`mmf.physics.seq`¶

`ScarfII`(varargin, *kwargs)	Scarf II (hyperbolic) potential:
`S`(varargin, *kwargs)	Scarf II (hyperbolic) potential:
`HO_radial`(varargin, *kwargs)	Radial Harmonic Oscillator.
`R`(x, n, a, b)	Return the Romanovski polynomial

Inheritance diagram for mmf.physics.seq:

Inheritance diagram of mmf.physics.seq

Numerical Solutions to the Schrodinger equation.

This module contains some exact solutions to the Schrodinger equation for use in testing.

class mmf.physics.seq.ScarfII(*varargin, **kwargs)[source]¶

Bases: mmf.objects.StateVars

Scarf II (hyperbolic) potential:

ScarfII(A=1,
        B=0,
        a=1,
        m=1,
        hbar=1)

$V(x) = \frac{\hbar^2}{2m a^2}\Biggl[A^2 + \left(B^2 - A^2 - A\right) \sech^{2}\frac{x}{a} + B\left(2A + 1\right) \sech\frac{x}{a}\; \tanh \frac{x}{a}\Biggr].$

Notes

The depth of the potential is controlled by A while the asymmetry is controlled by B (though the spectrum is unchanged).

from mmf.physics.seq import ScarfII
x = np.linspace(-3,3,100)
for B in [-1,0,1,2]:
   V = ScarfII(A=1, B=B)
   plt.plot(x, V(x), label='B=%i'%(B,))

plt.legend()

(Source code, png, hires.png, pdf)

Examples

from mmf.physics.seq import ScarfII
V = ScarfII(A=3, B=2)
x = np.linspace(-10, 10, 200)
v = V(x)
En = V.E()
plt.plot(x, v, 'k')
for n, E in enumerate(En):
   x_ = x[np.where(v <= E)][[0, -1]]
   line = plt.plot(x_, x_*0 + E, lw=2)
   colour = line[0].get_color()
   psi = V.psi(n)(x)
   psi = psi/np.max(abs(psi))
   plt.plot(x, 2*psi, colour + '-')

(Source code, png, hires.png, pdf)

Attributes

class mmf.physics.seq.S(*varargin, **kwargs)[source]¶

Bases: mmf.objects.StateVars

Scarf II (hyperbolic) potential:

S(A=1,
  B=0,
  a=1,
  m=1,
  hbar=1)

ScarfII(A=1,
        B=0,
        a=1,
        m=1,
        hbar=1)

$V(x) = \frac{\hbar^2}{2m a^2}\Biggl[A^2 + \left(B^2 - A^2 - A\right) \sech^{2}\frac{x}{a} + B\left(2A + 1\right) \sech\frac{x}{a}\; \tanh \frac{x}{a}\Biggr].$

Notes

The depth of the potential is controlled by A while the asymmetry is controlled by B (though the spectrum is unchanged).

from mmf.physics.seq import ScarfII
x = np.linspace(-3,3,100)
for B in [-1,0,1,2]:
   V = ScarfII(A=1, B=B)
   plt.plot(x, V(x), label='B=%i'%(B,))

plt.legend()

(Source code, png, hires.png, pdf)

Examples

from mmf.physics.seq import ScarfII
V = ScarfII(A=3, B=2)
x = np.linspace(-10, 10, 200)
v = V(x)
En = V.E()
plt.plot(x, v, 'k')
for n, E in enumerate(En):
   x_ = x[np.where(v <= E)][[0, -1]]
   line = plt.plot(x_, x_*0 + E, lw=2)
   colour = line[0].get_color()
   psi = V.psi(n)(x)
   psi = psi/np.max(abs(psi))
   plt.plot(x, 2*psi, colour + '-')

(Source code, png, hires.png, pdf)

Attributes

State Variables:

A¶

Dimensionless parameter (depth).

B¶

Dimensionless parameter (controls asymmetry).

a¶

Length scale

m¶

<no description>

hbar¶

class mmf.physics.seq.HO_radial(*varargin, **kwargs)[source]¶

Bases: mmf.objects.StateVars

Radial Harmonic Oscillator.

HO_radial(d=3,
          hbar=1,
          m=1,
          omega=1)

Notes

In d-dimensions, the Hamiltonian is simply a sum over d independent oscillators, $\smash{\op{H}}^{(d)} = \sum_{i=1}^{d} \op{H}_{i}$ , where each $\op{H}_{i}$ is an independent oscillator in coordinate $x_{i}$ . The energies are additive and thus:

$E_{n_{1},n_{2},\cdots,n_{d}} = \left(\sum_{i} n_{i} + \frac{d}{2}\right)\hbar\omega.$

It is useful to consider this as a central potential problem in the coordinate $r$ . In this case we recover a 1-d problem with an additional set of quantum numbers corresponding to angular momentum $l$ . This follows from the expression of the Laplacian in d-dimensions:

$\nabla^{\field{R}^d} = \frac{1}{r^{d-1}}\diff{}{r}r^{d-1}\diff{}{r} + \frac{1}{r^2}\nabla^{S^{d-1}}$

where $\nabla^{S^{d-1}}$ is the Laplacian defined on the unit sphere $S^{d-1}$ . This can be brought into a simple form by introducing the radial momentum operator:

$\op{p}_{r} = \frac{-i}{r^{(d-1)/2}}\pdiff{}{r}r^{(d-1)/2}.$

In terms of this we have:

$\frac{1}{r^{d-1}}\diff{}{r}r^{d-1}\diff{}{r} = -\op{p}_{r}^2 - \frac{(d-1)(d-3)}{4r^2}.$

Finally, introducing the appropriate d-dimensional spherical harmonics $\mathcal{Y}_{l}(\vect{r})$ we have

$\frac{1}{r^2}\nabla^{S^{d-1}}\mathcal{Y}_{l}(\vect{r}) = -l(l+d-2)\mathcal{Y}_{l}(\vect{r}).$

Combining these gives the effective radial Hamiltonianfootnote{One can define raising a lowering operators to analyse this~cite{Arik:2008ix}.}

$\begin{aligned} \smash{\op{H}}^{(d)} &= \frac{1}{2m}\left( \op{p}_{r}^2 + \frac{\nu_{l}^2 - \frac{1}{4}}{\op{r}^2} + m^2\omega^2 \op{r}^2 \right), & \nu_{l} &= l + \frac{d-2}{2}. \end{aligned}$

The energy levels $E_{l,j}$ and degeneracies $g_{l,j}$ are thus

$\begin{aligned} E_{l,j} &= \left(l + 2j + \frac{d}{2}\right)\hbar\omega, & g_{l,j} &= \binom{d+l+2j-1}{l+2j} = \frac{\Gamma(d+l+2j)}{\Gamma(l+2j+1)\Gamma(d)}. \end{aligned}$

with normalized radial eigenfunctions are $\Psi^{(l)}_{j}(r) = R_{j}(r/L)$ where

$R^{l}_{j}(r) = (-1)^{j}\sqrt{\frac{2\Gamma(j+1)}{\Gamma(l+j+d/2)}} e^{-r^2/2} r^{\nu+1/2}L_{j}^{\nu_l}(r^2)$

and $L_{j}^{\nu_l}$ are the generalized Laguerre polynomials.

Attributes

mmf.physics.seq.R(x, n, a, b)[source]¶

Return the Romanovski polynomial

$R_{m}^{(a,b)}(x) &= \frac{1}{w^{(a,b)}(x)} \diff[m]{}{x}(1+x^2)^{m}w^{(a,b)}(x)\\ w^{(a,b)}(x) &= (1+x^2)^{-a}e^{b\tan^{-1}(x)}$

Notes

Computed using the recurrence

$R_{m}^{(a,b)}(x) &= [b + 2(m - a)x]R_{m-1}^{(a,b)}(x) + 2(m - 1)(m - a)(1+x^2)R_{m-2}^{(a-1,b)}(x),\\ R_{0}^{(a,b)} &= 1,\\ R_{1}^{(a,b)} &= 2(1-a)x + b.$

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