15, 240, 241
infinite, 259
SU(N +1), 233
Supersymmetric partners, 5
Supersymmetry, viii, 5, 133
SWKB
approximation, 147
Symmetry group, 13, 52, 228, 232, 241, 255
SO(N ), 7, 8
System Hamiltonian, 39
T
Tangent function, 111, 208
Taylor series, 40, 260
Tensor representation, 16, 21, 23, 25, 26, 32,
35, 36, 48
Tensor space, 16, 17
Tensor Young tableau, 19–23, 26
Three-dimensional case, 43
Transformation equations, 46, 261
Transformation operators, 42
Transposition, 241
Tricomi equation, 159, 169
Turning points, 133, 134, 137, 139, 142
Two-body system, 39, 49
Two-to-one correspondence, 31, 32
U
U(1), 228, 233
Uncertainty relation, 5, 6, 78
Unit matrix, 53, 251
Unit operator, 259
Unit vector, 41, 46
Unitary matrices
irreducible, 30
Unitary representation
irreducible, 6, 27, 77, 252
V
Vector space, 15, 247, 258
Index 295
W
Wave equations, vii, 3, 5, 7–9, 97, 149, 237
higher dimensional, 4, 8
Wavefunction, 6, 49, 50, 52–55, 57, 61, 72, 75,
76, 79, 82, 84, 85, 97–101, 106,
108–110, 117, 119, 120, 122, 128,
130, 133, 134, 139, 144–147, 149,
158, 159, 168, 169, 182, 189, 204,
207, 211, 217, 218, 220
radial, 67, 120
Weak potential, 62, 159, 162, 177, 182, 184,
196
Weight vector, 52
Wigner n-j symbols, 14
WKB, 130
analysis, 132
approximation, 129, 131, 133
frame, 146
improvement, 6
integral, 129
method, 129, 130
quantization rule, 132
theory, 132
Y
Young
diagram, 37
operator, 17, 18, 36
pattern, 17, 18, 21, 23, 24, 35–38, 45, 48,
49
Z
Zero component, 51
Zero modes, 231–233
Zero momentum, 7, 109, 111, 116, 203, 217
Zero-point energy, 106