# in one and two dimensions

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## 8 matching pages

##### 1: Bibliography P

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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.
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##### 2: 1.6 Vectors and Vector-Valued Functions

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►In either case
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►The geometrical image $C$ of a path $\mathbf{c}$ is called a

*simple closed curve*if $\mathbf{c}$ is one-to-one, with the exception $\mathbf{c}(a)=\mathbf{c}(b)$. … ►###### 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

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###### Example 1: In one and two dimensions any $q(x)$ with a ‘Dip, or Well’ has a partly discrete spectrum

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The Bound State of Weakly Coupled Schrödinger Operators in One and Two Dimensions.
Annals of Physics 97 (2), pp. 279–288.
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##### 5: Bibliography C

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Bound states in one and two spatial dimensions.
J. Math. Phys. 44 (2), pp. 406–422.
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##### 6: 22.19 Physical Applications

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►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
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►Classical motion in one dimension is described by Newton’s equation
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►Two types of oscillatory motion are possible.
For an initial displacement with $$, bounded oscillations take place near one of the two points of stable equilibrium $x=\pm \sqrt{1/\beta}$.
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►Whittaker (1964, Chapter IV) enumerates the complete class of one-body classical mechanical problems that are solvable this way.
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##### 7: Bibliography B

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Quantum mechanics of one- and two-electron atoms.
Springer-Verlag, Berlin.
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Quantum Mechanics of One- and Two-electron Atoms.
Rosetta edition, Plenum Publishing Corp., New York.
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One- and two-electron photoejection from ${\mathrm{H}}^{-}$: A multichannel $J$-matrix calculation.
Phys. Rev. A 14, pp. 2159–2173.
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Criterion for Existence of a Bound State In One Dimension.
American Journal of Physics 68 (2), pp. 160–161.
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Two cold atoms in a harmonic trap.
Found. Phys. 28 (4), pp. 549–559.
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##### 8: 1.2 Elementary Algebra

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►The last two equations require ${a}_{j}>0$ for all $j$.
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►For matrices $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$ of the same dimensions,
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*distributive*if $\mathbf{B}$ and $\mathbf{C}$ have the same dimensions … ►Two vectors $\mathbf{u}$ and $\mathbf{v}$ are*orthogonal*if … ►and for the corresponding eigenvectors one has to solve the linear system …