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in one and two dimensions


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1: Bibliography P
  • G. Pólya (1949) Remarks on computing the probability integral in one and two dimensions. In Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, 1945, 1946, pp. 63–78.
  • 2: 1.6 Vectors and Vector-Valued Functions
    In either case … The geometrical image C of a path 𝐜 is called a simple closed curve if 𝐜 is one-to-one, with the exception 𝐜 ( a ) = 𝐜 ( b ) . …
    Green’s Theorem
    otherwise, one is the negative of the other. …
    Green’s Theorem (for Volume)
    3: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
    Example 1: In one and two dimensions any q ( x ) with a ‘Dip, or Well’ has a partly discrete spectrum
    4: Bibliography S
  • B. Simon (1976) The Bound State of Weakly Coupled Schrödinger Operators in One and Two Dimensions. Annals of Physics 97 (2), pp. 279–288.
  • 5: Bibliography C
  • K. Chadan, N. N. Khuri, A. Martin, and T. T. Wu (2003) Bound states in one and two spatial dimensions. J. Math. Phys. 44 (2), pp. 406–422.
  • 6: 22.19 Physical Applications
    With appropriate scalings, Newton’s equation of motion for a pendulum with a mass in a gravitational field constrained to move in a vertical plane at a fixed distance from a fulcrum is … Classical motion in one dimension is described by Newton’s equation … Two types of oscillatory motion are possible. For an initial displacement with 1 / β | a | < 2 / β , bounded oscillations take place near one of the two points of stable equilibrium x = ± 1 / β . … Whittaker (1964, Chapter IV) enumerates the complete class of one-body classical mechanical problems that are solvable this way. …
    7: Bibliography B
  • H. A. Bethe and E. E. Salpeter (1957) Quantum mechanics of one- and two-electron atoms. Springer-Verlag, Berlin.
  • H. A. Bethe and E. E. Salpeter (1977) Quantum Mechanics of One- and Two-electron Atoms. Rosetta edition, Plenum Publishing Corp., New York.
  • J. T. Broad and W. P. Reinhardt (1976) One- and two-electron photoejection from H : A multichannel J -matrix calculation. Phys. Rev. A 14, pp. 2159–2173.
  • K. R. Brownstein (2000) Criterion for Existence of a Bound State In One Dimension. American Journal of Physics 68 (2), pp. 160–161.
  • T. Busch, B. Englert, K. Rzażewski, and M. Wilkens (1998) Two cold atoms in a harmonic trap. Found. Phys. 28 (4), pp. 549–559.
  • 8: 1.2 Elementary Algebra
    The last two equations require a j > 0 for all j . … For matrices 𝐀 , 𝐁 and 𝐂 of the same dimensions, … distributive if 𝐁 and 𝐂 have the same dimensionsTwo vectors 𝐮 and 𝐯 are orthogonal if … and for the corresponding eigenvectors one has to solve the linear system …