`mmf.physics.potentials.two_body`¶

`DoubleExponential`([v0, r0, mu0, f_v, f_r, ...])	Double Exponential potential:
`SoonYongPotential2`([v0, A0, V1, A1, hbar, ...])	Extended Potential used by Soon Yong.
`Potential`([abs_tol, rel_tol, Rmin])	Instances are callable and return compact potentials V(r).
`PoschlTeller`([v0, r0, hbar, M, abs_tol, ...])	Extended Poschl-Teller (Rosen-Morse) potential used by Soon
`BargmannPotential`([a, re, hbar, M])	Bargmann Potential where the effective range expansion is exact
`PotentialV0R0`([abs_tol, rel_tol, Rmin])	Mixing for two-parameter potentials with a parameter $v_0$ governing the overall strength and a parameter $r_0$ governing the effective range.
`DoubleGaussian`([v0, r0, mu0, f_v, f_r, ...])	Double Gaussian potential:
`SoonYongPotential`	Extended Poschl-Teller (Rosen-Morse) potential used by Soon

Inheritance diagram for mmf.physics.potentials.two_body:

Inheritance diagram of mmf.physics.potentials.two_body

Code for extracting properties of two-body systems.

Phase Shifts¶

We consider here short-range potentials such that for $V(r>R) = 0$ . In this case, the potential may be described in terms of the phase shift of the scattering states.

class mmf.physics.potentials.two_body.DoubleExponential(v0=1.1909151179666313, r0=0.6666666666666666, mu0=None, f_v=-2.0, f_r=2.0, hbar=1.0, M=0.5, abs_tol=1e-12, rel_tol=1e-12, Rmin=0.1)[source]¶

Bases: mmf.physics.potentials.two_body.PotentialV0R0

Double Exponential potential:

$V(r) = -v_0 \frac{\hbar^2}{M} \mu_0^2 \left( \exp[-\mu_0 r] + f_v \exp[-\mu_0 r f_r] \right)$

Note that $\mu_0 = 4/r_0$ here.

>>> v = DoubleExponential(mu0=14.0)
>>> v.get_ar()              # One of Stefano's values to check.
(5852295157..., 0.3099...)
>>> v.set_ainv_re(ainv=0.0, re=1.0)
>>> v.v0, v.r0
(1.190915117966..., 0.9218838916...)

Methods

`__call__`(r)
`compute_R`()	Return R >= self.Rmin such that `abs(self(R)) <
`dVdV`(r)	Return V’(r)/V(r). Should be overloaded for speed and/or
`gE`(E[, R, method])	Return k*cot(delta) as a function of energy E for the specified
`get_ar`([R, method])	Return (a,re) where a is the s-wave scattering length and re is the s-wave effective range.
`get_deq`(f, jac, U0[, method])	Return a “deq” object capable of integrating the system
`get_deq_E`(E[, method])	Return solution to energy dependent differential equation for (u,du) for finding the s-wave phase shift.
`get_deq_ar`([method])	Return the differential equation (u,du,uu,duu) for finding the s-wave scattering length and the s-wave effective range.
`get_deq_gE`(E[, method])	Return solution to energy dependent differential equation
`get_deq_gE_old`(E[, method, version])	Old version in terms of fp and fm.
`get_mu0`()
`plot`([Rmax])	Plot potential.
`plot_gE`(Erange[, Rmax])	Plot potential.
`plot_u`(E[, Rmax])	Plot radial wavefunction.
`set_ainv_re`(ainv, re)	Uses a non-linear root-finder to set the parameters v0 and r0 so that the scattering length is 1.0/ainv and the effective range is re.
`set_mu0`(mu0)

__init__(v0=1.1909151179666313, r0=0.6666666666666666, mu0=None, f_v=-2.0, f_r=2.0, hbar=1.0, M=0.5, abs_tol=1e-12, rel_tol=1e-12, Rmin=0.1)[source]¶

get_mu0()[source]¶

mu0¶

set_mu0(mu0)[source]¶

class mmf.physics.potentials.two_body.SoonYongPotential2(v0=-1.0, A0=6.2527, V1=0.0, A1=3.0, hbar=0.15915494309189535, M=0.5, kF=1.0, abs_tol=1e-12, rel_tol=1e-12, Rmin=0.1)[source]¶

Bases: mmf.physics.potentials.two_body.Potential

Extended Potential used by Soon Yong.

V(r) = hbar**2/M*(v0*mu0**2/(cosh(mu0*r))**2

V1*mu1**2/(cosh(mu1*r))**2)

His units are set by kF = (3*pi*pi*N)**(1./3.) / L for a box of length L mu = A*kF

As a test, at unitarity, v0 = 1.0 and re = 2/mu for two-body scattering where m here is the individual mass of the particles.

>>> V = SoonYongPotential2(v0=-1.0,V1=0.0)
>>> (a,re) = V.get_ar()
>>> mu = V.A0*V.kF
>>> abs(a) > 6e8
True
>>> abs(re - 2.0/mu) < 1e-9
True

# kF_SY = (3.*pi)**(2./3.) # EFG_SY = kF_SY**2/2.0*3.0/5.0 # EFG = EFG_SY/4.0/pi**2 # re = 2.0*L/(6.2527*(3.0*pi)**(2./3.)) # # V = -2*hbar^2*mu^2*v0/m/cosh(mu*r)**2 # v0 = 1.0 => ainv = 0, re = 2.0/mu # v0 = 1.30119 => kfa_SY = 1.0

Methods

`__call__`(r)
`compute_R`()	Return R >= self.Rmin such that `abs(self(R)) <
`dVdV`(r)	Return V’(r)/V(r).
`gE`(E[, R, method])	Return k*cot(delta) as a function of energy E for the specified
`get_ar`([R, method])	Return (a,re) where a is the s-wave scattering length and re is the s-wave effective range.
`get_deq`(f, jac, U0[, method])	Return a “deq” object capable of integrating the system
`get_deq_E`(E[, method])	Return solution to energy dependent differential equation for (u,du) for finding the s-wave phase shift.
`get_deq_ar`([method])	Return the differential equation (u,du,uu,duu) for finding the s-wave scattering length and the s-wave effective range.
`get_deq_gE`(E[, method])	Return solution to energy dependent differential equation
`get_deq_gE_old`(E[, method, version])	Old version in terms of fp and fm.
`plot`([Rmax])	Plot potential.
`plot_gE`(Erange[, Rmax])	Plot potential.
`plot_u`(E[, Rmax])	Plot radial wavefunction.
`set_ainv`([ainv])	Set the coefficient v0 so that the scattering length is a.
`set_ainv_re`([ainv, re])	Set the coefficients v0 and A0 so that the scattering length is
`set_ainv_re_VV`([ainv, re])	Set the coefficients v0 and V1 so that the scattering length is

__init__(v0=-1.0, A0=6.2527, V1=0.0, A1=3.0, hbar=0.15915494309189535, M=0.5, kF=1.0, abs_tol=1e-12, rel_tol=1e-12, Rmin=0.1)[source]¶

dVdV(r)[source]¶

Return V’(r)/V(r).

V:=v0/cosh(mu0*r)^2+V1/cosh(mu1*r)^2; with(codegen); C(simplify(diff(V,r)/V),optimized);

set_ainv(ainv=0.0)[source]¶

Set the coefficient v0 so that the scattering length is a.

>>> V = SoonYongPotential2()
>>> V.v0 = -0.5
>>> V.set_ainv(0.0)
>>> np.allclose(V.v0, -1.0)
True

set_ainv_re(ainv=0.0, re=0.07)[source]¶

Set the coefficients v0 and A0 so that the scattering length is 1.0/ainv and the effective range is re.

>>> V = SoonYongPotential2(kF=1.0)
>>> V.V1 = 0.0
>>> V.set_ainv_re(ainv=0.0,re=1.0)
>>> np.allclose(V.A0, 2./V.kF)
True
>>> np.allclose(V.v0, -1.0)
True

set_ainv_re_VV(ainv=0.0, re=0.07)[source]¶

Set the coefficients v0 and V1 so that the scattering length is 1.0/ainv and the effective range is re.

>>> V = SoonYongPotential2()
>>> V.A0 = 6.2527
>>> V.A1 = 3.0
>>> V.v0 = -1.5
>>> V.V1 = 0.6
>>> V.set_ainv_re_VV(ainv=0.0,re=0.0)
>>> np.allclose(V.v0, -1.4789900537007772)
True
>>> np.allclose(V.V1, 0.61932931745837538)
True

class mmf.physics.potentials.two_body.Potential(abs_tol=1e-12, rel_tol=1e-12, Rmin=None)[source]¶

Bases: object

Instances are callable and return compact potentials V(r).

The class allows one to specify parameters. The important property is that when parameters are modified, the class ensures that the radius returned by R = get_R() is such that V(R) < abs_tol.

The parameter M should be the reduced mass of the system.

Methods

`__call__`(r)	Return V(r).
`compute_R`()	Return R >= self.Rmin such that `abs(self(R)) <
`dVdV`(r)	Return V’(r)/V(r). Should be overloaded for speed and/or
`gE`(E[, R, method])	Return k*cot(delta) as a function of energy E for the specified
`get_ar`([R, method])	Return (a,re) where a is the s-wave scattering length and re is the s-wave effective range.
`get_deq`(f, jac, U0[, method])	Return a “deq” object capable of integrating the system
`get_deq_E`(E[, method])	Return solution to energy dependent differential equation for (u,du) for finding the s-wave phase shift.
`get_deq_ar`([method])	Return the differential equation (u,du,uu,duu) for finding the s-wave scattering length and the s-wave effective range.
`get_deq_gE`(E[, method])	Return solution to energy dependent differential equation
`get_deq_gE_old`(E[, method, version])	Old version in terms of fp and fm.
`plot`([Rmax])	Plot potential.
`plot_gE`(Erange[, Rmax])	Plot potential.
`plot_u`(E[, Rmax])	Plot radial wavefunction.

Construct the potential object.

Methods

`__call__`(r)	Return V(r).
`compute_R`()	Return R >= self.Rmin such that `abs(self(R)) <
`dVdV`(r)	Return V’(r)/V(r). Should be overloaded for speed and/or
`gE`(E[, R, method])	Return k*cot(delta) as a function of energy E for the specified
`get_ar`([R, method])	Return (a,re) where a is the s-wave scattering length and re is the s-wave effective range.
`get_deq`(f, jac, U0[, method])	Return a “deq” object capable of integrating the system
`get_deq_E`(E[, method])	Return solution to energy dependent differential equation for (u,du) for finding the s-wave phase shift.
`get_deq_ar`([method])	Return the differential equation (u,du,uu,duu) for finding the s-wave scattering length and the s-wave effective range.
`get_deq_gE`(E[, method])	Return solution to energy dependent differential equation
`get_deq_gE_old`(E[, method, version])	Old version in terms of fp and fm.
`plot`([Rmax])	Plot potential.
`plot_gE`(Erange[, Rmax])	Plot potential.
`plot_u`(E[, Rmax])	Plot radial wavefunction.

__init__(abs_tol=1e-12, rel_tol=1e-12, Rmin=None)[source]¶: Construct the potential object.

R¶: Return a radius for which self(R) <= abs_tol.

compute_R()[source]¶: Return R >= self.Rmin such that abs(self(R)) < self.abs_tol.

dVdV(r)[source]¶: Return V’(r)/V(r). Should be overloaded for speed and/or accuracy.

gE(E, R=None, method='odeint')[source]¶: Return k*cot(delta) as a function of energy E for the specified scattering problem by matching at a specified radius R.

get_ar(R=None, method='odeint')[source]¶: Return (a,re) where a is the s-wave scattering length and re is the s-wave effective range. Integrates the wavefunction to radius R. Answer is independent of R if V(r>R) = 0.

get_deq(f, jac, U0, method='odeint')[source]¶: Return a “deq” object capable of integrating the system defined by U’ = f(r,U) and diff(U’,U) = jac(r,U) with initial condition U(0.0) = U0.

get_deq_E(E, method='odeint')[source]¶

Return solution to energy dependent differential equation for (u,du) for finding the s-wave phase shift.

Uses the following system: du’(r) = 2*m/hbar**2*(V(r)-E)*u(r) u’(r) = du(r)

with initial conditions

u(0) = 0.0 du(0) = 1.0

Call (u,du) = deq.integrate(R) to integrate to R. Note that deq maintains state, so subsequent calls must have increasing R.

This is to be used by calling the method integrate(R).

get_deq_ar(method='odeint')[source]¶

Return the differential equation (u,du,uu,duu) for finding the s-wave scattering length and the s-wave effective range.

Uses the following system: u’(r) = du(r) du’(r) = 2*m/hbar**2*V(r)*u(r) uu’(r) = duu(r) duu’(r) = 2*m/hbar**2*(V(r)*uu(r)-u(r))

with initial conditions

u(0) = 0.0 du(0) = 1.0 uu(0) = 0.0 duu(0) = 0.0

Call (u,du,uu,duu) = deq.integrate(R) to integrate to R. Note that deq maintains state, so subsequent calls must have increasing R.

get_deq_gE(E, method='odeint')[source]¶

Return solution to energy dependent differential equation for (Ap, Am) for finding the s-wave phase shift which is

$g(E) = \kappa \frac{A_+ - A_-}{A_+ + A_-}$

where $E = -\kappa^2$ and the s-wave radial wavefunction has the asymptotic form

$u(r>R) = A_+ e^{\kappa r} + A_-e^{-\kappa r}.$

We solve the radial Schrödinger equation

$u''(r) = [\tilde{V}(r) - \tilde{E}]u(r)$

where $\tilde{V} = 2mV(r)/\hbar^2$ and $\tilde{E} = 2mE(r)/\hbar^2$ with initial conditions $u(0)=0$ , and $u'(0)=1$ using the following transformation

$A_{\pm}(r) = \frac{1}{2}\left(u(r) \pm \frac{u'(r)}{\kappa}\right) e^{\mp\kappa r}$

These should be constant at the asymptotic values once $r>R$ is larger than the support of the potential. These functions satisfy the following set of coupled first-order equations:

$\diff{}{r} \begin{pmatrix} A_+(r)\\ A_-(r) \end{pmatrix} = \frac{\tilde{V}(r)}{2\kappa}\begin{pmatrix} 1 & e^{-2\kappa r}\\ -e^{2\kappa r} & -1 \end{pmatrix} \begin{pmatrix} A_+(r)\\ A_-(r) \end{pmatrix}$

with initial conditions $A_{\pm}(0) = \pm\frac{1}{2\kappa}$ .

get_deq_gE_old(E, method='odeint', version=1)[source]¶

Old version in terms of fp and fm.

version=0

fp' = dfp'
dfp' = (kappa**2 + VV)*fp - V'/V*(kappa*fp - dfp)
fm' = dfm
dfm' = (kappa**2 + VV)*fm + V'/V*(kappa*fm + dfm)

fp(0) = 1/2/kappa
dfp(0) = 1/2
fm(0) = -1/2/kappa
dfm(0) = 1/2

version=1

fp' =  ((2*kappa**2 + VV)*fp + VV*fm)/2/kappa
fm' = -((2*kappa**2 + VV)*fm + VV*fp)/2/kappa

fp(0) = 1/2/kappa
fm(0) = -1/2/kappa

Call (fp,dfp,fm,dfm) = deq.integrate(R) to integrate to R. Note that deq maintains state, so subsequent calls must have increasing R.

This is to be used by calling the method integrate(R).

Examples

>>> V = DoubleGaussian(f_v=0.0)
>>> V._gE_neg(E=-1.0, version=0)
-1.874095805...
>>> V._gE_neg(E=-1.0, version=1)
-1.874095805...
>>> V._gE_neg(E=-1.0, version=2)
-1.874095805...

plot(Rmax=None)[source]¶: Plot potential.

plot_gE(Erange, Rmax=None)[source]¶: Plot potential.

plot_u(E, Rmax=None)[source]¶: Plot radial wavefunction.

class mmf.physics.potentials.two_body.PoschlTeller(v0=-1.0, r0=0.31986181969389227, hbar=1.0, M=0.5, abs_tol=1e-12, rel_tol=1e-12, Rmin=0.1)[source]¶

Bases: mmf.physics.potentials.two_body.PotentialV0R0

Extended Poschl-Teller (Rosen-Morse) potential used by Soon Yong, Alex, Joe, and Stefano:

$V(r) &= v_0 \frac{\hbar^2}{M}\frac{\mu_0^2}{\cosh^2(\mu_0 r)}\\ V(r) &= v_0 \frac{\hbar^2}{M}\frac{4}{r_0^2}\sech^2\frac{2r}{r_0}\\$

Units are set by $k_F = (3\pi^2N_+)^{1/3}/L$ for a box of length $L$ and $\mu = A k_F$ . As a test, at unitarity, $v_0 = 1$ and $r_e = 2/\mu_0$ for two-body scattering where $m$ here is the individual mass of the particles (M=0.5 by default).

>>> V = PoschlTeller(v0=-1.0)
>>> (a,re) = V.get_ar()
>>> mu = 2.0/V.r0
>>> abs(a) > 6e8
True
>>> abs(re - 2.0/mu) < 1e-9
True

It is resonant $(a=\infty)$ for $v_0 = -n(2n - 1)$ for $n \in [1,2,3,...]$ .

>>> def infty(n):
...    return abs(PoschlTeller.a_(-n*(2.0*n-1.0)))
>>> infty(1) > 1e8
True
>>> infty(2) > 1e8
True

At these resonances one has $r_e/r_0 = H(2n-1)$ where $H(n)$ is the Harmonic Number $H(n) = \sum_{j=1}^{n}j^{-1}$ .

>>> def H(n):
...     return (1./numpy.arange(1,n+1)).sum()
>>> def zero(n):
...    return abs(PoschlTeller.re_(-n*(2.0*n-1.0))-H(2*n-1))
>>> zero(1) < 1e-10
True
>>> zero(2) < 1e-10
True

The scattering length is zero at the following points where the effective range diverges. I am not sure yet what these correspond to:

v0 = [0.0,-4.4438633257625231,-12.913326147200248,
-25.421577034814653,-41.955714059176017,
-62.508749555456298,...]

# kF_SY = (3.*pi)**(2./3.)
# EFG_SY = kF_SY**2/2.0*3.0/5.0
# EFG = EFG_SY/4.0/pi**2
# re = 2.0*L/(6.2527*(3.0*pi)**(2./3.))
#
# V = -2*hbar^2*mu^2*v0/m/cosh(mu*r)**2
# v0 = 1.0 => ainv = 0, re = 2.0/mu
# v0 = 1.30119 => kfa_SY = 1.0

Methods

`__call__`(r)
`compute_R`()	Return R >= self.Rmin such that `abs(self(R)) <
`dVdV`(r)	Return $V'(r)/V(r)$
`gE`(E[, R, method])	Return k*cot(delta) as a function of energy E for the specified
`get_ar`([R, method])	Return (a,re) where a is the s-wave scattering length and re is the s-wave effective range.
`get_deq`(f, jac, U0[, method])	Return a “deq” object capable of integrating the system
`get_deq_E`(E[, method])	Return solution to energy dependent differential equation for (u,du) for finding the s-wave phase shift.
`get_deq_ar`([method])	Return the differential equation (u,du,uu,duu) for finding the s-wave scattering length and the s-wave effective range.
`get_deq_gE`(E[, method])	Return solution to energy dependent differential equation
`get_deq_gE_old`(E[, method, version])	Old version in terms of fp and fm.
`k_cot_delta_analytic`(E[, l])	This potential has some analytic properties (see Alex’s thesis and
`plot`([Rmax])	Plot potential.
`plot_gE`(Erange[, Rmax])	Plot potential.
`plot_u`(E[, Rmax])	Plot radial wavefunction.
`set_ainv`([ainv])	Set the coefficient v0 so that the inverse scattering length is ainv.
`set_ainv_re`([ainv, re])	Set the coefficients v0 and r0 so that the inverse scattering length is ainv and the effective range is re.

__init__(v0=-1.0, r0=0.31986181969389227, hbar=1.0, M=0.5, abs_tol=1e-12, rel_tol=1e-12, Rmin=0.1)[source]¶

a_ = <numpy.lib.function_base.vectorize object at 0x9975910>[source]¶

are_ = <numpy.lib.function_base.vectorize object at 0x99758b0>[source]¶

dVdV(r)[source]¶

Return $V'(r)/V(r)$

V:=v0/cosh(mu0*r)^2+V1/cosh(mu1*r)^2;
with(codegen);
C(simplify(diff(V,r)/V),optimized);

k_cot_delta_analytic(E, l=None)[source]¶

This potential has some analytic properties (see Alex’s thesis and references therein):

System Message: WARNING/2 (k\cot\delta_k &= -k\frac{\Im{Z}}{\Re{Z}},\\ Z &= \frac{2^{-i\tilde{k}}\Gamma(i\tilde{k})} {\Gamma\left(\frac{1+\lambda + i\tilde{k}}{2}\right) \Gamma\left(\frac{2-\lambda + i\tilde{k}}{2}\right)},\\ \tilde{k} &= k/\mu, \qquad l = \frac{1 + \sqrt{1 - 8v_0}}{2}})

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Examples

>>> V = PoschlTeller(v0=-1.0,r0=2.0/6.2527)
>>> V.gE(E=1.0)
0.1599309...
>>> V.k_cot_delta_analytic(E=1.0)
0.1599309...

mu0[source]¶: Stefano, Alex, etc. use this parameter instead, so we provide an alias.

re_ = <numpy.lib.function_base.vectorize object at 0x9975950>[source]¶

set_ainv(ainv=0.0)[source]¶

Set the coefficient v0 so that the inverse scattering length is ainv.

>>> V = PoschlTeller()
>>> V.v0 = -0.5
>>> V.set_ainv(0.0)
>>> np.allclose(V.v0, -1.0)
True

set_ainv_re(ainv=0.0, re=0.07)[source]¶

Set the coefficients v0 and r0 so that the inverse scattering length is ainv and the effective range is re.

>>> V = PoschlTeller(r0=1.0)
>>> re = 1.0
>>> V.set_ainv_re(ainv=0.0,re=re)
>>> np.allclose(V.r0, re)
True
>>> np.allclose(V.v0, -1.0)
True

class mmf.physics.potentials.two_body.BargmannPotential(a=1.0, re=0.4, hbar=0.15915494309189535, M=0.5, **kwarg)[source]¶

Bases: mmf.physics.potentials.two_body.Potential

Bargmann Potential where the effective range expansion is exact with only a and r_e.

>> V = BargmannPotential(a=1.0,re=0.01) >> (a,re) = V.get_ar() >> (a,re) (1.0, 0.1)

Methods

`__call__`(r)
`compute_R`()	Return R >= self.Rmin such that `abs(self(R)) <
`dVdV`(r)	Return V’(r)/V(r).
`gE`(E[, R, method])	Return k*cot(delta) as a function of energy E for the specified
`get_ar`([R, method])	Return (a,re) where a is the s-wave scattering length and re is the s-wave effective range.
`get_deq`(f, jac, U0[, method])	Return a “deq” object capable of integrating the system
`get_deq_E`(E[, method])	Return solution to energy dependent differential equation for (u,du) for finding the s-wave phase shift.
`get_deq_ar`([method])	Return the differential equation (u,du,uu,duu) for finding the s-wave scattering length and the s-wave effective range.
`get_deq_gE`(E[, method])	Return solution to energy dependent differential equation
`get_deq_gE_old`(E[, method, version])	Old version in terms of fp and fm.
`plot`([Rmax])	Plot potential.
`plot_gE`(Erange[, Rmax])	Plot potential.
`plot_u`(E[, Rmax])	Plot radial wavefunction.

__init__(a=1.0, re=0.4, hbar=0.15915494309189535, M=0.5, **kwarg)[source]¶

dVdV(r)[source]¶

Return V’(r)/V(r).

V:=v0*beta*exp(-2*b*r)/(1+beta*exp(-2*b*r))^2; with(codegen); C(simplify(diff(V,r)/V),optimized);

class mmf.physics.potentials.two_body.PotentialV0R0(abs_tol=1e-12, rel_tol=1e-12, Rmin=None)[source]¶

Bases: mmf.physics.potentials.two_body.Potential

Mixing for two-parameter potentials with a parameter $v_0$ governing the overall strength and a parameter $r_0$ governing the effective range. The potential can have additional parameters, but these two will be adjusted to set the scattering-length and range.

Methods

`__call__`(r)	Return V(r).
`compute_R`()	Return R >= self.Rmin such that `abs(self(R)) <
`dVdV`(r)	Return V’(r)/V(r). Should be overloaded for speed and/or
`gE`(E[, R, method])	Return k*cot(delta) as a function of energy E for the specified
`get_ar`([R, method])	Return (a,re) where a is the s-wave scattering length and re is the s-wave effective range.
`get_deq`(f, jac, U0[, method])	Return a “deq” object capable of integrating the system
`get_deq_E`(E[, method])	Return solution to energy dependent differential equation for (u,du) for finding the s-wave phase shift.
`get_deq_ar`([method])	Return the differential equation (u,du,uu,duu) for finding the s-wave scattering length and the s-wave effective range.
`get_deq_gE`(E[, method])	Return solution to energy dependent differential equation
`get_deq_gE_old`(E[, method, version])	Old version in terms of fp and fm.
`plot`([Rmax])	Plot potential.
`plot_gE`(Erange[, Rmax])	Plot potential.
`plot_u`(E[, Rmax])	Plot radial wavefunction.
`set_ainv_re`(ainv, re)	Uses a non-linear root-finder to set the parameters v0 and r0 so that the scattering length is 1.0/ainv and the effective range is re.

Construct the potential object.

Methods

`__call__`(r)	Return V(r).
`compute_R`()	Return R >= self.Rmin such that `abs(self(R)) <
`dVdV`(r)	Return V’(r)/V(r). Should be overloaded for speed and/or
`gE`(E[, R, method])	Return k*cot(delta) as a function of energy E for the specified
`get_ar`([R, method])	Return (a,re) where a is the s-wave scattering length and re is the s-wave effective range.
`get_deq`(f, jac, U0[, method])	Return a “deq” object capable of integrating the system
`get_deq_E`(E[, method])	Return solution to energy dependent differential equation for (u,du) for finding the s-wave phase shift.
`get_deq_ar`([method])	Return the differential equation (u,du,uu,duu) for finding the s-wave scattering length and the s-wave effective range.
`get_deq_gE`(E[, method])	Return solution to energy dependent differential equation
`get_deq_gE_old`(E[, method, version])	Old version in terms of fp and fm.
`plot`([Rmax])	Plot potential.
`plot_gE`(Erange[, Rmax])	Plot potential.
`plot_u`(E[, Rmax])	Plot radial wavefunction.
`set_ainv_re`(ainv, re)	Uses a non-linear root-finder to set the parameters v0 and r0 so that the scattering length is 1.0/ainv and the effective range is re.

__init__(abs_tol=1e-12, rel_tol=1e-12, Rmin=None)¶: Construct the potential object.

set_ainv_re(ainv, re)[source]¶

Uses a non-linear root-finder to set the parameters v0 and r0 so that the scattering length is 1.0/ainv and the effective range is re.

>>> V = PoschlTeller()
>>> V.set_ainv_re(ainv=0.0,re=1.0)
>>> np.allclose(V.r0, 1.0)
True
>>> np.allclose(V.v0, -1.0)
True

class mmf.physics.potentials.two_body.DoubleGaussian(v0=0.7860976072874742, r0=0.6666666666666666, mu0=None, f_v=-4.0, f_r=2.0, hbar=1.0, M=0.5, abs_tol=1e-12, rel_tol=1e-12, Rmin=0.1)[source]¶

Bases: mmf.physics.potentials.two_body.PotentialV0R0

Double Gaussian potential:

$V(r) &= -v_0 \frac{\hbar^2}{M} \mu_0^2 \left( \exp[-(\mu_0 r/2)^{2}] + f_v \exp[-(\mu_0 r f_r/2)^2] \right)\\ &= -v_0 \frac{\hbar^2}{M} \mu_0^2 \left( \exp[-(\mu_0 r/2)^{2}] + f_v \exp[-(\mu_0 r f_r/2)^2] \right)\\$

Note that $\mu_0 = 4/r_0$ here.

>>> v = DoubleGaussian(mu0=12.0)
>>> v.get_ar()              # One of Stefano's values to check.
(-2887043864..., 0.3293...)
>>> v.set_ainv_re(ainv=0.0, re=1.0)
>>> v.v0, v.r0
(0.786097607..., 1.012180386...)

Methods

`__call__`(r)
`compute_R`()	Return R >= self.Rmin such that `abs(self(R)) <
`dVdV`(r)	Return V’(r)/V(r). Should be overloaded for speed and/or
`gE`(E[, R, method])	Return k*cot(delta) as a function of energy E for the specified
`get_ar`([R, method])	Return (a,re) where a is the s-wave scattering length and re is the s-wave effective range.
`get_deq`(f, jac, U0[, method])	Return a “deq” object capable of integrating the system
`get_deq_E`(E[, method])	Return solution to energy dependent differential equation for (u,du) for finding the s-wave phase shift.
`get_deq_ar`([method])	Return the differential equation (u,du,uu,duu) for finding the s-wave scattering length and the s-wave effective range.
`get_deq_gE`(E[, method])	Return solution to energy dependent differential equation
`get_deq_gE_old`(E[, method, version])	Old version in terms of fp and fm.
`get_mu0`()
`plot`([Rmax])	Plot potential.
`plot_gE`(Erange[, Rmax])	Plot potential.
`plot_u`(E[, Rmax])	Plot radial wavefunction.
`set_ainv_re`(ainv, re)	Uses a non-linear root-finder to set the parameters v0 and r0 so that the scattering length is 1.0/ainv and the effective range is re.
`set_mu0`(mu0)

__init__(v0=0.7860976072874742, r0=0.6666666666666666, mu0=None, f_v=-4.0, f_r=2.0, hbar=1.0, M=0.5, abs_tol=1e-12, rel_tol=1e-12, Rmin=0.1)[source]¶

get_mu0()[source]¶

mu0¶

set_mu0(mu0)[source]¶

mmf.physics.potentials.two_body.SoonYongPotential¶: alias of PoschlTeller

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