If we consider ( ) 2,kr tΨ G to be the probability density for finding, at time t, particle 1 at particle k at and particle N at ,,1 … Gr …G ,kr ,Nr G a normalization condition analogous to Eq.(29.2) holds as (35.2) 1V* =ΨΨ∫ d The integral is defined in a space with 3N dimensions called the configuration space of the system, where a configuration point is given by 3N position coordinates 1 1 1, , , , , , , , , ,k k k N N N ,x y z x y z x y z… … and the volume element is NNNkkkNk dzdydxdzdydxdzdydxdddd ………… 1111 VVVV == Physical observables of the system will be represented by Hermitian operators, obtained from the classical representation in terms of the position and momentum variables as indicated by Postulate 2. Thus the many-particle Hamiltonian results from Eq.(2.32) as ( ) (∑ ∑ = ≠ ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ++∇−= k rrUtrU m H ,, 2 ˆ GGG= ) ) (35.3) where only acts on the coordinates of the kth particle, of mass The one-particle potential energy is due to external fields where this particle moves and k∇ .km( trU kk ,G( )jkkj rrU GG , gives the mutual interaction between the particles k and j. It is assumed that the position and momentum operators of different particles commute, and so do all the operators describing single-particle observables if they refer to different particles. The evolution of a many-particle system will be then formulated as a generalization of Eq.(30.6) in terms of given by Eqs.(35.1) and (35.3) HˆandΨ Ψ=∂ Ψ∂ H t i ˆ= (35.4) If the Hamiltonian (35.3) does not depend explicitly on the time, it is usual to express the time dependence of the many-particle state function by means of the evolution operator defined as ( )tUˆ ( ) ( ) ( )NkNk rrrtUtrrr G…G…GG…G…G ,,,,ˆ,,,,, 11 αΨ=Ψ (35.5) which is formally derived from Eq.(35.4) as ( ) ( ) ( )ˆˆ ˆU ti U t i HU t t t ˆα α α ∂∂ ⎡ ⎤Ψ = Ψ =⎣ ⎦∂ ∂= = Ψ It follows that (35.6) ( ) =/ˆˆ tHietU −= which is the analogue of Eq.(30.12). The evolution operator is unitary, Eq.(II.22), since ( ) ( )†ˆ ˆU t U t Iˆ= (35.7) Substituting Eq.(35.5), one obtains from Eq.(35.4) that the stationary states of the Nparticle systems are given by the energy eigenvalue equation ( ) ( )1 1ˆ , , , , , , , ,k N k NH r r r E r r rα α αΨ = ΨG G G G G G… … … … (35.8) The solution to the N-particle problem is facilitated if we take into account the symmetries of many systems of physical interest, which allow us to define constants of motion and criteria for simplifying the form (35.3) of the Hamiltonian operator. An alternative to complex explicit calculations is to associate with specific symmetries time independent unitary operators which transform the state functions into so that Eq.(35.4) gives ˆ ,U Ψ ˆ ,U Ψ ( ) ( )†ˆ ˆ ˆ ˆ ˆ ˆ ˆi U i U UH UHU Ut t∂ ∂ΨΨ = = Ψ = Ψ∂ ∂= = ˆ This is again a Schrödinger equation for the state Uˆ Ψ with the new Hamiltonian †ˆ ˆ ˆ ˆH UHU′ = If the Hamiltonian remains unchanged by the transformation, we have (35.9) †ˆ ˆ ˆ ˆ ˆ ˆor , 0H UHU U H⎡ ⎤= ⎣ ⎦ = According to the evolution equation (29.43), this implies that so that U is a constant of motion. If the unitary transformation which leaves the Hamiltonian invariant, is a function of an observable, its constancy implies that the given observable is a constant in any state of the system.
If we consider ( ) 2,kr tΨ G to be the probability density for finding, at time t, particle 1 at particle k at and particle N at ,,1 … Gr …G ,kr ,Nr G a normalization condition analogous to Eq.(29.2) holds as (35.2) 1V* =ΨΨ∫ d The integral is defined in a space with 3N dimensions called the configuration space of the system, where a configuration point is given by 3N position coordinates 1 1 1, , , , , , , , , ,k k k N N N ,x y z x y z x y z… … and the volume element is NNNkkkNk dzdydxdzdydxdzdydxdddd ………… 1111 VVVV == Physical observables of the system will be represented by Hermitian operators, obtained from the classical representation in terms of the position and momentum variables as indicated by Postulate 2. Thus the many-particle Hamiltonian results from Eq.(2.32) as ( ) (∑ ∑ = ≠ ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ++∇−= k rrUtrU m H ,, 2 ˆ GGG= ) ) (35.3) where only acts on the coordinates of the kth particle, of mass The one-particle potential energy is due to external fields where this particle moves and k∇ .km( trU kk ,G( )jkkj rrU GG , gives the mutual interaction between the particles k and j. It is assumed that the position and momentum operators of different particles commute, and so do all the operators describing single-particle observables if they refer to different particles. The evolution of a many-particle system will be then formulated as a generalization of Eq.(30.6) in terms of given by Eqs.(35.1) and (35.3) HˆandΨ Ψ=∂ Ψ∂ H t i ˆ= (35.4) If the Hamiltonian (35.3) does not depend explicitly on the time, it is usual to express the time dependence of the many-particle state function by means of the evolution operator defined as ( )tUˆ ( ) ( ) ( )NkNk rrrtUtrrr G…G…GG…G…G ,,,,ˆ,,,,, 11 αΨ=Ψ (35.5) which is formally derived from Eq.(35.4) as ( ) ( ) ( )ˆˆ ˆU ti U t i HU t t t ˆα α α ∂∂ ⎡ ⎤Ψ = Ψ =⎣ ⎦∂ ∂= = Ψ It follows that (35.6) ( ) =/ˆˆ tHietU −= which is the analogue of Eq.(30.12). The evolution operator is unitary, Eq.(II.22), since ( ) ( )†ˆ ˆU t U t Iˆ= (35.7) Substituting Eq.(35.5), one obtains from Eq.(35.4) that the stationary states of the Nparticle systems are given by the energy eigenvalue equation ( ) ( )1 1ˆ , , , , , , , ,k N k NH r r r E r r rα α αΨ = ΨG G G G G G… … … … (35.8) The solution to the N-particle problem is facilitated if we take into account the symmetries of many systems of physical interest, which allow us to define constants of motion and criteria for simplifying the form (35.3) of the Hamiltonian operator. An alternative to complex explicit calculations is to associate with specific symmetries time independent unitary operators which transform the state functions into so that Eq.(35.4) gives ˆ ,U Ψ ˆ ,U Ψ ( ) ( )†ˆ ˆ ˆ ˆ ˆ ˆ ˆi U i U UH UHU Ut t∂ ∂ΨΨ = = Ψ = Ψ∂ ∂= = ˆ This is again a Schrödinger equation for the state Uˆ Ψ with the new Hamiltonian †ˆ ˆ ˆ ˆH UHU′ = If the Hamiltonian remains unchanged by the transformation, we have (35.9) †ˆ ˆ ˆ ˆ ˆ ˆor , 0H UHU U H⎡ ⎤= ⎣ ⎦ = According to the evolution equation (29.43), this implies that so that U is a constant of motion. If the unitary transformation which leaves the Hamiltonian invariant, is a function of an observable, its constancy implies that the given observable is a constant in any state of the system.
For systems which obey quantum mechanics, the formulation of the problem of treating large numbers of particles is, of course, somewhat different than it is for classical systems. The microscopic description of the system is provided by a wave function which (in the absence of spin) is a function of the classical coordinates { q i }. The mathematical model is provided by a Hamiltonian operator H which is often obtained from the corresponding classical Hamiltonian by the replacement p i → ( ħ /i)(∂/∂ q i ). In other cases the form of the Hamiltonian operator is simply postulated. The microscopic dynamics are provided by the Schrödinger equation i ħ (∂Ψ/∂ t ) = H Ψ which requires as initial condition the knowledge of the wave function Ψ({ q i }, t ) at some initial time t 0 . (Boundary conditions on Ψ({ q i }, t ) must be specified as part of the description of the model as well.) The results of experiments in quantum mechanics are characterized by operators, usually obtained, like the Hamiltonian, from their classical forms and termed observables. Operators associated with observables must be Hermitian. In general, the various operators corresponding to observables do not commute with one another. It is possible to find sets of commuting operators whose mutual eigenstates span the Hilbert space in which the wave function is confined by the Schrödinger equation and the boundary conditions. A set of such (time independent) eigenstates, termed ψ ν ( q ), is a basis for the Hilbert space.
We now may bring the time-dependent state functions ( )tx,Ψ into the picture, taking into account, as we already did in Eq.(29.9), that only the state function but not its position coordinate x varies with time, so that ( ) ( ) t tx dt txd ∂ Ψ∂≡Ψ ,, (30.3) Upon insertion of the expression (29.24) for the expectation value, Eq.(30.2) becomes ˆˆ ˆ ˆ ˆd d fi f i f dx i f f dx i dt dt t t t ⎛ ⎞∂Ψ ∂Ψ ∂= Ψ Ψ = Ψ +Ψ + Ψ Ψ⎜ ⎟∂ ∂ ∂⎝ ⎠∫ ∫ ∫= = = = dx∗ (30.4) On the other hand, as the Hamiltonian Hˆ is a Hermitian operator, we have ( )ˆ ˆˆ ˆ ˆˆ ˆ ˆ, f ff H i Hf f H dx i dt t ∞ ∞ ∂ ∂⎡ ⎤ + = −Ψ Ψ +Ψ Ψ + Ψ Ψ⎣ ⎦ ∂ ∂∫ ∫= = x (30.5) ( ) dx t fidxHffH Ψ∂ ∂Ψ+⎥⎦ ⎤⎢⎣ ⎡ ΨΨ+ΨΨ−= ∫∫ ∗∗ ˆˆˆˆˆ = From Eqs.(30.4) and (30.5) it follows that the evolution equation (30.2) is true only if Ψ=∂ Ψ∂ H t i ˆ= (30.6) and this is a basic result called the time-dependent Schrödinger equation. This equation describes the evolution of quantum systems in terms of time-dependent state functions that obey Eq.(30.6), but neglect the time dependence of the quantum operators. Such a description is known as the Schrödinger picture. The fifth postulate states that the timedependent Schrödinger equation must always be satisfied. Postulate 5. The evolution of the state function of a quantum system is given by the time-dependent Schrödinger equation (30.6), where Hˆ is the Hamiltonian of the system.
Components of a field function are functions of coordinates and time and satisfy some definite differential equations, which are called field equations. Field equations can be given a very general variational or Lagrangian form, if one accepts, that a field as the dynamic system must be characterized by a definite Lagrangian function. For a mechanical system this function is written down as the sum over all the material points of the system. For a continuous system like a wave field, Lagrangian function is expressed by the space integral from Lagrangian function density: L(t) = ∫ Ld3x. (4.1) However, the variational principle deals not with the Lagrangian function but with an action S obtained from it by means of integration over x0 = ct S = c ∫ L(t)dt. (4.2) Therefore, noncovariant expression (4.1) in Lagrangian formalism is in fact intermediate and it is sufficient to consider the Lagrangian function density L, which depends on the four space-time coordinates x in an implicit way. Hereinafter, we shall simply call L by the Lagrangian. Analogously, the Hamiltonian function density H will be named the Hamiltonian. The Lagrangian depends on components of a field wavefunction ψk and on their first derivatives with respect to the space-time coordinates ∂ψk/∂x ν . In so doing we assign the meaning of generalized field coordinates to ψk, and the meaning of generalized field velocities to ∂ψk/∂x ν . The demand of lacking derivatives with respect to field functions above the first order in the Lagrangian is caused by the following reason. Dynamics of all known quantum fields is described either by equations of the first or the second order. On the one hand, these equations follow from the principle of a least action: δS = 0, (4.3) and under action variation the order of derivatives, entering L, is raised by one. The Lagrangian must be a relativistically covariant quantity, namely, a relativistic scalar L′(x′) = L(x). Thanks to the invariance under the space-time translations, we conclude that the Lagrangian must not manifestly contain any space-time coordinates x. Since dynamical observables (four-dimensional vector of the energy-momentum, the moment of momentum tensor, and so on) are real quantities, then the Lagrangian must be also a real quantity. We shall also impose the locality condition on the Lagrangian, that is, we assume that the Lagrangian depends only on the state of fields in the infinitely small vicinity of the point Particles, Fields, and Quantum x. So: L(x) = L(ψk, ∂ψk ∂xν ) = L (ψk(x), ψk;ν (x)) . General variation of an action, related with varying both field wave functions and of integration domain boundaries, is equal to: δS = δ ∫ L(x)dx = ∫ L(x′)dx′ − ∫ L(x)dx, (4.4) where dx = dx0dr. For L′(x′) we have the expression: L′(x′) = L (ψ′k(x′), ψ′k;ν(x′)) = L(x) + δL(x) = L(x) + δL(x) + dLdxλ δxλ, (4.5) In Eq. (4.5) we denote a form variation of corresponding quantities by the symbol δ. This way δL(x) represents varying the Lagrangian only on the account of the variation of field functions and field function derivatives: δL(x) = ∂L ∂ψk δψk + ∂L ∂ψk;ν δψk;ν , (4.6) and δψk(x) does a variation of a field function form: δψk(x) = ψ ′ k(x) − ψk(x).
Inadequate descriptions of atoms and molecules by the methods of classical physics led researchers to propose new ways to describe physical reality, giving birth to a totally new science, quantum mechanics. The methods of quantum mechanics are based on the introduction of a wave function, whose physical meaning is related to the probability of finding a certain particle, at a certain time in a volume element, positioned between x and x + dx in the x = direction, between y and y + dy in the y = direction, and between z and z + dz in the z = direction at certain time t. This wave function Ψ satisfies the Schrödinger equation, (−2mℏ2∇2+v)Ψ=EΨ,ℏ=2πh, or for short, HΨ = EΨ, where H, the Hamiltonian operator, is defined by the expression H=−2mℏ2∇2+V; h is Planck’s constant; ∇2 is the sum of the partial second derivatives with respect to x, y, and z; m is the mass of the particle; and V is the potential energy of the system. The Hamiltonian H represents the quantum equivalent of the sum of the kinetic energy and potential energy, with V being the potential energy operator and 2m−ℏ2∇2 the kinetic energy operator. Finally, E is the total energy of the system and is a number, not an operator.
The operator is contained in the expression (31.1) of the central field Hamiltonian, and this can be shown using Eq.(31.6), where the component operators are replaced by their forms (29.16) written in Cartesian coordinates, and this gives ⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂−∂ ∂⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂−∂ ∂−⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂−∂ ∂⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂−∂ ∂−= z x x z z x x z y z z y y z z yL 222ˆ == ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂−∂ ∂⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂−∂ ∂− x y y x x y y x2= It is a straightforward matter to show that this relation may also be written as ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂+∂ ∂+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂+∂ ∂+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂+∂ ∂−= 2 2 yx z xz y zy xL = ⎥⎦ ⎤⎢⎣ ⎡ ∂ ∂+∂ ∂+∂ ∂+∂∂ ∂+∂∂ ∂+∂∂ ∂+ z z y y x x xz zx zy yz yx xy 222222 222 2= We now use the expansion ( )2 2ˆ ˆr p x y z x y zx y z x y z⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂⋅ = − + + + +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠G G = ⎥⎦ ⎤⎢⎣ ⎡ ∂ ∂+∂ ∂+∂ ∂+∂∂ ∂+∂∂ ∂+∂∂ ∂+∂ ∂+∂ ∂+∂ ∂−= z z y y x x xz zx zy yz yx xy z z y y x x so that the addition of the last two relations yields ( ) ( ) 2 2 222 2 2 2 2 22 2 2ˆ ˆLˆ r p x y z x y zx y zx y z⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂+ ⋅ = − + + + + + + +⎜ ⎟ ⎜ ⎟∂ ∂ ∂∂ ∂ ∂ ⎝ ⎠⎝ ⎠G G = = ( )pripr ˆˆˆ 22 GG=G ⋅+= (31.10) It follows that the momentum can be expressed as ( ) ( ) ( ) rrr r r L r priprL r p ∂ ∂−⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂−=⎥⎦ ⎤⎢⎣ ⎡ ⋅−⋅+= 2 ˆ1ˆˆˆˆˆ1ˆ ==GG=GGG (31.11) where the radial part results from the substitution of Eq. (I.20), Appendix I, which gives 1 1ˆ sinr p i i e e e r r rθ ϕθ θ ϕ ⎛ ⎞∂ ∂ ∂= − ∇ = − + +⎜ ⎟∂ ∂ ∂⎝ ⎠ G G G G= = and hence ˆ ˆr p i r r ∂⋅ = − ∂ G G = Therefore the Hamiltonian (31.1) becomes ( )rUL rrrr r rm H + ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ −∂ ∂+⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂−= 222 2 ˆ111 2 ˆ = = and reduces to the equivalent form ( )rUL rr r rrm H +⎥⎦ ⎤⎢⎣ ⎡ −⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ 2 ˆ11 2 ˆ = = (31.12) used for a particle bound by a central force field. 31.2. EIGENVALUE EQUATIONS IN A CENTRAL FIELD Spherically symmetric systems, where the potential energy ( )rU is independent of the direction of ,rG are usually treated using the spherical coordinates ( )ϕθ ,,r related to the Cartesian coordinates as referred to in Appendix I, Eq.(I.19). As the angle θ is defined by the inclination to the z-axis, it is mathematically convenient to take the direction of the component of ( zyx ,, ) L G that is compatible with (in the sense introduced in Chapter 34) to be the z-axis. Therefore we consider the eigenvalue equations of the operators and given by ˆ ˆ, zH L 2Lˆ ( ) ( )ˆ , , , ,H r rθ ϕ ε θ ϕΨ = Ψ ( ) ( )ϕθϕθ ,,,,ˆ rLrL zz Ψ=Ψ (31.13) ( ) ( )ϕθϕθ ,,,,ˆ 22 rLrL Ψ=Ψ The appropriate form of the Hamiltonian Hˆ is conveniently written using the standard expression (I.23) of the Laplace operator in spherical coordinates as 2∇ ( )rU rrr r rrm H +⎥⎦ ⎤⎢⎣ ⎡ ∂ ∂+⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ ∂ ∂+⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ ∂ ∂−= 2 sin 1sin sin 11 2 ˆ ϕθθθθθ = (31.14) Using the transformation (I.19) from Cartesian to spherical coordinates and the chain rule for differentiation, we find sin sin sin cosx y z r r x y x y z x y y θ ϕ θ ϕϕ ϕ ϕ ϕ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= + + = − + = −∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂x so that the operator takes the form zLˆ ϕ∂ ∂=⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂−∂ ∂= ix y y x i Lz ==ˆ (31.15) Furthermore, from Eqs.(31.12) and (31.14) it follows that is in fact expressed by that part of Hamiltonian which operates on the angular variables 2Lˆ ,θ ϕ only, that is, ⎥⎦ ⎤⎢⎣ ⎡ ∂ ∂+⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ ∂ ∂−= 2 sin 1sin sin 1ˆ ϕθθθθθ=L (31.16) The energy eigenfunctions in central field, which are simultaneously eigenfunctions of and are subjected, in addition to the usual bound state requirement (30.21), to the single-valueness condition zLˆ 2ˆ ,L ( ) ( )πϕθϕθ 2,,,, +Ψ=Ψ rr since the points ϕ and πϕ 2+ are in fact the same physical point. The solution to the time-independent Schrödinger equation (31.13) in central field can be found by a process of separation of variables, yielding three ordinary differential equations in the variables ,r θ and ,ϕ respectively, to which a physical significance will be assigned. It is common practice to separate the variables in two stages, first substituting the trial solution ( ) ( ) ( ), , ,r R r Yθ ϕΨ = θ ϕ (31.17) into the energy eigenvalue equation. If the Hamiltonian is expanded in spherical coordinates, as given by Eq.(31.14), Eq.(31.13) becomes ( )2 22 2 2 21 1 1 sin2 2 sin sin 21R Y Yr U r r m R r r m Y ε θθ θ θ θ ϕ ⎡ ⎤⎡ ⎤∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞⎡ ⎤+ − = − +⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎣ ⎦∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦ = = (31.18) One side is independent of θ and ,ϕ and the other side is independent of r, so they must be separately equal to some constant, which is conveniently chosen to be equal to In this way we obtain a radial equation, given by .2/2 mL ( ) ( ) ( )rRrR mr LrU dr dr dr d rm ε=⎥⎦ ⎤⎢⎣ ⎡ ++⎟⎠ ⎞⎜⎝ ⎛− 2 2 1 2 = (31.19) where was replaced by for obvious reasons, and an angular equation, having the so-called spherical harmonics r∂∂ / drd / ( ),Y θ ϕ as solutions, which reads 2 2 ˆor sin 1sin sin 1 ==⎥⎦ ⎤⎢⎣ ⎡ ∂ ∂+⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ ∂ ∂− ϕθθθθθ= (31.20) The form of the spherical harmonics is obtained as a result of the second stage of the separation process, where we assume that ( ) ( ) ( )ϕθϕθ Φ= PY , so that Eq.(31.20) becomes 2 2 1sinsinsin ϕθθθθ θ ∂ Φ∂ Φ−=⎥⎦ ⎤⎢⎣ ⎡ +⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ ∂ ∂ === LP P (31.21) Since one side is independent of ϕ while the other is independent of ,θ they must each be equal to a constant, which we call We thus obtain the differential equations .2zL d L dϕ Φ 2− = Φ= (31.22) where again ϕ∂∂ / has been replaced by ϕdd / , and 0 sin sin sin 1 22 ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −+⎟⎠ ⎞⎜⎝ ⎛ PLL d dP d d z θθθθθ === (31.23) where θ∂∂ / has been replaced by θdd / . Therefore the energy equation (31.13) completely separates, in central field, into a radial equation (31.19) and two angular equations (31.22) and (31.23). The radial equation, giving the energy levels ε of the system, takes the form ( ) ( ) ( ) ( )rRrR mr LrUrR dr d rdr d m ε=⎥⎦ ⎤⎢⎣ ⎡ ++⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ +− 2 2 2 2 = (31.24) which can be simplified by making the substitution ( ) ( )rrRr =ξ (31.25) and using the identity 2 1r dd d r dr r rdr dr ξ ξ⎛ ⎞+ =⎜ ⎟⎝ ⎠ r Thus Eq.(31.24) reduces to ( ) ( ) ( ) ( )rr mr LrU dr rd m εξξξ =⎥⎦ ⎤⎢⎣ ⎡ ++− 2 22 = (31.26) which is formally identical to the time-independent Schrödinger equation in one dimension (30.17), provided the central force potential ( )rU is replaced by an effective potential energy of the form ( ) ( ) 2 2 2mr LrUrUeff += The repulsive potential, associated with the angular momentum, modifies the central potential as illustrated before, see Figure 3.3, for ( ) rCrU /−= . Further progress with the solution of the radial equation depends on the particular form of the potential function . We shall consider the significant example of one-electron atoms in the next chapter.