Particle in a spherically symmetric potential
Encyclopedia
In quantum mechanics
, the dynamics of a particle in a spherically symmetric potential has a Hamiltonian
of the following form:
where
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,
and 
. When this is done, the time-independent Schrödinger equation
for the system is separable
.
s of the Schrödinger equation
have the form
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
The differential equation which characterizes the function
is called the radial equation.
The spherical harmonics
satisfy
Substituting this into the Schrödinger equation
we get a one-dimensional eigenvalue equation,
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:
If subsequently the substitution
is made into
the radial equation becomes
which is precisely a Schrödinger equation for the function u(r) with an effective potential given by
where the radial coordinate r ranges from 0 to
. The correction to the potential V(r) is called the centrifugal barrier term.
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 function
s and Laguerre polynomials.
, replace everywhere E with 
). Introducing the dimensionless variable
the equation becomes a Bessel equation for J defined by
(whence the notational choice of J):
which regular solutions for positive energies are given by so-called Bessel functions of the first kind'
so that the solutions written for R are the so-called Spherical Bessel function
.
The solutions of Schrödinger equation in polar coordinates for a particle of mass
in vacuum are labelled by three quantum numbers: discrete indices l and m, and k varying continuously in 
:
where
, 
are the spherical Bessel function and 
are the spherical harmonics.
These solutions represent states of definite angular momentum, rather than of definite (linear) momentum, which are provided by plane waves
.
for 
and 
elsewhere. That is, inside a sphere of radius 
the potential is equal to V0 and it is zero outside the sphere. A potential with such a finite discontinuity is called a square potential.
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 first constraint comes from the fact that Neumann N and Hankel H functions are singular 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:
with A a constant to be determined later. Note that for bound states,
.
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):
Second constraint on continuity of ψ at
along with normalization allows the determination of constants A and B. Continuity of the derivative (or logarithmic derivative
for convenience) requires quantization of energy.
inside the sphere and 
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 
the kth zero of 
, we have:
So that one is reduced to the computations of these zeros
, typically by using a table or calculator, as these zeros are not solvable for the general case.
In the special case
(spherical symmetric orbitals), the spherical Bessel function is 
, which zeros can be easily given as 
. Their energy eigenvalues are thus:
In this article it is shown that an N-dimensional isotropic harmonic oscillator has the energies
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,
Introducing
and recalling that
, we will show that the radial Schrödinger equation has the normalized solution,
where the function
is a generalized Laguerre polynomial in γr2 of order k (i.e., the highest power of the polynomial is proportional to γkr2k).
The normalization constant Nnl is,
The eigenfunction Rn,l(r) belongs to energy En and is to be multiplied by the spherical harmonic
, where
This is the same result as given in this article if we realize that
.
Then we normalize the generalized Laguerre functions to unity. This normalization is with
the usual volume element r2 dr.
First we scale
the radial coordinate
and then the equation becomes
with
.
Consideration of the limiting behaviour of v(y) at the origin and at infinity suggests the following substitution for v(y),
This substitution transforms the differential equation to
where we divided through with
, which can be done so long as y is not zero.
If the substitution
is used, 
, and the differential operators become
The expression between the square brackets multiplying f(y) becomes the differential equation characterizing the generalized Laguerre equation (see also Kummer's equation):
with
.
Provided
is a non-negative integral number, the solutions of
this equations are generalized (associated) Laguerre polynomials
From the conditions on k follows: (i)
and (ii) n and l are either both odd or both even. This leads to the condition on l given above.
Remembering that
, we get the normalized radial solution
The normalization condition for the radial wavefunction is
Substituting
, gives 
and the equation becomes
By making use of the orthogonality properties of the generalized Laguerre polynomials, this equation simplifies to
Hence, the normalization constant can be expressed as
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
where we used the definition of the double factorial. Hence, the normalization constant is also given by
:
where
The mass m0, 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 m0 is very close to the mass of the electron me for all hydrogenic atoms. In the remaining of the article we make the approximation
m0 = me. Since me 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,
Substitute
and 
into the radial Schrödinger equation given above. This gives an equation in which all natural constants are hidden,
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 state
s, in contrast to the class (ii) solutions that are known as scattering states.
For negative W the quantity
is real and positive. The scaling of y, i.e., substitution of 
gives the Schrödinger equation:
For
the inverse powers of x are negligible and a solution for large x is 
. The other solution, 
, is physically non-acceptable. For 
the inverse square power dominates and a solution for small x is xl+1. The other solution, x-l, is physically non-acceptable.
Hence, to obtain a full range solution we substitute
The equation for fl(x) becomes,
Provided
is a non-negative integer, say k, this equation has polynomial solutions written as
which are generalized Laguerre polynomials of order k. We will take the convention for generalized Laguerre polynomials
of Abramowitz and Stegun.
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
The principal quantum number
n satisfies
, or 
.
Since
, the total radial wavefunction is
with normalization constant
which belongs to the energy
In the computation of the normalization constant use was made of the integral
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, the dynamics of a particle in a spherically symmetric potential has a Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
of the following form:
where
denotes the mass of the particle in the potential.In its quantum mechanical formulation, it amounts to solving the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
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,
and . When this is done, the time-independent Schrödinger equationSchrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
for the system is separable
Separation of variables
In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....
.
Structure of the eigenfunctions
The eigenfunctionEigenfunction
In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
s of the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
have the form
in which the spherical polar angles
Spherical angle
A spherical angle is a particular dihedral angle; it is the angle between two intersecting arcs on a sphere, and is measured by the angle between the planes containing the arcs ....
θ and φ represent the colatitude
Colatitude
In spherical coordinates, colatitude is the complementary angle of the latitude, i.e. the difference between 90° and the latitude.-Astronomical use:The colatitude is useful in astronomy because it refers to the zenith distance of the celestial poles...
and azimuthal angle, respectively. The last two factors of ψ are often grouped together as spherical harmonics
Spherical harmonics
In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...
, so that the eigenfunctions take the form
The differential equation which characterizes the function
is called the radial equation.Derivation of the radial equation
The kinetic energy operator in spherical polar coordinates isThe spherical harmonics
Spherical harmonics
In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...
satisfy
Substituting this into the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
we get a one-dimensional eigenvalue equation,
Relationship with 1-D Schrödinger equation
Note that the first term in the kinetic energy can be rewrittenThis follows because both sides of this equation can be shown by application of the product rule
Product rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...
to be equal to a third form of this operator:
If subsequently the substitution
is made into
the radial equation becomes
which is precisely a Schrödinger equation for the function u(r) with an effective potential given by
where the radial coordinate r ranges from 0 to
. 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 harmonicSpherical HarmonicSpherical Harmonic is a science fiction novel from the Saga of the Skolian Empire by Catherine Asaro. It tells the story of Dyhianna Selei , the Ruby Pharaoh of the Skolian Imperialate, as she strives to reform her government and reunite her family in the aftermath of a devastating interstellar...
s, which serves as the basis for other cases. 
(finite) for 
and 0 elsewhere, or a particle in the spherical equivalent of the square wellParticle in a boxIn quantum mechanics, the particle in a box model describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems...
, useful to describe scattering and bound stateBound stateIn physics, a bound state describes a system where a particle is subject to a potential such that the particle has a tendency to remain localised in one or more regions of space...
s in a nucleusAtomic nucleusThe nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...
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 atomHydrogen-like atomA hydrogen-like ion is any atomic nucleus with one electron and thus is isoelectronic with hydrogen. Except for the hydrogen atom itself , these ions carry the positive charge e, where Z is the atomic number of the atom. Examples of hydrogen-like ions are He+, Li2+, Be3+ and B4+...
s.
We outline the solutions in these cases, which should be compared to their counterparts in cartesian coordinates, cf. particle in a box
Particle in a box
In quantum mechanics, the particle in a box model describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems...
. This article relies heavily on Bessel function
Bessel function
In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
s and Laguerre polynomials.
Vacuum case
Let us now consider V(r) = 0 (if, replace everywhere E with ). Introducing the dimensionless variablethe equation becomes a Bessel equation for J defined by
(whence the notational choice of J):which regular solutions for positive energies are given by so-called Bessel functions of the first kind'
so that the solutions written for R are the so-called Spherical Bessel function.The solutions of Schrödinger equation in polar coordinates for a particle of mass
in vacuum are labelled by three quantum numbers: discrete indices l and m, and k varying continuously in :where
, are the spherical Bessel function and are the spherical harmonics.These solutions represent states of definite angular momentum, rather than of definite (linear) momentum, which are provided by plane waves
.Sphere with square potential
Let us now consider the potential for and elsewhere. That is, inside a sphere of radius the potential is equal to V0 and it is zero outside the sphere. A potential with such a finite discontinuity is called a square potential.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 singular 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:
with A a constant to be determined later. Note that for bound states,
.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):
Second constraint on continuity of ψ at
along with normalization allows the determination of constants A and B. Continuity of the derivative (or logarithmic derivativeLogarithmic derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formulawhere f ′ is the derivative of f....
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 inside the sphere and 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 the kth zero of , we have:So that one is reduced to the computations of these zeros
, typically by using a table or calculator, as these zeros are not solvable for the general case.In the special case
(spherical symmetric orbitals), the spherical Bessel function is , which zeros can be easily given as . Their energy eigenvalues are thus:3D isotropic harmonic oscillator
The potential of a 3D isotropic harmonic oscillator isIn this article it is shown that an N-dimensional isotropic harmonic oscillator has the energies
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,
Introducing
and recalling that
, we will show that the radial Schrödinger equation has the normalized solution,where the function
is a generalized Laguerre polynomial in γr2 of order k (i.e., the highest power of the polynomial is proportional to γkr2k).The normalization constant Nnl is,
The eigenfunction Rn,l(r) belongs to energy En and is to be multiplied by the spherical harmonic
, whereThis is the same result as given in this article if we realize that
.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 with
the usual volume element r2 dr.
First we scale
Nondimensionalization
Nondimensionalization is the partial or full removal of units from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis...
the radial coordinate
and then the equation becomes
with
.Consideration of the limiting behaviour of v(y) at the origin and at infinity suggests the following substitution for v(y),
This substitution transforms the differential equation to
where we divided through with
, which can be done so long as y is not zero.Transformation to Laguerre polynomials
If the substitution
is used, , and the differential operators becomeThe expression between the square brackets multiplying f(y) becomes the differential equation characterizing the generalized Laguerre equation (see also Kummer's equation):
with
.Provided
is a non-negative integral number, the solutions ofthis equations are generalized (associated) Laguerre polynomials
Laguerre polynomials
In mathematics, the Laguerre polynomials, named after Edmond Laguerre ,are the canonical solutions of Laguerre's equation:x\,y + \,y' + n\,y = 0\,which is a second-order linear differential equation....
From the conditions on k follows: (i)
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
, we get the normalized radial solutionThe normalization condition for the radial wavefunction is
Substituting
, gives and the equation becomesBy making use of the orthogonality properties of the generalized Laguerre polynomials, this equation simplifies to
Hence, the normalization constant can be expressed as
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
where we used the definition of the double factorial. Hence, the normalization constant is also given by
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 lawCoulomb's law
Coulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...
:
where
- ε0 is the permittivity of the vacuum,
- Z is the atomic numberAtomic numberIn chemistry and physics, the atomic number is the number of protons found in the nucleus of an atom and therefore identical to the charge number of the nucleus. It is conventionally represented by the symbol Z. The atomic number uniquely identifies a chemical element...
(eZ is the charge of the nucleus), - e is the elementary chargeElementary chargeThe elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
(charge of the electron), - r is the distance between the electron and the nucleus.
The mass m0, introduced above, is the reduced mass
Reduced mass
Reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. This is a quantity with the unit of mass, which allows the two-body problem to be solved as if it were a one-body problem. Note however that the mass determining the gravitational force is not...
of the system. Because the electron mass is about 1836 smaller than the mass of the lightest nucleus (the proton), the value of m0 is very close to the mass of the electron me for all hydrogenic atoms. In the remaining of the article we make the approximation
m0 = me. Since me 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,
Substitute
and into the radial Schrödinger equation given above. This gives an equation in which all natural constants are hidden,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 state
Bound state
In physics, a bound state describes a system where a particle is subject to a potential such that the particle has a tendency to remain localised in one or more regions of space...
s, in contrast to the class (ii) solutions that are known as scattering states.
For negative W the quantity
is real and positive. The scaling of y, i.e., substitution of gives the Schrödinger equation:For
the inverse powers of x are negligible and a solution for large x is . The other solution, , is physically non-acceptable. For the inverse square power dominates and a solution for small x is xl+1. The other solution, x-l, is physically non-acceptable.Hence, to obtain a full range solution we substitute
The equation for fl(x) becomes,
Provided
is a non-negative integer, say k, this equation has polynomial solutions written aswhich are generalized Laguerre polynomials of order k. We will take the convention for generalized Laguerre polynomials
of Abramowitz and Stegun.
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
The principal quantum number
Principal quantum number
In atomic physics, the principal quantum symbolized as n is the firstof a set of quantum numbers of an atomic orbital. The principal quantum number can only have positive integer values...
n satisfies
, or .Since
, the total radial wavefunction iswith normalization constant
which belongs to the energy
In the computation of the normalization constant use was made of the integral