Quantum mechanics of the 1∕x2 potential

Essin, Andrew M.; Griffiths, David J.

doi:10.1119/1.2165248

Cited by 154 publications

(172 citation statements)

References 19 publications

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“…For future reference notice also that with the convention 0rθφ = + det e a µ = +1/(r 2 sin θ) the above imply 22) and so…”

Section: Jhep09(2017)007mentioning

confidence: 99%

“…This has as solutions the usual spherical Bessel functions 22) and B = 0 if we demand ψ be bounded at r = 0. Specializing to j = 1 2 the appropriate solutions are f + = A + j 0 (kr), f − = B − j 1 (kr), g + = B + j 1 (kr) and g − = A − j 0 (kr).…”

Section: B21 Charged-shell Modelmentioning

confidence: 99%

“…otherwise hit for the inverse-square potential, see for example, [22]) of smooth solutions at the origin.…”

Section: Jhep09(2017)007mentioning

confidence: 99%

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Point-particle effective field theory III: relativistic fermions and the Dirac equation

Burgess¹,

Hayman²,

Rummel³

et al. 2017

J. High Energ. Phys.

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We formulate point-particle effective field theory (PPEFT) for relativistic spinhalf fermions interacting with a massive, charged finite-sized source using a first-quantized effective field theory for the heavy compact object and a second-quantized language for the lighter fermion with which it interacts. This description shows how to determine the near-source boundary condition for the Dirac field in terms of the relevant physical properties of the source, and reduces to the standard choices in the limit of a point source. Using a first-quantized effective description is appropriate when the compact object is sufficiently heavy, and is simpler than (though equivalent to) the effective theory that treats the compact source in a second-quantized way. As an application we use the PPEFT to parameterize the leading energy shift for the bound energy levels due to finite-sized source effects in a model-independent way, allowing these effects to be fit in precision measurements. Besides capturing finite-source-size effects, the PPEFT treatment also efficiently captures how other short-distance source interactions can shift bound-state energy levels, such as due to vacuum polarization (through the Uehling potential) or strong interactions for Coulomb bound states of hadrons, or any hypothetical new short-range forces sourced by nuclei.

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“…For future reference notice also that with the convention 0rθφ = + det e a µ = +1/(r 2 sin θ) the above imply 22) and so…”

Section: Jhep09(2017)007mentioning

confidence: 99%

Section: B21 Charged-shell Modelmentioning

confidence: 99%

See 1 more Smart Citation

Point-particle effective field theory III: relativistic fermions and the Dirac equation

Burgess¹,

Hayman²,

Rummel³

et al. 2017

J. High Energ. Phys.

View full text Add to dashboard Cite

show abstract

“…8 As many have observed [28][29][30][31][32][33][34][35][36][37][38][39][40] because of this the Schrödinger Hamiltonian can fail to be self-adjoint, depending the boundary conditions that hold at r = . Selecting a choice of boundary condition to secure its self-adjointness -not a unique construction -is known as constructing its self-adjoint extension [41][42][43][44][45][46][47][48].…”

Section: Jhep07(2017)072mentioning

confidence: 99%

Point-particle effective field theory II: relativistic effects and Coulomb/inverse-square competition

Burgess

Hayman

Rummel

et al. 2017

J. High Energ. Phys.

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We apply point-particle effective field theory (PPEFT) to compute the leading shifts due to finite-sized source effects in the Coulomb bound energy levels of a relativistic spinless charged particle. This is the analogue for spinless electrons of calculating the contribution of the charge-radius of the source to these levels, and our calculation disagrees with standard calculations in several ways. Most notably we find there are two effective interactions with the same dimension that contribute to leading order in the nuclear size, one of which captures the standard charge-radius contribution. The other effective operator is a contact interaction whose leading contribution to δE arises linearly (rather than quadratically) in the small length scale, , characterizing the finite-size effects, and is suppressed by (Zα) 5 . We argue that standard calculations miss the contributions of this second operator because they err in their choice of boundary conditions at the source for the wave-function of the orbiting particle. PPEFT predicts how this boundary condition depends on the source's charge radius, as well as on the orbiting particle's mass. Its contribution turns out to be crucial if the charge radius satisfies (Zα) 2 a B , where a B is the Bohr radius, because then relativistic effects become important for the boundary condition. We show how the problem is equivalent to solving the Schrödinger equation with competing Coulomb, inverse-square and delta-function potentials, which we solve explicitly. A similar enhancement is not predicted for the hyperfine structure, due to its spin-dependence. We show how the charge-radius effectively runs due to classical renormalization effects, and why the resulting RG flow is central to predicting the size of the energy shifts (and is responsible for its being linear in the source size). We discuss how this flow is relevant to systems having much larger-than-geometric cross sections, such as those with large scattering lengths and perhaps also catalysis of reactions through scattering with monopoles. Experimental observation of these effects would require more precise measurement of energy levels for mesonic atoms than are now possible.

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“…To handle these problems the concept of self-adjoint extension is reviewed in references [25,26,28]. Also see references [31,32].…”

Section: B Physical Viewmentioning

confidence: 99%

Factorizing the time evolution operator

Quijas

Aguilar

2007

Phys. Scr.

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There is a widespread belief in the quantum physical community, and in textbooks used to teach Quantum Mechanics, that it is a difficult task to apply the time evolution operator e itĤ/ on an initial wave function. That is to say, because the hamiltonian operator generally is the sum of two operators, then it is a difficult task to apply the time evolution operator on an initial wave function ψ(x, 0), for it implies to apply terms like (â +b) n . A possible solution of this problem is to factorize the time evolution operator and then apply successively the individual exponential operator on the initial wave function. However, the exponential operator does not directly factorize, i. e. eâ +b = eâeb. In this work we present a useful procedure for factorizing the time evolution operator when the argument of the exponential is a sum of two operators, which obey specific commutation relations. Then, we apply the exponential operator as an evolution operator for the case of elementary unidimensional potentials, like the particle subject to a constant force and the harmonic oscillator. Also, we argue about an apparent paradox concerning the time evolution operator and non-spreading wave packets addressed previously in the literature.

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Quantum mechanics of the 1∕x2 potential

Cited by 154 publications

References 19 publications

Point-particle effective field theory III: relativistic fermions and the Dirac equation

Point-particle effective field theory III: relativistic fermions and the Dirac equation

Point-particle effective field theory II: relativistic effects and Coulomb/inverse-square competition

Factorizing the time evolution operator

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