Particle in a spherically symmetric potential


Particle in a spherically symmetric potential

In quantum mechanics, the particle in a spherically symmetric potential describes the dynamics of a particle in a potential which has spherical symmetry. The Hamiltonian for such a system has the form

:hat{H} = frac{hat{p}^2}{2m_0} + V(r)

where m_0 denotes the mass of the particle in the potential.

In its quantum mechanical formulation, it amounts to solving the Schrödinger equation with the potential "V(r)" which depend only on "r", the modulus of r. Due to the spherical symmetry of the system it is useful to use spherical coordinates "r", heta and phi. When this is done, the time independent Schrödinger equation for the system is separable.

Structure of the eigenfunctions

The eigenfunctions of the Schrödinger equation have the form:psi(r, heta, phi) = R(r)Theta( heta)Phi(phi) ,;in which the spherical polar angles θ and φ represent the colatitude and azimuthal angle, respectively. The last two factors of ψ are often grouped together as spherical harmonics, so that the eigenfunctions take the form:psi(r, heta, phi) = R(r)Y_{lm}( heta,phi).,

The differential equation which characterizes the function R(r) ,; is called the radial equation.

Derivation of the radial equation

The kinetic energy operator in spherical polar coordinates is:frac{hat{p}^2}{2m_0} = -frac{hbar^2}{2m_0} abla^2 = - frac{hbar^2}{2m_0,r^2}left [ frac{partial}{partial r}Big(r^2 frac{partial}{partial r}Big) - hat{l}^2 ight] .The spherical harmonics satisfy :hat{l}^2 Y_{lm}( heta,phi)equiv left{ -frac{1}{sin^2 heta} left [sin hetafrac{partial}{partial heta} Big(sin hetafrac{partial}{partial heta}Big)+frac{partial^2}{partial phi^2} ight] ight} Y_{lm}( heta,phi) = l(l+1)Y_{lm}( heta,phi).

Substituting this into the Schrödinger equation we get a one-dimensional eigenvalue equation,:left { - {hbar^2 over 2m_0 r^2} {dover dr}left(r^2{dover dr} ight) +{hbar^2 l(l+1)over 2m_0r^2}+V(r) ight } R(r)=ER(r)

Relationship with 1-D Schrödinger equation

Note that the first term in the kinetic energy can be rewritten : T_requiv frac{1}{r^2} frac{d}{dr}r^2frac{d}{dr} = frac{1}{r}frac{d^2}{dr^2}r. This follows because both sides of this equation can be shown by application of the product rule to be equal to a third form of this operator:: T_r =frac{1}{r^2} frac{d}{dr}(r^2frac{d}{dr})= frac{d^2}{dr^2}+frac{2}{r}frac{d}{dr}. If subsequently the substitution u(r) stackrel{mathrm{def{=} rR(r) is made into : - {hbar^2 over 2m_0 r^2} {d R(r)over dr}left(r^2{dover dr} ight) +{hbar^2 l(l+1)over 2m_0r^2}R(r)+V(r) R(r)=ER(r),R(r)={u(r)over r};; {dRover dr}={1over r}{du(r)over dr}-{u(r)over r^2};; r^2{dR(r)over dr}=r{du(r)over dr}-u(r);; {dover dr}(r^2 {dRover dr})={dover dr}(r{du(r)over dr}-u(r))={du(r)over dr}+r{d^2 u(r)over dr^2}-{du(r)over dr}=r{d^2 u(r)over dr^2},: - {hbar^2 over 2m_0 r^2} r{d^2 u(r)over dr^2} +{hbar^2 l(l+1)over 2m_0r^2}{u(r)over r}+V(r) {u(r)over r}=E{u(r)over r},: - {hbar^2 over 2m_0 } {d^2 u(r)over dr^2} +{hbar^2 l(l+1)over 2m_0r^2}u(r)+V(r) u(r)=Eu(r),

the radial equation becomes:-{hbar^2 over 2m_0} {d^2 u(r) over dr^2} + V_{mathrm{eff(r) u(r) = E u(r)

which is precisely a Schrödinger equation for the function "u(r)" with an effective potential given by:V_{mathrm{eff(r) = V(r) + {hbar^2l(l+1) over 2m_0r^2},

where the radial coordinate "r" ranges from 0 to infty. The correction to the potential "V(r)" is called the centrifugal barrier term.

Solutions for potentials of interest

Five special cases arise, of special importance:
#"V(r)=0", or solving the vacuum in the basis of spherical harmonics, which serves as the basis for other cases.
#V(r)=V_0 (finite) for r and 0 elsewhere, or a particle in the spherical equivalent of the square well, useful to describe scattering and bound states in a nucleus or quantum dot.
# As the previous case, but with an infinitely high jump in the potential on the surface of the sphere.
#"V(r)~r²" for the three-dimensional isotropic harmonic oscillator.
#"V(r)~1/r" to describe bound states of hydrogen-like atoms.

We outline the solutions in these cases, which should be compared to their counterparts in cartesian coordinates, cf. particle in a box. This article relies heavily on Bessel functions and Laguerre polynomials.

Vacuum case

Let us now consider "V(r)=0" (if V_0, replace everywhere "E" with E-V_0). Introducing the dimensionless variable : ho stackrel{mathrm{def{=} kr, qquad k stackrel{mathrm{def{=} sqrt{2m_0Eoverhbar^2} the equation becomes a Bessel equation for "J" defined by J( ho) stackrel{mathrm{def{=} sqrt ho R(r) (whence the notational choice of "J"): : ho^2{d^2Jover d ho^2}+ ho{dJover d ho}+left [ ho^2-left(l+frac{1}{2} ight)^2 ight] J=0

which regular solutions for positive energies are given by so-called "Bessel functions of the first kind" J_{l+1/2}( ho) so that the solutions written for "R" are the so-called "Spherical Bessel function" R(r) = j_l(kr) stackrel{mathrm{def{=} sqrt{pi/(2kr)} J_{l+1/2}(kr).

The solutions of Schrödinger equation in polar coordinates for a particle of mass m_0 in vacuum are labelled by three quantum numbers: discrete indices "l" and "m", and "k" varying continuously in [0,infty] ::psi(mathbf{r})=j_l(kr)Y_{lm}( heta,phi)where k stackrel{mathrm{def{=} sqrt{2m_0E}/hbar, j_l are the spherical Bessel function and Y_{lm} are the spherical harmonics.

These solutions represent states of definite angular momentum, rather than of definite (linear) momentum, which are provided by plane waves exp(i mathbf{k}cdotmathbf{r}).

Sphere with square potential

Let us now consider the potential V(r)=V_0 for r and V(r)=0 elsewhere. That is, inside a sphere of radius r_0 the potential is equal to "V"0 and it is zero outside the sphere. A potential with such a finite discontinuity is called a "square potential."A. Messiah, "Quantum Mechanics", vol. I, p. 78, North Holland Publishing Company, Amsterdam (1967). Translation from the French by G.M. Temmer]

We first consider bound states, i.e., states which display the particle mostly inside the box (confined states). Those have an energy "E" less than the potential outside the sphere, i.e., they have negative energy, and we shall see that there are a discrete number of such states, which we shall compare to positive energy with a continuous spectrum, describing scattering on the sphere (of unbound states). Also worth noticing is that unlike Coulomb potential, featuring an infinite number of discrete bound states, the spherical square well has only a finite (if any) number because of its finite range (if it has finite depth). The resolution essentially follows that of the vacuum with normalization of the total wavefunction added, solving two Schrödinger equations — inside and outside the sphere — of the previous kind, i.e., with constant potential. Also the following constraints hold:
#The wavefunction must be regular at the origin.
#The wavefunction and its derivative must be continuous at the potential discontinuity.
#The wavefunction must converge at infinity.

The first constraint comes from the fact that Neumann "N" and Hankel "H" functions are nonsingular at the origin. The physical argument that "ψ" must be defined everywhere selected Bessel function of the first kind "J" over the other possibilities in the vacuum case. For the same reason, the solution will be of this kind inside the sphere:

:R(r)=Aj_lleft(sqrt{2m_0(E-V_0)overhbar^2}r ight),qquad r

with "A" a constant to be determined later. Note that for bound states, V_0.

Bound states bring the novelty as compared to the vacuum case that "E" is now negative (in the vacuum it was to be positive). This, along with third constraint, selects Hankel function of the first kind as the only converging solution at infinity (the singularity at the origin of these functions does not matter since we are now outside the sphere):

:R(r)=Bh^{(1)}_lleft(isqrt{-2m_0Eoverhbar^2}r ight),qquad r>r_0

Second constraint on continuity of ψ at r=r_0 along with normalization allows the determination of constants "A" and "B". Continuity of the derivative (or logarithmic derivative for convenience) requires quantization of energy.

Sphere with infinite square potential

In case where the potential well is infinitely deep, so that we can take V_0=0 inside the sphere and infty outside, the problem becomes that of matching the wavefunction inside the sphere (the spherical Bessel functions) with identically zero wavefunction outside the sphere. Allowed energies are those for which the radial wavefunction vanishes at the boundary. Thus, we use the zeros of the spherical Bessel functions to find the energy spectrum and wavefunctions. Calling u_{l,k} the "k"th zero of j_l, we have:

:E_l={u_{l,k}^2hbar^2over2m_0r_0^2}

So that one is reduced to the computations of these zeros u_{l,k} and to their ordering them (as illustrated graphically below) (note that zeros of "j" are the same as those of "J").

3D isotropic harmonic oscillator

The potential of a is: V(r) = frac{1}{2} m_0 omega^2 r^2. In this article it is shown that an "N"-dimensional isotropic harmonic oscillator has the energies:E_n = hbar omega( n + frac{N}{2})quadhbox{with}quad n=0,1,ldots,infty,,i.e., "n" is a non-negative integral number; ω is the (same) fundamental frequency of the "N" modes of the oscillator. In this case "N = 3", so that the radial Schrödinger equation becomes,

:left [-{hbar^2 over 2m_0} {d^2 over dr^2} + {hbar^2l(l+1) over 2m_0r^2}+ frac{1}{2} m_0 omega^2 r^2 - hbaromega(n+ frac{3}{2}) ight] u(r) = 0.Introducing:gamma equiv frac{m_0omega}{hbar}and recalling that u(r) = r R(r),, we will show that the radial Schrödinger equation has the normalized solution,

:R_{n,l}(r) =N_{nl} , r^{l} , e^{-frac{1}{2}gamma r^2}; L^{(l+frac{1}{2})}_{frac{1}{2}(n-l)}(gamma r^2),where the function L^{(alpha)}_k(gamma r^2) is a generalized Laguerre polynomial in "γr ²" of order "k" (i.e., the highest power of the polynomial is proportional to "γkr2k").

The normalization constant "Nnl" is,

: N_{nl} = left [frac{ 2^{n+l+2} ,gamma^{l+frac{3}{2 } {pi^{frac{1}{2} ight] ^{frac{1}{2 left [frac{ [frac{1}{2}(n-l)] !; [frac{1}{2}(n+l)] !}{(n+l+1)!} ight] ^{frac{1}{2 .

The eigenfunction "Rn,l(r)" belongs to energy "En" and is to be multiplied by the spherical harmonic Y_{lm}( heta, phi),, where: l = n, n-2, ldots, l_mathrm{min}quadhbox{with}quad l_mathrm{min} =egin{cases}1 & mathrm{if}; n; mathrm{odd} \0 & mathrm{if}; n; mathrm{even}end{cases}

This is the same result as given in if we realize that gamma = 2 u,.

Derivation

First we transform the radial equation by a few successive substitutions to the generalized Laguerre differential equation, which has known solutions: the generalized Laguerre functions.Then we normalize the generalized Laguerre functions to unity. This normalization is withthe usual volume element "r"² d"r".

First we scale the radial coordinate:y = sqrt{gamma}r quad hbox{with}quad gamma equiv frac{m_0omega}{hbar},and then the equation becomes:left [{d^2 over dy^2} - {l(l+1) over y^2} - y^2 + 2n - 3 ight] v(y) = 0 with v(y) = u left(y / sqrt{gamma} ight).

Consideration of the limiting behaviour of "v(y)" at the origin and at infinity suggests the following substitution for "v(y)",:v(y) = y^{l+1} e^{-y^2/2} f(y).This substitution transforms the differential equation to:left [{d^2 over dy^2} + 2 left(frac{l+1}{y}-y ight)frac{d}{dy} + 2n - 2l ight] f(y) = 0,where we divided through with y^{l+1} e^{-y^2/2}, which can be done so long as "y" is not zero.

Transformation to Laguerre polynomials

If the substitution x = y^2, is used, y = sqrt{x}, and the differential operators become:frac{d}{dy} = frac{dx}{dy}frac{d}{dx} = 2 y frac{d}{dx} = 2 sqrt{x} frac{d}{dx}, extrm{and}:frac{d^2}{dy^2} = frac{d}{dy} left( 2 y frac{d}{dx} ight) = 4 x frac{d^2}{dx^2} + 2 frac{d}{dx}.

The expression between the square brackets multiplying "f(y)" becomes the differential equation characterizing the generalized Laguerre equation (see also Kummer's equation)::xfrac{d^2g}{dx^2} + Big( (l+ frac{1}{2}) + 1 - xBig) frac{dg}{dx} + frac{1}{2}(n-l) g(x) = 0with g(x) equiv f(sqrt{x}) ,;.Provided k equiv (n-l)/2 , is a non-negative integral number, the solutions ofthis equations are generalized (associated) Laguerre polynomials: g(x) = L_k^{(l+frac{1}{2})}(x).From the conditions on "k" follows: (i) n ge l, and (ii) "n" and "l" are either both odd or both even. This leads to the condition on "l" given above.

Recovery of the normalized radial wavefunction

Remembering that u(r) = r R(r),, we get the normalized radial solution:R_{n,l}(r) =N_{nl} , r^{l} , e^{-frac{1}{2}gamma r^2}; L^{(l+frac{1}{2})}_{frac{1}{2}(n-l)}(gamma r^2).

The normalization condition for the radial wavefunction is:int^infty_0 r^2 |R(r)|^2 , dr = 1.Substituting q = gamma r^2 ,;, gives dq = 2 gamma r , dr ,; and the equation becomes

:frac{N^2_{nl{2gamma^{l+{3 over 2}int^infty_0 q^{l + {1 over 2 e^{-q} left [ L^{(l+frac{1}{2})}_{frac{1}{2}(n-l)}(q) ight ] ^2 , dq = 1.

By making use of the orthogonality properties of the generalized Laguerre polynomials, this equation simplifies to:frac{N^2_{nl{2gamma^{l+{3 over 2} cdot frac{Gamma [frac{1}{2}(n+l+1)+1] }{ [frac{1}{2}(n-l)] !} = 1.

Hence, the normalization constant can be expressed as:N_{nl} = sqrt{frac{2 , gamma^{l+{3over 2 , (frac{n-l}{2})! }{Gamma(frac{n+l}{2}+frac{3}{2})

Other forms of the normalization constant can be derived by using properties of the gamma function, while noting that "n" and "l" are both of the same parity. This means that "n+l" is always even, so that the gamma function becomes:Gamma left [{1 over 2} + left( frac{n+l}{2} + 1 ight) ight] = frac{sqrt{pi}(n+l+1)!!}{2^{frac{n+l}{2}+1 = frac{sqrt{pi}(n+l+1)!}{2^{n+l+1} [frac{1}{2}(n+l)] !},

where we used the definition of the double factorial. Hence, the normalization constant is also given by: N_{nl} = left [ frac{2^{n+l+2} ,gamma^{l+{3 over 2, [{1 over 2}(n-l)] !; [{1 over 2}(n+l)] !}{;pi^{1 over 2} (n+l+1)! } ight ] ^{1 over 2}= sqrt{2} left( frac{gamma}{pi} ight )^{1 over 4} ,({2 gamma})^{ell over 2} , sqrt{frac{2 gamma (n-l)!!}{(n+l+1)!! .

Hydrogen-like atoms

A hydrogenic (hydrogen-like) atom is a two-particle system consisting of a nucleus and an electron. The two particles interact through the potential given by Coulomb's law:

:V(r) = -frac{1}{4 pi epsilon_0} frac{Ze^2}{r}

where
* ε0 is the permittivity of the vacuum,
* "Z" is the atomic number ("eZ" is the charge of the nucleus),
* "e" is the elementary charge (charge of the electron),
* "r" is the distance between the electron and the nucleus.

The mass "m"0, introduced above, is the reduced mass of the system. Because the electron mass is about 1836 smaller than the mass of the lightest nucleus (the proton), the value of "m"0 is very close to the mass of the electron "m"e for all hydrogenic atoms. In the remaining of the article we make the approximation "m"0 = "m"e. Since "m"e will appear explicitly in the formulas it will be easy to correct for this approximation if necessary.

In order to simplify the Schrödinger equation, we introduce the following constants that define the atomic unit of energy and length, respectively,: E_ extrm{h} = m_ extrm{e} left( frac{e^2}{4 pi varepsilon_0 hbar} ight)^2 quadhbox{and}quad a_{0} = 4pivarepsilon_0hbar^2}over{m_ extrm{e} e^2.

Substitute y = Zr/a_0, and W = E/(Z^2 E_ extrm{h}), into the radial Schrödinger equation given above. This gives an equation in which all natural constants are hidden,: left [ -frac{1}{2} frac{d^2}{dy^2} + frac{1}{2} frac{l(l+1)}{y^2} - frac{1}{y} ight] u_l = W u_l .Two classes of solutions of this equation exist: (i) "W" is negative, the corresponding eigenfunctions are square integrable and the values of "W" are quantized (discrete spectrum).(ii) "W" is non-negative. Every real non-negative value of "W" is physically allowed (continuous spectrum), the corresponding eigenfunctions are non-square integrable. In the remaining part of this article only class (i) solutions will be considered. The wavefunctions are known as bound states, in contrast to the class (ii) solutions that are known as "scattering states".

For negative "W" the quantity alpha equiv 2sqrt{-2W} is real and positive. The scaling of "y", i.e., substitution of x equiv alpha y gives the Schrödinger equation::left [ frac{d^2}{dx^2} -frac{l(l+1)}{x^2}+frac{2}{alpha x} - frac{1}{4} ight] u_l = 0,quad hbox{with}quad x ge 0. For x ightarrow infty the inverse powers of "x" are negligible and a solution for large "x" is exp [-x/2] . The other solution, exp [x/2] , is physically non-acceptable. For x ightarrow 0 the inverse square power dominates and a solution for small "x" is "x""l"+1. The other solution, "x"-"l", is physically non-acceptable.Hence, to obtain a full range solution we substitute:u_l(x) = x^{l+1} e^{-x/2}f_l(x).,The equation for "f""l"("x") becomes,:left [ xfrac{d^2}{dx^2} + (2l+2-x) frac{d}{dx} +( u -l-1) ight] f_l(x) = 0 quadhbox{with}quad u = (-2W)^{-frac{1}{2.Provided u-l-1 is a non-negative integer, say "k", this equation has polynomial solutions written as: L^{(2l+1)}_{k}(x),qquad k=0,1,ldots , which are generalized Laguerre polynomials of order "k". We will take the convention for generalized Laguerre polynomialsof Abramowitz and Stegun. [Abramowitz_Stegun_ref|22|775] Note that the Laguerre polynomials given in many quantum mechanical textbooks, for instance the book of Messiah, are those of Abramowitz and Stegun multiplied by a factor ("2l+1+k")! The definition given in this Wikipedia article coincides with the one of Abramowitz and Stegun.

The energy becomes: W = -frac{1}{2n^2}quad hbox{with}quad n equiv k+l+1 . The principal quantum number "n" satisfies n ge l+1, or l le n-1.Since alpha = 2/n, the total radial wavefunction is:R_{nl}(r) = N_{nl} left(frac{2Zr}{na_0} ight)^{l}; e^{-{ extstyle frac{Zr}{na_0}; L^{(2l+1)}_{n-l-1}left(frac{2Zr}{na_0} ight),

with normalization constant:N_{nl} = left [left(frac{2Z}{na_0} ight)^3 cdot frac{(n-l-1)!}{2n [(n+l)!] } ight] ^{1 over 2}

which belongs to the energy : E = - frac{Z^2}{2n^2}E_ extrm{h},qquad n=1,2,ldots .

In the computation of the normalization constant use was made of the integral [ H. Margenau and G. M. Murphy, "The Mathematics of Physics and Chemistry", Van Nostrand, 2nd edition (1956), p. 130. Note that convention of the Laguerre polynomial in this book differs from the present one. If we indicate the Laguerre in the definition of Margenau and Murphy with a bar on top, we have ar{L}^{(k)}_{n+k} = (-1)^k (n+k)! L^{(k)}_n.] :int_0^infty x^{2l+2} e^{-x} left [ L^{(2l+1)}_{n-l-1}(x) ight] ^2 dx =frac{2n (n+l)!}{(n-l-1)!} .

References


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