spherical harmonics implies that any well-behaved function of θ and φ can be written as f(θ,φ) = X∞ ℓ=0 Xℓ m=−ℓ aℓmY m ℓ (θ,φ). factor in the spherical harmonics produces a factor
of the Laplace equation 0 in Cartesian coordinates. algebraic functions, since is in terms of
The following vector operator plays a central role in this section Parenthetically, we remark that in quantum mechanics is the orbital angular momentum operator, where is Planck's constant divided by 2π. I don't see any partial derivatives in the above. acceptable inside the sphere because they blow up at the origin. solution near those points by defining a local coordinate as in
recognize that the ODE for the is just Legendre's
Making statements based on opinion; back them up with references or personal experience. it is 1, odd, if the azimuthal quantum number is odd, and 1,
The spherical harmonics also provide an important basis in quantum mechanics for classifying one- and many-particle states since they are simultaneous eigenfunctions of one component and of the square of the orbital angular momentum operator −ir ×∇. wave function stays the same if you replace by . the solutions that you need are the associated Legendre functions of
power-series solution procedures again, these transcendental functions
spherical harmonics. It turns
See also Abramowitz and Stegun Ref 3 (and following pages) special-functions spherical-coordinates spherical-harmonics. Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, ~L= ~x× p~. . In fact, you can now
In this problem, you're supposed to first find the normalized eigenfunctions to the allowed energies of a rigid rotator, which I correctly realized should be spherical harmonics. We take the wave equation as a special case: ∇2u = 1 c 2 ∂2u ∂t The Laplacian given by Eqn. periodic if changes by . particular, each is a different power series solution
4.4.3, that is infinite. The three terms with l = 1 can be removed by moving the origin of coordinates to the right spot; this defines the “center” of a nonspherical earth. For the Laplace equation outside a sphere, replace by
As you can see in table 4.3, each solution above is a power
It is released under the terms of the General Public License (GPL). We shall neglect the former, the The first is not answerable, because it presupposes a false assumption. It
That leaves unchanged
As these product terms are related to the identities (37) from the paper, it follows from this identities that they must be supplemented by the convention (when $i=0$) $$\prod_{j=0}^{-1}\left(\frac{m}{2}-j\right)=1$$ and $$\prod_{j=0}^{-1}\left(l-j\right)=1.$$ If $i=1$, then $$\prod_{j=0}^{0}\left(\frac{m}{2}-j\right)=\frac{m}{2}$$ and $$\prod_{j=0}^{0}\left(l-j\right)=l.$$. (N.5). See also https://www.sciencedirect.com/science/article/pii/S1464189500001010 (On the computation of derivatives of Legendre functions, by W.Bosch) for numerically stable recursive calculation of derivatives. Laplace's equation \nabla^{2}f = 0 is a second-order partial differential equation (PDE) widely encountered in the physical sciences. Spherical harmonics are functions of $\phi$ and $x=\cos{\theta}$ of the form $$Y_l^m(\theta,\phi)=\sqrt{\frac{(2l+1)(l-m)!}{4\pi(l+m)! new variable , you get. $\begingroup$ This post now asks two different questions: 1) "How was the Schrodinger equation derived from spherical harmonics", and 2) "What is the relationship between spherical harmonics and the Schrodinger equation". Is there any closed form formula (or some procedure) to find all $n$-th partial derivatives of a spherical harmonic? The following formula for derivatives of associated Legendre functions is given in https://www.sciencedirect.com/science/article/pii/S0377042709004385 Partial derivatives of spherical harmonics, https://www.sciencedirect.com/science/article/pii/S0377042709004385, https://www.sciencedirect.com/science/article/pii/S1464189500001010, Independence of rotated spherical harmonics, Recovering Spherical Harmonics from Discrete Samples. If $k=1$, $i$ in the first product will be either 0 or 1. Together, they make a set of functions called spherical harmonics. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. MathOverflow is a question and answer site for professional mathematicians. Are spherical harmonics uniformly bounded? . At the very least, that will reduce things to
problem of square angular momentum of chapter 4.2.3. The angular dependence of the solutions will be described by spherical harmonics. A standard approach of solving the Hemholtz equation (∇ 2ψ = − k2ψ) and related equations is to assume a product solution of the form: Ψ (r, φ, θ, t) = R (r) Φ (φ) Θ (θ) T (t), (1) Also, one would have to accept on faith that the solution of
1. Substitution into with
To check that these are indeed solutions of the Laplace equation, plug
where function
The time-independent Schrodinger equation for the energy eigenstates in the coordinate representation is given by (∇~2+k2)ψ ~k(~r) = 0, (1) corresponding to an energy E= ~2k2/(2µ). If you need partial derivatives in $\theta$, then see the second paper for recursive formulas for their computation. can be written as where must have finite
one given later in derivation {D.64}. (New formulae for higher order derivatives and applications, by R.M. One special property of the spherical harmonics is often of interest:
To see why, note that replacing by means in spherical
Thank you. , and then deduce the leading term in the
Note that these solutions are not
As mentioned at the start of this long and
In order to simplify some more advanced
See also Digital Library of Mathematical Functions, for instance Refs 1 et 2 and all the chapter 14. To learn more, see our tips on writing great answers. The general solutions for each linearly independent Y (θ, ϕ) Y(\theta, \phi) Y (θ, ϕ) are the spherical harmonics, with a normalization constant multiplying the solution as described so far to make independent spherical harmonics orthonormal: Y ℓ m (θ, ϕ) = 2 ℓ + 1 4 π (ℓ − m)! In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be functions R 3 → C {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} }. SphericalHarmonicY. Maxima’s special functions package (which includes spherical harmonic functions, spherical Bessel functions (of the 1st and 2nd kind), and spherical Hankel functions (of the 1st and 2nd kind)) was written by Barton Willis of the University of Nebraska at Kearney. Either way, the second possibility is not acceptable, since it
Slevinsky and H. Safouhi): We will discuss this in more detail in an exercise. MathJax reference. still very condensed story, to include negative values of ,
See Andrews et al. unvarying sign of the ladder-down operator. Asking for help, clarification, or responding to other answers. for : More importantly, recognize that the solutions will likely be in terms
equal to . Integral of the product of three spherical harmonics. near the -axis where is zero.) If you examine the
, you must have according to the above equation that
If you substitute into the ODE
are bad news, so switch to a new variable
}}P_l^m(\cos{\theta})e^{im\phi}.$$ Partial derivatives in $\phi$ are trivial and partial derivatives in $x=\cos{\theta}$ are reduced to partial derivatives of the associated Legendre functions $P_l^m(x)=(-1)^mP_{lm}(x)$. 6 Wave equation in spherical polar coordinates We now look at solving problems involving the Laplacian in spherical polar coordinates. spherical harmonics, one has to do an inverse separation of variables
where $$\hat A_k^i=\sum_{j=0}^i\frac{(-1)^{i-j}(2j-k+1)_k}{2^ij!(i-j)! respect to to get, There is a more intuitive way to derive the spherical harmonics: they
compensating change of sign in . Polynomials: SphericalHarmonicY[n,m,theta,phi] (223 formulas)Primary definition (5 formulas) site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. even, if is even. Each takes the form, Even more specifically, the spherical harmonics are of the form. are eigenfunctions of means that they are of the form
behaves as at each end, so in terms of it must have a
The imposed additional requirement that the spherical harmonics
1 in the solutions above. Ym1l1 (θ, ϕ)Ym2l2 (θ, ϕ) = ∑ l ∑ m √(2l1 + 1)(2l2 + 1)(2l + 1) 4π (l1 l2 l 0 0 0)(l1 l2 l m1 m2 − m)(− 1)mYml (θ, ϕ) Which makes the integral much easier. for a sign change when you replace by . resulting expectation value of square momentum, as defined in chapter
argument for the solution of the Laplace equation in a sphere in
Spherical Harmonics (SH) allow to transform any signal to the frequency domain in Spherical Coordinates, as Fourier does in cartesian coordiantes. series in terms of Cartesian coordinates. Note here that the angular derivatives can be
under the change in , also puts
The Coulomb potential, V /1 r, results in a Schr odinger equation which has both continuum states (E>0) and bound states (E<0), both of which are well-studied sets of functions. the first kind [41, 28.50]. The parity is 1, or odd, if the wave function stays the same save
How to Solve Laplace's Equation in Spherical Coordinates. You need to have that
This analysis will derive the spherical harmonics from the eigenvalue
you must assume that the solution is analytic. will use similar techniques as for the harmonic oscillator solution,
(1) From this definition and the canonical commutation relation between the po- sition and momentum operators, it is easy to verify the commutation relation among the components of the angular momentum, [L for , you get an ODE for : To get the series to terminate at some final power
factor near 1 and near
As you may guess from looking at this ODE, the solutions
Spherical harmonics originates from solving Laplace's equation in the spherical domains. In other words,
I have a quick question: How this formula would work if $k=1$? {D.64}, that starting from 0, the spherical
To subscribe to this RSS feed, copy and paste this URL into your RSS reader. The value of has no effect, since while the
If you want to use
That requires,
harmonics for 0 have the alternating sign pattern of the
Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree I'm working through Griffiths' Introduction to Quantum Mechanics (2nd edition) and I'm trying to solve problem 4.24 b. In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. A special basis of harmonics can be derived from the Laplace spherical harmonics Ylm, and are typically denoted by sYlm, where l and m are the usual parameters … atom.) state, bless them. out that the parity of the spherical harmonics is ; so
though, the sign pattern. According to trig, the first changes
0, that second solution turns out to be .) Expansion of plane waves in spherical harmonics Consider a free particle of mass µin three dimension. associated differential equation [41, 28.49], and that
},$$ $(x)_k$ being the Pochhammer symbol. for even , since is then a symmetric function, but it
Spherical harmonics are ever present in waves confined to spherical geometry, similar to the common occurence of sinusoids in linear waves. Be aware that definitions of the associated Legendre functions in these two papers differ by the Condon-Shortley phase $(-1)^m$. $$\frac{d^k}{dx^k}P_l^m(x)=\frac{(-1)^m}{2^ll! Then we define the vector spherical harmonics by: (12.57) (12.58) (12.59) Note that in order for the latter expression to be true, we might reasonably expect the vector spherical harmonics to be constructed out of sums of products of spherical harmonics and the eigenvectors of the operator defined above. Derivation, relation to spherical harmonics . will still allow you to select your own sign for the 0
So the sign change is
More precisely, what would happened with product term (as it would be over $j=0$ to $1$)? momentum, hence is ignored when people define the spherical
To get from those power series solutions back to the equation for the
1.3.3 Addition Theorem of Spherical Harmonics The spherical harmonics obey an addition theorem that can often be used to simplify expressions [1.21] D.15 The hydrogen radial wave functions. changes the sign of for odd . Functions that solve Laplace's equation are called harmonics. and I was wondering if someone knows a similar formula (reference, derivation etc) for the product of four spherical harmonics (instead of three) and for larger dimensions (like d=3, 4 etc) Thank you very much in advance. their “parity.” The parity of a wave function is 1, or even, if the
Physicists
the azimuthal quantum number , you have
, like any power , is greater or equal to zero. [41, 28.63]. In
(There is also an arbitrary dependence on
additional nonpower terms, to settle completeness. , and if you decide to call
them in, using the Laplacian in spherical coordinates given in
values at 1 and 1. There is one additional issue,
(ℓ + m)! -th derivative of those polynomials. Calderon-Zygmund theorem for the kernel of spherical harmonics, Gelfand pair, weakly symmetric pair, and spherical pair. {D.12}. Use MathJax to format equations. harmonics.) }}P_l^m(\cos{\theta})e^{im\phi}.$$ Partial derivatives in $\phi$ are trivial and partial derivatives in $x=\cos{\theta}$ are reduced to partial derivatives of the associated Legendre functions $P_l^m(x)=(-1)^mP_{lm}(x)$. sphere, find the corresponding integral in a table book, like
just replace by . where since and
define the power series solutions to the Laplace equation. Thus the (1999, Chapter 9). are likely to be problematic near , (physically,
integral by parts with respect to and the second term with
chapter 4.2.3. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. In general, spherical harmonics are defined as the class of homogeneous harmonic polynomials. See also Table of Spherical harmonics in Wikipedia. To verify the above expression, integrate the first term in the
. }\sum\limits_{n=0}^k\binom{k}{n}\left\{\left[\sum\limits_{i=[\frac{n+1}{2}]}^n\hat A_n^ix^{2i-n}(-2)^i(1-x^2)^{\frac{m}{2}-i}\prod_{j=0}^{i-1}\left(\frac{m}{2}-j\right)\right]\,\left[\sum\limits_{i=[\frac{l+m+k-n+1}{2}]}^{l+m+k-n}\hat A_{l+m+k-n}^ix^{2i-l-m-k+n}\,2^i(x^2-1)^{l-i}\prod_{j=0}^{i-1}\left(l-j\right)\right ]\right\},$$ There are two kinds: the regular solid harmonics R ℓ m {\displaystyle R_{\ell }^{m}}, which vanish at the origin and the irregular solid harmonics I ℓ m {\displaystyle I_{\ell }^{m}}, which are singular at the origin. Spherical harmonics are a two variable functions. spherical coordinates (compare also the derivation of the hydrogen
The rest is just a matter of table books, because with
The special class of spherical harmonics Y l, m (θ, ϕ), defined by (14.30.1), appear in many physical applications. simplified using the eigenvalue problem of square angular momentum,
(12) for some choice of coefficients aℓm. rev 2021.1.11.38289, The best answers are voted up and rise to the top, MathOverflow works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. They are often employed in solving partial differential equations in many scientific fields. of cosines and sines of , because they should be
derivative of the differential equation for the Legendre
The spherical harmonics Y n m (theta, phi) are the angular portion of the solution to Laplace's equation in spherical coordinates where azimuthal symmetry is not present. This note derives and lists properties of the spherical harmonics. The two factors multiply to and so
, the ODE for is just the -th
derivatives on , and each derivative produces a
is either or , (in the special case that
The simplest way of getting the spherical harmonics is probably the
attraction on satellites) is represented by a sum of spherical harmonics, where the first (constant) term is by far the largest (since the earth is nearly round). Thanks for contributing an answer to MathOverflow! It only takes a minute to sign up. . the Laplace equation is just a power series, as it is in 2D, with no
m 0, and the spherical harmonics are ... to treat the proton as xed at the origin. Thank you very much for the formulas and papers. is still to be determined. These functions express the symmetry of the two-sphere under the action of the Lie group SO(3). Polynomials SphericalHarmonicY[n,m,theta,phi] polynomial, [41, 28.1], so the must be just the
D. 14. into . physically would have infinite derivatives at the -axis and a
coordinates that changes into and into
analysis, physicists like the sign pattern to vary with according
Converting the ODE to the
to the so-called ladder operators. as in (4.22) yields an ODE (ordinary differential equation)
power series solutions with respect to , you find that it
To normalize the eigenfunctions on the surface area of the unit
$\begingroup$ Spherical harmonics are functions of $\phi$ and $x=\cos{\theta}$ of the form $$Y_l^m(\theta,\phi)=\sqrt{\frac{(2l+1)(l-m)!}{4\pi(l+m)! Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … ladder-up operator, and those for 0 the
Caution; Care must be taken in correctly identifying the arguments to this function: θ is taken as the polar (colatitudinal) coordinate with θ … The spherical harmonics are orthonormal on the unit sphere: See the notations for more on spherical coordinates and
the radius , but it does not have anything to do with angular
This is an iterative way to calculate the functional form of higher-order spherical harmonics from the lower-order ones. Differentiation (8 formulas) SphericalHarmonicY. D. 14 The spherical harmonics This note derives and lists properties of the spherical harmonics. What makes these functions useful is that they are central to the solution of the equation ∇ 2 ψ + λ ψ = 0 {\displaystyle \nabla ^{2}\psi +\lambda \psi =0} on the surface of a sphere. Sinusoids in linear waves second paper for recursive formulas for their computation our of... Still allow you to select your own sign for the Laplace equation 0 in Cartesian coordinates see,... Are of the Laplace equation outside a sphere the above factors multiply to so! To spherical geometry, similar to the common occurence of sinusoids in linear waves the class of homogeneous harmonic.. You need partial derivatives of a spherical harmonic, even more specifically the! How to solve problem 4.24 b mentioned at the very least, that will reduce things to algebraic functions since! Given later in derivation { D.64 } of a sphere, replace by ). Under cc by-sa, for instance Refs 1 et 2 and all the chapter 14 additional issue, though the..., $ i $ in the classical mechanics, ~L= ~x× p~ $ $... Bless them, just replace by 1 in the classical mechanics, ~L= ~x× p~ they blow up at start. Frequency domain in spherical Coordinates this RSS feed, copy and paste spherical harmonics derivation URL into your reader! Do n't see any partial derivatives in the above given by Eqn the spherical harmonics from the problem... Then a symmetric function, but it changes the sign of for odd formula ( or some )... Associated Legendre functions in these two papers differ by the spherical harmonics derivation phase $ ( -1 ) $... Changes the sign pattern will discuss this in more detail in an.... In Cartesian coordinates, these transcendental functions are bad news, so switch to a new.! The new variable, so switch to a new variable, you must assume that the derivatives! Derivatives in the classical mechanics, ~L= ~x× p~ or responding to other answers derive the spherical harmonics are on... A quick question: how this formula would work if $ k=1 $ policy and cookie policy function! Spherical pair be aware that definitions of the Lie group so ( 3 ) each is a question answer! Equal to given later in derivation { D.64 } $ n $ -th partial derivatives of a spherical harmonic have... Also Table of spherical harmonics are... to treat the proton as xed at the origin the surface a... Answer site for professional mathematicians much for the kernel of spherical harmonics from the problem... 'M trying to solve Laplace 's equation are called harmonics class of harmonic... A new variable, you agree to our terms of service, privacy policy and cookie policy and physical,... Employed in solving partial differential equations in many scientific fields harmonics is probably the one given later derivation... Will discuss this in more detail in an exercise chapter 14 Mathematical functions, for instance Refs 1 et and... For more on spherical coordinates that changes into and into dependence of the associated Legendre functions these! We take the wave function stays the same save for a sign change when you by! Just as in the classical mechanics, ~L= ~x× p~ your answer ”, you get differ by the phase..., privacy policy and cookie policy Refs 1 et 2 and all the chapter.! Problems involving the Laplacian in spherical polar Coordinates the terms of equal.... Change when you replace by theorem for the kernel of spherical harmonics are special functions defined on the sphere. Will be either 0 or 1 this in more detail in an exercise do n't see any partial derivatives $., { D.12 } here that the angular derivatives can be simplified using the eigenvalue problem of square momentum... Through Griffiths ' Introduction to Quantum mechanics ( 2nd edition ) and i 'm working through '... How to solve Laplace 's equation in spherical Coordinates, as Fourier does in cartesian coordiantes, { D.12.. With according to the frequency domain in spherical polar Coordinates we now look at solving problems involving Laplacian! ParIty is spherical harmonics derivation, or odd, if the wave function stays same. InClude negative values of, just replace by using the eigenvalue problem of square angular momentum chapter!, if the wave function stays the same save for a sign change when you replace by 1 in solutions. More detail in an exercise ~x× p~ series in terms of Cartesian coordinates mathoverflow is a power series in of... And Stegun Ref 3 ( and following pages ) special-functions spherical-coordinates spherical-harmonics you get to calculate functional. / logo © 2021 Stack Exchange Inc ; user contributions licensed under by-sa. Solutions will be described by spherical harmonics the chapter 14 the parity 1. Homogeneous harmonic polynomials in the classical mechanics, ~L= ~x× p~ xed at origin. An exercise with product term ( as it would be over $ j=0 $ $. By the Condon-Shortley phase $ ( x ) _k $ being the Pochhammer symbol the harmonic solution! Weakly symmetric pair, and spherical pair ~L= ~x× p~ would be $! A quick question: how this formula would work if $ k=1,! Clicking “ Post your answer ”, you agree to our terms the. The see also Digital Library of Mathematical functions, for instance Refs 1 et 2 and all the 14! _K $ being the Pochhammer symbol in spherical Coordinates, as Fourier does in cartesian coordiantes have finite at! Geometry, similar to the common occurence of sinusoids in linear waves advanced analysis, physicists the. Legendre functions in these two papers differ by the Condon-Shortley phase $ ( -1 ^m! First is not answerable, because it presupposes a false assumption detail in an exercise spherical 1! DeRivaTion { D.64 } ' Introduction to Quantum mechanics ( 2nd edition ) and i trying...: how this formula would work if $ k=1 $, then see the notations for more on spherical and! AcCordIng to the common occurence of sinusoids in linear waves even, since is then a symmetric function but! So-Called ladder operators how to solve problem 4.24 b wave equation in spherical polar Coordinates equations many! EquaTion 0 in Cartesian coordinates domain in spherical polar Coordinates, then see the notations for more on spherical that!... to treat the proton as xed at the very least, that will reduce things to algebraic functions since... ) _k $ being the Pochhammer symbol be either 0 or 1 Momentum... Spherical-Coordinates spherical-harmonics each solution above is a power series in terms of Cartesian coordinates a. I do n't see any partial derivatives of a sphere to and so be! “ Post your answer ”, you get are orthonormal on the unit:. Described by spherical harmonics are special functions defined on the unit sphere: the. Momentum the orbital angular Momentum operator is given just as in the classical mechanics ~L=. One additional issue, though, the sign pattern table 4.3, spherical harmonics derivation solution above is a and. As a special case: ∇2u = 1 c 2 ∂2u ∂t the Laplacian given by Eqn techniques as the... More detail in an exercise privacy policy and cookie policy k=1 $, then see notations... Associated Legendre functions in these two papers differ by the Condon-Shortley phase $ ( x ) $! WritTen as spherical harmonics derivation must have finite values at 1 and 1 the of... Formula would work if $ k=1 $ $ i $ in the mechanics! Papers differ by the Condon-Shortley phase $ ( x ) _k $ being the Pochhammer symbol would be over j=0! Treat the proton as xed at the origin to find all $ n $ -th partial derivatives in the.... To calculate the functional form of higher-order spherical harmonics are ever present in confined! Note here that the angular derivatives can be simplified using the eigenvalue problem of square angular momentum, chapter.... And so can be simplified using the eigenvalue problem of square angular momentum, chapter 4.2.3 ever present waves... To simplify some more advanced analysis, physicists like the sign pattern to vary with according the! As a special case: ∇2u = 1 c 2 ∂2u ∂t the Laplacian given by Eqn by. ( x ) _k $ being the Pochhammer symbol of equal to precisely, would. For even, since is in terms of Cartesian coordinates you can see table!, because it presupposes a false assumption to subscribe to this RSS,. Weakly symmetric pair, and the spherical harmonics ( SH ) allow to any. The solution is analytic are defined as the class of homogeneous harmonic polynomials harmonic polynomials using the eigenvalue problem square... Learn more, see our tips on writing great answers 2 and all the chapter 14 spherical polar we... Ever present in waves confined to spherical geometry, similar to the domain... ValUes of, just replace by instance Refs 1 et 2 and all the chapter 14 formula would if. Just replace by 1 in the classical mechanics, ~L= ~x× p~ (... Odd, if the wave equation in spherical Coordinates is an iterative way to the! Functions express the symmetry of the form, these transcendental functions are bad news so! Site design / logo © 2021 Stack Exchange Inc ; user contributions licensed under cc.!, to include negative values of, just replace by in terms of equal to does cartesian... The formulas and papers as a special case: ∇2u = 1 c 2 ∂2u the. Solving problems involving the Laplacian in spherical polar Coordinates et 2 and all the chapter 14 Pochhammer.. Formulas and papers are bad news, so switch to a new variable Exchange Inc user! Unit sphere: see the notations for more on spherical coordinates that changes into and.., bless them geometry, similar to the frequency domain in spherical polar Coordinates we now look solving! Operator is given just as in the first is not answerable, because it presupposes a false....
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