Hydrogenic atoms. Chapter The Bohr atom

Transcription

1 Chapter Hydrogenic atoms One of the many goals of the astronomical spectroscopist is to interpret the details of spectral features in order to deduce the properties of the radiating or absorbing matter. It is essential to understand the internal working nature of the atom in order to deduce the physical conditions of the matter giving rise to observed spectral features. The simplest atom is hydrogen, since it comprises a single proton and a single electron. In general, hydrogenic atoms (those with a single electron) serve to clearly illustrate atomic structure while avoiding the complication of multiple interacting electrons. In addition to neutral hydrogen, H, hydrogenic atoms include the ions He +, Li +, Be +3, etc. We begin with the semi-classic Bohr model. We then briefly address the wave nature of matter and some basic quantum mechanics, including wave functions. The Schrödinger equation and resulting stationary states of hydrogen are presented. We then incorporate electron spin, spin-orbit coupling, and relativistic energies, all of which lead to the Dirac model and fine structure. We complete our discussion of bound states with radiative corrections (vacuum polarization) and isotope shifts. Finally, we present the formalism and wave function for unbound states.. The Bohr atom The hydrogenic atomic model proposed by Niels Bohr (93) is a semi-quantum mechanical approach to what is essentially classical physics. The Bohr model is based upon two postulates:. The electron moves in circular planer orbits about the nucleus such that the total angular momentum of the atom (electron plus nucleus) must be an integer number of Planck s constant, h.. The orbit of an electron is a stationary state; an atom emits or absorbs electromagnetic radiation oy when its electron changes states; the energy

2 CHAPTER. HYDROGENIC ATOMS of the photon equals the energy difference of the initial and final states. Bohr s first postulate is that the total angular momentum is a constant of motion. It is expressed (mn Ldφ = ωr N + m e ωre) dφ = nh, (.) where L is the total angular momentum of the atom, which is purely azimuthal (the orbit is confined to a plane). The geometric configuration is illustrated in Figure.a, where r n = r N + r e for a nucleus of mass m N and electron of mass m e orbiting about a common center of mass with angular frequency ω = dφ/dt. Since the orbits are postulated to be circular, ω, r e, and r N are constants. Applying m N r N = m e r e, we have L = ˆmωr n = n h, (.) where h = h/π, and where ˆm = m e /( + m e /m N ), is the reduced mass of the electron. Assuming a Coulomb potential, V (r) = Ze /r n, where Ze is the (a) (b) n= 8 n= 5 r N r n r e ω = dφ/ dt m e n= 4 n= 3 n= n= m N mass center r 4 E E 4 bound free r n n = Figure.: (a) The geometric configuration of the Bohr model in arbitrary state n illustrating the relationships between r n, r N, r e, and the center of mass (offset from the nucleus is exaggerated). (b) A schematic of the Bohr model with the relative sizes of the first five orbits in units of the Bohr radius, a. The limit n = is not to scale. When electrons move from one bound orbit to another, photons with energies given by the energy difference of the orbits are absorbed (upward transition) or emitted (downward transition); a bound-bound transition between n = 4 and n = is shown, for which the photon has energy E γ = E 4 E. Ionization from n = is also shown as a bound-free transition. a

3 .. THE BOHR ATOM 3 charge of the nucleus, the total energy of the atom in state n is where E n = Ze r n, r n = h n ˆme Z = a n Z ( me ), (.3) ˆm a = h m e e = cm =.5977 Å, (.4) is the Bohr radius for an infinite mass nucleus, and where the ratio m e / ˆm explicitly accounts for the reduced electron mass. For hydrogen ˆm/m e = For helium ˆm/m e = We also can define the reduced Bohr radius â = (m e / ˆm)a. Substituting r n into E n (Eq..3), we obtain the binding energy for orbit n, E n = ˆme4 Z h n = e Z ˆm a n = m ec m e (Zα) ( ) ˆm, (.5) where the last form provides the energy in terms of the rest energy of the electron, m e c, and the fine structure constant α = e / hc. We define the Rydberg constant, R = m ee 4 h = e a = m ec α =.7987 erg = ev, (.6) for an infinite mass nucleus, i.e., ˆm = m e. In general, for a hydrogenic atom with nuclear charge Ze, We can thus rewrite Eq..5 in the simplified form n m e R Z = Z ˆm m e R. (.7) E n = R Z n. (.8) For hydrogen, Z = and ˆm/m e =.99946, we obtain R H = ev... Energy Structure In general, atomic energy structure is elucidated using various quantities, including the transition energy, the excitation energy and potential, the ionization energy, and the ground state ionization potential. Here, we briefly define each of these terms. Transition energy According to Bohr s second postulate, the electron is induced to move from a lower bound orbit to a higher orbit via the absorption of a photon. Alternatively, if the electron spontaneously transitions from a higher orbit to a lower orbit, a photon is emitted. When discussing transitions, we adopt the general convention

4 4 CHAPTER. HYDROGENIC ATOMS of denoting an orbit lower than orbit n as n, though contextual exceptions to the convention will arise. An example bound-bound transition between n = and n = 4, is illustrated in Figure.. The photon energy, E γ, is the difference between the upper level and lower level binding energies, E γ = E n n = E n n E n E n (.9) = hνn n = hνn n = hc/λn n = hc/λ n n where νn n and νn n are the frequencies [Hz] of the emitted and absorbed photons, which correspond to the wavelengths λ n n and λ n n, respectively. Note that the transition energy for a upward transition, En n is equivalent to the transition energy for a downward transition, En n. Invoking Eq..8, the transition energy between states n and n is written En n = En n = R Z Excitation energy and potential [ n ] n. (.) The excitation energy is the excess stored internal energy of the atom when the electron orbits in some state n above the ground state (n = ). The energy required to raise the internal energy of the atom to some state n above the ground state is called the excitation potential. Both these quantities are obtained from Eq.. according to [ χ n = E n E = R Z ] n. (.) The excitation energy and excitation potential is commoy expressed in units of electron volts [ev]. Note that the excitation energy of the ground state is always ev and that χ n increases with increasing n. An alternative method for computing the energy of emitted or absorbed photons and/or the transition energy is to employ the difference of the excitation energies, Ionization energy hνn n = hνn n = En n = En n χ n χ n. (.) The minimum energy required to liberate, or free, an electron originally bound in orbit n is called the ionization energy and is expressed where the superscript i denotes ionization. E I n = E E n = E n = R Z n, (.3) For quantities such as the transition energy, etc., we adopt the convention that the subscript denotes the initial state and the superscript denotes the final state. For example, En n En n ( ), where the arrow indicates the direction of the transition, in this case emission.

5 .. THE BOHR ATOM 5 Incident photons with energies E γ = hν = hc/λ E I n can ionize the atom from state n. The photon frequencies and wavelengths must satisfy ν R Z hcn, or λ. (.4) hn R Z Photons with E γ = E I n will liberate the electron from the atom, but the electron will have no kinetic energy in the center of mass frame of the atom. For E γ > E I n, the additional energy above and beyond the ionization energy imparts kinetic energy (/)m e v to the free electron. Ground state ionization potential The ground-state ionization potential is the minimum energy required to ionize the atom from the ground state, n =. As such, it is equal to the ionization energy of the ground state. For hydrogenic atoms, the ground-state ionization potential is χ I = E E = E = R Z. (.5).. Grotrian diagrams and spectra In Table., expressions for the orbital energy, E n, transition energy, E n n, excitation potential, χ n, ionization energy, E I n, and the ground state ionization potential, χ I, are listed for hydrogenic atoms. Note that the ionization energy from level n is simply the negative of the orbital binding energy, and that the ionization potential from the ground state is simply the Rydberg constant. Table.: Hydrogenic Atom: Energies and Potentials Energy/Potential Symbol Expression Equation Energy E n = R Z /n.8 Transition Energies En n,en n = R Z(/n /n ). Excitation Potential χ n = R Z ( /n ). Ionization Energy En I = R Z /n.3 Ionization Potential χ I = R Z.5 Energy diagrams called Grotrian diagrams (after Walter Grotrian, who first introduced them in 98) are a useful visual aid for elucidating the energy structure and transition energies of the atom. Grotrian diagrams of neutral hydrogen (H ) and the singly ionized helium ion (He + ) are shown in Figures.a and.b. The left axes are the excitation energies (Eq..) and the right axes are the binding energies (Eq..8). Note that the relative energy structures for the hydrogenic atoms are in direct proportion by the scaling of Z ( ˆm/m e ). The excitation energy for the ground state is χ =, which corresponds to the binding energy E = R Z. As n increases above the ground state, the excitation potential increases (always positive), and reaches a maximum

6 6 CHAPTER. HYDROGENIC ATOMS χ n [ev] Hydrogen (H ) E n [ev] (3.464) n=(.36) 3 (3.54) n=5 (.544) (.75) n=4 (.85) (.88) n=3 (.5) (.) (.) Lyα 5 Lyβ 5 Lyγ 97 Lyδ 949 Limit Hα Hβ Hγ Limit n= ( 3.4) (a) n=( 3.598) χ n [ev] Singly Ionized Helium (He + ) E n [ev] (53.874) n=(.544) (5.4) n=5 (.77) 5 (5.7) n=4 ( 3.4) (48.37) n=3 ( 6.46) (4.84) (.) α 33 β 56 γ 43 δ 37 Limit Limit α β γ n=( 3.64) (b) n=( 54.48) Figure.: Grotrian energy diagrams for neutral hydrogen (panel a) and singly ionized helium (panel b). The left axes gives the excitation energy, χ n [ev], and the right axes give the electron energy, E n [ev]. The horizontal lines represent the respective binding energies for a given n. (a) For H, the first four transitions are shown as vertical lines for the n =, or Lyman series, as is the Lyman limit. Also shown are the first three transitions of the n =, or Balmer series, as is the Balmer limit. The corresponding wavelengths of the photons are provided (in angströms) for each transition. (b) For He + ; the same diagram as for H, except that the energies are all scaled by Z ( ˆm/m e). (The energies in parenthesis are computed from the expressions listed in Table.; they do not include relativistic and higher-order corrections and are not in full agreement with observations.) χ = E = R Z. The binding energy of the electron is always negative; as n increases, the binding energy decreases in magnitude (increases, getting less negative) until it reaches a maximum value of zero at n =. Selected transitions are shown as vertical lines showing the electron transition; each is labeled with the photon wavelength corresponding to the transition energy. For each lower state, n, there is a series of energy differences between E n and the successive higher states. Consider H ; for all transitions with lower state n =, the series is called the Lyman series. For lower state n =, it is called the Balmer series. For the Lyman series, it is customary to denote the first transition in the series ( n = ) as Lyα, the second transition in the series ( n = ) as Lyβ, etc., in order of the Greek alphabet. For large n, the transitions are simply numbered by n, i.e., Ly8, etc. The notation for the Balmer series follows the same convention as the Lyman series except that H is the prefix, i.e., Hα ( n = ), Hβ ( n = ), Hγ ( n = 3), etc. The ionization threshold ( n= ), is known as the Lyman limit, whereas the series limit n= is called the Balmer limit. The Lyman series gives rise

7 .. THE BOHR ATOM 7 Hε Hδ Hγ Hβ Hα H (n=) Balmer Lyβ Lyα H (n=) Lyman β α ε δ γ β α He + (n=) He + (n=) Wavelength, Ang Figure.3: The spectra of the neutral hydrogen (H ) Lyman series and Balmer series (top two panels) and the corresponding spectral series for single ionized helium (He +, bottom panel). The first transitions are shown for each, including the series limits, which are shown as dotted lines. Note that, for the Bohr model, there are virtually identical transition wavelengths in the H Lyman series and the He + Balmer series. to ultraviolet spectral lines (9 6 Å), whereas the Balmer series gives rise to optical spectral lines ( Å). The H Lyman and Balmer series, and the corresponding n = and n = series for He + (sometimes referred to as the helium Lyman and helium Balmer series, respectively), are illustrated in Figure.3 as a function of transition wavelength. The Lyman and Balmer series of He + lie in the far ultraviolet (8 34 Å for n = ; 9 64 Å for n = ). Note that the spectral series converges to the series limit, which is indicated with the dotted (for presentation purposes, oy the first transitions are shown; thus there is a an artificial gap between the last shown transition and the series limit). Note that several of the spectral lines in the helium Balmer series have virtually identical wavelengths to those in the hydrogen Lyman series. It would seem that this causes confusion for identifying these spectral features in astronomical spectra. In practice, this is not a problem; in astrophysical conditions where H persists primarily in the n = ground state, helium is mostly neutral and what little He + there is will be in the ground state and thus not exhibit helium Balmer series transitions.

8 8 CHAPTER. HYDROGENIC ATOMS. The Schrödinger atom The Bohr model has several short comings: () the theoretical formalism is not a full quantum mechanical treatment of the atom, but a semi-classical model based upon an ad hoc hypothesis of quantized angular momentum, () there is no explanation for why the orbiting (accelerating) electron does not radiate and lose energy (the steady-state orbits violate fundamental principles of electromagnetic theory), (3) the predicted spectrum of hydrogen does not match the observed spectrum in that the transition energies are not precisely correct nor is the observed fine splitting of lines predicted, and (4) the model cannot account for the relative intensities of the different spectral lines in the observed spectrum. The key to further understanding the atom derives from the particle-wave duality of matter. The fundamental principle of the wave nature of matter, as proposed by Louis De Broglie in 94, is that the motion of a particle with mass m and velocity v is equivalent to a propagating wave with wavelength λ = h/mv. (.6) De Broglie s hypothesis suggested that the motion of an electron in an atom be investigated using the formalism of wave mechanics. One characteristic of waves is that they constructively or destructively interfere with themselves. If a wave is spatially bound in the interval x, it interferes with itself until it sets up a resonance standing wave pattern with nλ = x, where n is an integer. These considerations lead directly to the Bohr orbits. A wave confined to a circular path is periodically bound to a pathlength πr. This pathlength must equal an integer number of wavelengths, πr = nλ = n(h/mv), yielding πr = nh/mωr, or the electron wave will destructively interfere with itself. We immediately obtain mωr = n h, the consequence of Bohr s first postulate (Eq..). However, there is no a priori reason to assume circular orbits confined to a plane... The Schrödinger equation With three degrees of freedom for the electron, the wave model yields a wide variety of three dimensional orbital configurations that satisfy the condition of constructive interference. The effect is that the radius and angular momentum of bound electrons are quantized into discrete allowed stationary states. The single assumption of the wave model of the atom is that the particles obey the laws of wave mechanics, i.e., the particles are described by a wave function, Ψ(r, t) = Ψ(r,θ,φ,t), which is the solution to a wave equation. In 95, Erwin Schrödinger derived what is now commoy referred to as Schrödinger s equation, i h Ψ(r,t) = HΨ(r,t), (.7) t where the classical, non-relativistic Hamiltonian is the energy operator H = h ˆm + V (r,t), (.8)

9 .. THE SCHRÖDINGER ATOM 9 and where ( h / ˆm) is the kinetic energy operator, V (r,t) is the potential specific to the system under consideration, and ˆm is the reduced electron mass. Since kinetic energy can be written in terms of the momentum, p / ˆm, the kinetic energy operator can also be written in terms of the square of the momentum operator, p = i h. Written out, the Hamiltonian is ) L H = h ˆm [ r r ( r r h r ] + V (r,t), (.9) where L = r p is the angular momentum with p being the instantaneous linear momentum, and where L is the angular momentum operator [ ( L = h sin θ ) + ] sinθ θ θ sin θ φ. (.).. Properties of the wave function The resulting form of the wave function depends on the exact form of the potential, V (r, t). However, the wave function must be single valued, piece-wise smooth, and bounded. For atomic bound states, the wave function is time varying complex eigenfunction of three eigenvalues, n, l, and m, which are a consequence of the periodic boundary conditions in the azimuthal (l; φ π) and polar (m; θ π) spatial coordinates. The boundary condition on the radial spatial coordinate is that the wave function must vanish as r. Denoting the eigenstate m as a generic state n, the time-dependent wave equation can be written as a spatial part and a time-varying part, Ψ n (r,t) = ψ n (r)exp { i(e n / h)t}, (.) where ψ n (r) satisfies the time-independent Schrödinger equation, ] Hψ n (r) = [ h ˆm + V (r) ψ n = E n ψ n (r). (.) Eq.. shows that when the Hamiltonian operates on a wave function having eigenstate n, the result is the product of the energy for that eigenstate, E n, and the wave function, i.e., Hψ n (r) = E n ψ(r), where E n is called the eigenenergy. For bound states, i.e., E <, oy certain value of E n are allowed. Being periodically bounded, the ψ n (r) are complex functions. The E n are real and appear in the time-varying part of the wave functions, exp { i(e n / h)t} = cos(e n / h)t + isin(e n / h)t, (.3) which reflects the oscillation of the amplitude of ψ n (r) at arbitrary position r = (r,θ,φ) with frequency ω n = E n / h. That is, ψ n (r) is the wave function amplitude of the eigenstate at r and this amplitude oscillates at a frequency in proportion to the eigenenergy of the state. The wave functions are orthonormal and obey the condition ψ n (r) ψ n (r) ψn (r)ψ n(r)dv = δ n n, (.4)

10 CHAPTER. HYDROGENIC ATOMS where δ n n is the Dirac δ-function, which evaluates to δ n n = when n = n and δ n n = when n n. The expression ψ n (r) ψ n (r) is the Dirac notation for the integral over all space, where ψ n(r) = ψ n (r) and ψ n (r) = ψ n (r). Max Born postulated that the square of the wave function provides the probability of finding the electron at r at time t within the volume element dv centered on the nucleus, where the volume element is dv = r drdω = r dr sinθ dθ dφ. Thus, the probability density of the particle position is P n (r)dv = ψ n(r)ψ n (r)dv. (.5)..3 The Schrödinger wave function For hydrogenic atoms, a Coulomb potential, V (r) = Ze /r, is applied, where Ze is the charge of the nucleus. For E (unbound states), the effective potential yields a continuum of acceptable eigenfunctions. For E < (bound states), oy certain acceptable eigenfunctions and eigenvalues solve the Schrödinger equation. As such, the wave function is forced to have an integer number of nodes in the radial, and in both the azimuthal and polar coordinates. The bound-state eigenfunctions are of the form ψ m (r,θ,φ) = R (r)y lm (θ,φ), (.6) where R (r) is the radial component, and where Y lm (θ,φ) is the spherical harmonic function, which governs the azimuthal and polar modulations of R (r). The integers n, l, and m are the eigenvalues, called quantum numbers. The principle quantum number is n = n r + l +, (.7) where n r is the radial quantum number with n r =,,,...,. The value of n r gives the number of radial nodes of R (r). The eigenvalue l is the angular momentum quantum number; it provides the number of nodes in the azimuthal direction of the Y lm (θ,φ) functions. Examining Eq..7, we see that for a given n, l is a maximum for n r = and is equal to n. Since the minimum value of n is n = and n r n, the minimum value is l =. We have the condition l =,,,...,n. (.8) The eigenvalue m is known as the magnetic quantum number; its absolute value provides the number of nodes in the polar direction of the Y lm (θ,φ) functions. It is interpreted as the z-axis projection, L z of the orbital angular momentum, L for state m. These projections are limited by the value of l, so that m = l, l+,...,,...,l,l. (.9) A schematic of the allowed quantum numbers for n =,, and 3 is presented in Figure.4. In Table., we summarize the quantum numbers.

11 .. THE SCHRÖDINGER ATOM principle n,,3,... 3 angular momentum l,,,..., n magnetic m l,...,,...,+l Figure.4: A schematic chart of the allowed quantum numbers, m, based upon the spatial boundary conditions and wave interference properties of an orbiting electron. Table.: Schrödinger model bound state quantum numbers Description Allowed ranges Number of states n principle n l angular momentum l n n m magnetic l m l l + The radial component The full general expression for the radial component is, [ (Z ) 3 (n l )! R (r) = n[(n+l)!] 3 â ] / { ρ l exp ρ } where L l+ n+l (ρ) is the associated Laguerre polynomial, L l+ k= L l+ n+l (ρ), (.3) n l n+ (ρ) = ( ) k+ [(n+l)!] ρ k (n l k)!(l++k)! k!, (.3) and where ρ = Z m e r, â = a nâ ˆm. (.3) The interpretation of the wave function rests with Born s postulate as expressed in Eq..5. Born s postulate is that the meaning of the wave function is probabilistic; the amplitude of the wave function squared provides the probability of finding the particle in a given volume element. The quantity R (r) is the radial component of the probability density of the electron with state. When multiplied by the electron charge, e R (r) dr provides the radial charge density in the interval r r + dr. Multiplying the radial component of the probability density by r, we obtain the radial distribution function, D (r)dr = r R (r) dr, (.33)

12 CHAPTER. HYDROGENIC ATOMS.5 n = l = D (r) R (r) n = l = n = l =.5.5 n = 3 l = n = 3 l = n = 3 l = r / a µ r / a µ r / a µ Figure.5: The radial component to the charge density, e R (r), is plotted as dashed curves for atomic hydrogen for various selected states. Plotted as solid curves is the radial distribution function, D (r); the quantity D (r)dr provides the probability of finding the electron in state at radial distance r r + dr. The vertical scales are arbitrary. which provides the probability that the electron will be found between the distance r and r + dr from the nucleus. Note that this follows from R (r)r (r)r dr =. (.34) In Figure.5, the radial component to the charge density, e R (r), and the radial distribution function, D (r), are plotted for n =,, and 3 and their allowed l states. The azimuthal and polar components The Y lm (θ,φ) account for the azimuthal and polar periodicity of the wave properties of the bound electron (the constructive interference). The spherical harmonic functions are written where the azimuthal component is Y lm (θ,φ) = Θ lm (θ)φ m (φ), (.35) Φ m (φ) = π exp {imφ} = π [cos φ + isin φ]. (.36)

13 .. THE SCHRÖDINGER ATOM 3 To be single valued, the functions must obey Φ m (mπ) = Φ m (), which restricts m to integer values, m =, ±, ±, ±3,... The polar component is [ ] / (l+)(l m)! ( ) m Pl m (cos θ) m Θ lm (θ) = (l+m)! ( ) m Θ l m (θ) m <, (.37) where Pl m (cos θ) are the associated Legendre polynomials. Written in full, the spherical harmonics are [ ] / (l+)(l m)! ( ) m Pl m (cos θ)exp {imφ} m Y lm (θ,φ) = 4π(l+m)! ( ) m Y lm(θ,φ) m <, (.38) which obey the orthonormal property Y l m (θ,φ)y lm(θ,φ)dω = δ l lδ m m. (.39) l= l= n=3 x θ φ z y l= m= m= m= m= m= Figure.6: The spherical harmonic components to the wave functions, Y lm (θ, φ), for n = 3. The wave function, ψ m (r, θ, φ) = R (r) Y lm (θ, φ), has the general azimuthal and polar pattern shown, but with the amplitudes modulated by the radial component, R (r) (not shown). The azimuthal component, Φ m (φ), modulates the amplitude of the wave function in rotation about the z axis. The polar component, Θ lm (θ), modulates the amplitude of the wave function in proportion of the angle between the radial vector and the z axis. In locations where the Y lm (θ,φ) have nodes, the wave function is vanished. For n = 3, examples of the l =,, and and the m =,,,+,+ spherical harmonic functions are illustrated in Figure.6.

14 4 CHAPTER. HYDROGENIC ATOMS Probability distributions The probability distributions, P m (r) (see Eq..5) are illustrated in Figure.7 as labeled by their states (n,l,m). Oy principle states n =, n = 3, and n = 4 are shown for (l =,m = ), (l =,m = ), and (l =,m = ). The view is from the positive y axis looking at the zx plane. Note that there are n r = n (l + ) radial nodes; the amplitude of the radial component to the wave function modulates the amplitude of the harmonic functions. Note that the various states exhibit substantial spatial overlap. Figure.7: The probability distributions, P m, for principle quantum states n =, n = 3, and n = 4 are illustrated for the l, m states (l =, m = ), (l =, m = ), and (l =, m = ). Each distribution is labeled (n, l, m) and are viewed in the zx plane, where the z axis is vertical and the x axis is horizontal. These functions illustrate that the view of an electron as an orbiting particle must be abandoned in favor of the view that the charge density of the electron is distributed spatially in proportion to the square of the wave function. This is the wave mechanics interpretation of the stationary bound state.

15 .. THE SCHRÖDINGER ATOM 5 Angular momentum and multiplicity of states Whereas the constant of motion for the Bohr model is the azimuthal orbital angular moment, the constants of motion for the Schrödinger wave model are the total angular momentum vector, L, and the azimuthal component (z axis projection) of the angular momentum, L z, where, L = L = (L x + L y + L z) / = l(l+) h, L z = m h, (.4) where L z is the azimuthal or z component in Cartesian coordinates. (a) m (b) m 5/ (c) z 3/ 3/ / / 6 / L / / / 3/ 5/ Figure.8: A schematic of the angular momentum vector, L, and the z axis projection, L z. These two quantities are the constants of motion for the Schrödinger model of the atom. (a) The l = case for which L = h, showing the three possible allowed L z projections m =,, + in units of h. (b) The l = case, for which L = 6 h, showing the five possible allowed L z projection m =,,, +, + in units of h. (c) The precession of L around the z axis is illustrated for l = with m = + h. The angular momentum vector and its z projection are illustrated in Figure.8a for l = and in Figure.8b for l =. The vector L processes about the z axis as illustrated in Figure.8c. Its projection on the z axis can be any one of l + allowed projections. For l =, the magnitude of L is h and the possible projections are h,,+ h; for l =, the magnitude of L is 6 h and the possible projections are h, h,,+ h,+ h. The number of m stationary states that an electron can occupy for state is g = l +, which is called the multiplicity of states. The total multiplicity of states for level n is n n g n = g = (l + ) = n. (.4) l= l=

16 6 CHAPTER. HYDROGENIC ATOMS..4 Energy structure In general, the expectation value, y of a given operator, Y, is given by the integral over all space y = Ψ(r,t) Y Ψ(r,t) Ψ (r,t)y Ψ(r,t)dV. (.4) This principle will be instrumental in determining the energy structure and the relative rates of different electron transitions. The expectation value of the energy for eigenstate m is obtained using the Hamiltonian operator (Eq..8). We have E = Ψ m (r) H Ψ m (r), (.43) where in ψ m (r) is given by Eq... Performing the integration, we find that the energies of the stationary states, ψ m (r), depend oy on the principle quantum number, n, and that the energy for state n is identical to that of the Bohr model (Eq..5), namely E n = ˆme4 h Z n = R Z n. (.44) As shown in Eq..4, the multiplicity of states for level n is g n = n for the Schrödinger model, i.e., there are n possible lm states that can be occupied by an electron in level n. Eq..44 indicates that the energy of state n is independent of the lm state of the electron. The lm states are energy degenerate. Consequently, the non-relativistic wave mechanics approach to describing the bound stationary states of electrons in the hydrogen atom yields the same spectral features and the same energy structure as the Bohr model (..). However, we note that the physical interpretations of the energy states and transitions are very different for the wave model of the atom as compared to the Bohr model...5 Selection rules and Grotrian diagram The transition energies in the Schrödinger model are identical to those of the Bohr model, namely, En n = En n = E n E n. (.45) However, not all m states can transition to all n l m states, and visa versa. This is a consequence of conservation of angular momentum and polarization of the electric field of photons. The details are presented in.. Here, we simply state the so-called selection rules for the Schrödinger transitions. Selection rules The selection rules for which transitions are permitted are derived from the so-called dipole approximation (see Chapter ), which yields transition probabilities proportional to the dipole moment between two stationary states m

17 .. THE SCHRÖDINGER ATOM 7 and n l m. The non-vanishing terms of the dipole moment between stationary states provide the selection rules. Employing m = m m, and l = l l, we simply present the Schrödinger model dipole selection rules in Table.3. Table.3: Schrödinger model dipole selection rules Transition Rule l ± m,± Transitions in the Schrodinger model are restricted to those in which the angular momentum quantum number changes up or down according to l = l± and in which the magnetic quantum number does not change, i.e., m = m, or changes up or down according to m = m ±. Whether l = l + or l = l is independent of whether the transition is upward (absorption) or downward (emission). The same applies for the change in m. In reality, non-permitted transition do occur; these are called forbidden transitions. The transition probabilities and selection rules will be discussed in... Grotrian diagram Because the transition energies depend oy on n and are degenerate with l and m, and because the energies are identical to those of the Bohr model, the energy structure and predicted spectrum from the Schrödinger model are identical to those of the Bohr model (see Table. and Figure.3). However, because of the l and m energy degeneracy, more than one of several permitted transitions from upper state n to lower state n can give rise to spectral lines with the same frequency or wavelength. For this reason, a spectroscopic notation for the initial and final states of each transition was developed. Each n level is called a shell, and historically, observational spectroscopists named the n = level the K shell, the n = level the L shell, the n = 3 level the M shell, the n = 4 level the N shell, etc. For a given shell, n, each angular momentum state, l, is called a subshell. The subshells are also described by spectroscopic notation, s for l =, p for l =, d for l =, and f for l = 3. These respectively stand for sharp, principle, diffuse, and fundamental. A given subshell,, for example n = with l =, is denoted s. For n = and l =, the state is denoted p. The magnetic quantum number is not included in the spectroscopic notation. As an example, there is oy a single s state (m = ) and there are three p states (m =,,+). For the Schrödinger model, the permitted Lyα transition is denoted p s, whereas the Lyα transition s s is forbidden by the dipole selection rules. In Figure.9, we present the Grotrian diagram for n =,, and 3. The downward permitted transitions for Lyα (p s) and Hα (3p s, 3s p, and 3d p) are indicated and labeled using spectroscopic notation. Note that there

18 8 CHAPTER. HYDROGENIC ATOMS ev s p d ev.88 n = 3 3s p 3p s 3d p.5. n = 3.4 p s. n = s Figure.9: Grotrian diagram for the first three principle n levels for the Schrödinger model. The l states increase from left to right, where s denotes l =, p denotes l =, and d denotes l =. The excitation energy is given on the left and the binding energy is given on the right in units or electron volts [ev]. The dipole selection rules dictate that permitted transitions obey l = ± and m =, ± (m states are not shown in the Grotrian diagram). The permitted p s Lyα transition and Hα multiplet (3p s, 3s-p, and 3d p) are illustrated. Not illustrated is the permitted 3p s Lyβ transition. p d are three possible channels by which an Hα photon can be emitted or absorbed. However, the transition probabilities of these three transition (m states not shown) depend upon the initial and final lm states. Thus, given an ensemble of hydrogen atoms, the final emission or absorption intensity of the Hα spectral line will be a weighted average of the transition probabilities of the three transitions. This will be discussed further in...3 The Dirac atom As revolutionary and powerful as the Schrödinger model is, it still has substantial shortcomings: () the Hamiltonian accounted for non-relativistic energies oy, () as with the Bohr model, the predicted spectrum of hydrogen does not match the observed spectrum, including mismatches with the observed transition energies and the lack of a prediction of fine splitting of the lines, (3) a property known as electron spin had been discovered and experiments demonstrated that it played a role in the energy structure of atoms; spin was not incorporated into the Schrödinger model.

19 .3. THE DIRAC ATOM 9.3. Spin In 9, Otto Stern and Walther Gerlach discovered that silver atoms beamed through a magnetic field were deflected into two discrete beams (they were predicting oy a single deflection in proportion to the magnetic moment of the atom). In 95, Samual Gouldsmit and George Uhlenbeck showed that when atoms are placed in a magnetic field, the spectral lines split in proportion to the field strength (the phenomenon is now called the Zeeman effect). These results suggested that the electron has an intrinsic magnetic moment proportional to an intrinsic angular momentum, or spin. The experiments indicate spin has a multiplicity of states g s = s + with s = /. Thus, g s = ; there are two spin states. The result is that the total magnetic moment of the atom is due to the combined magnetic moment of the electron orbit and the intrinsic magnetic moment of the electron. Thus, we introduce two additional quantum numbers, s = / and m s = ±/, and the spin wave function, denoted χ ms (which obeys all orthonormal properties). When m s = +/, the function χ +/ is referred to as a spin up state, and when m s = /, the function χ / is referred to as a spin down state. For clarification, the magnetic quantum number (z component of the orbital angular momentum) will be written m l in the context of the Dirac model. A schematic of the electron spin vector and its precession is illustrated in Figure.. Similar to the orbital angular momentum vector, L, which has magnitude L = l(l + ) h with z component L z = m l h, the electron spin vector, S, has magnitude S = s(s + ) h = 3/4 h with z component magnitude S z = m s h = ±(/) h. (a) m s (b) +z 3/ / { m 3/4 s = +/ 3/4 / / Figure.: (a) The spin vector has magnitude p 3/4 h and precesses about the z axis with projection m s h, where m s = ±/. (b) Schematic of the S vector precession for the spin up state.

20 CHAPTER. HYDROGENIC ATOMS.3. Spin-orbit coupling The orbital angular momentum and the spin combine such that the z component of the orbital angular momentum, L z, no longer retains its status as a constant of motion. The new constants of motion are the total angular momentum (orbital plus spin), denoted J, and the z component of J, i.e., J z. This new quantum number is the result of spin-orbit coupling, J = L + S. (.46) For hydrogenic atoms, the magnitude of J is J = j(j + ) h, where the total angular momentum quantum number is given by j = { s l = l ± s l, (.47) where s = /, and where the z component is in multiples J z = m j h, with m j = m l + m s = j, j+,...,j,j. (.48) Since l takes on n possible integer values from l = to n, we see that j takes on n possible values, j = /,3/,...,n /. Note that for a single electron, j l and m j must always be a half integer. orbital angular momentum l,,,..., n total angular momentum = j { / l l + s l > / / 3/ 3/ 5/ z axis projection m j +/ j, j+,...,+j / +/ / +3/ / +3/ 3/ +/ +5/ / 3/ +/ 3/ +/ 5/ / +3/ Figure.: A schematic chart, analogous to Figure.4, of the total angular momentum quantum numbers, j, and z projection, m j, for the l =,, and states of the n = 3 level due to spin-orbit coupling. Note there is n = 3 unique j states, /, 3/, and 5/. Relative to the Schrödinger model, the number of states for level n are doubled, so that the multiplicity of states for level n is g n = n. In Table.3, we summarize the quantum numbers. In Figure., the resulting quantum states for the total angular momentum are illustrated for l =,, and 3. Due to the multiplicity of states of the electron, g s = s +, the multiplicity of states for state is g = (s + )(l + ). Effectively, the two spin states of the electron doubles the number of allowed stationary states, which doubles the multiplicity of states for level n (see Eq..4) from n to n g n = (s + )(l + ) = n. (.49) l=

21 .3. THE DIRAC ATOM (a) m j (b) m j 3/ / J L S 3/ / / 5/4 3/4 / Figure.: (a) The total angular momentum vector, J, as the vector sum of L and S. For this example, l = and the electron is in the spin up state, leading to j = 3/. The z axis projection m j = +3/ was chosen for this illustration. The new constant of motion, J, precesss azimuthally about the z axis with projection m j, whereas L precesses about J. Note that the projection of L on the z axis, L z, is no longer a constant of motion; however S and m s remain constants of motion. (b) Schematic of the vector addition emphasizing the magnitudes of the vectors. The total angular momentum, J, is represented by the thick line with the solid arrow and has magnitude p j(j + ) h = p 5/4 h. The orbital angular momentum and electron spin are shown as thin lines with open arrows, where the magnitude of the angular momentum is p l(l + ) h = h and the magnitude of the spin is p 3/4 h, as always. The other possible projections of J on the z axis (not shown) are m j = 3/, /, and +/. Figure.3: Dirac model bound-state quantum numbers Description Allowed ranges Number of states n principle n l orbital angular momentum l n n j total angular momentum / j n / n m l orbital magnetic l m l l l + m j total magnetic j m j j j + m s spin m s = ±/ All values are integers or integer multiples of /. The spin-orbit coupling (vector addition) rules, given by Eqs..47 and.48, are schematically illustrated in Figure.a for the state l = and spin up state resulting in j = 3/ and m j = +3/. The vector J precesses about the z axis in one of j + = 4 possible projections m j h, with m j = 3/, /,+/, or +3/. Oy the m j = +3/ case is illustrated. The L vector precesses about the J vector and its z component, L z, is no longer a constant of motion; the constant of motion is J z. The spin, S, adds to L, but with oy two possible z axis projections m s h, where m s = /,+/. The magnitudes of the J, L, and S vectors are illustrated Figure.b for l = and j = +3/. The magnitude of the orbital angular momentum vector is l(l + ) h = h. The magnitude of the spin vector is always s(s + ) h = 3/4 h. The magnitude of the total angular momentum vector (illustrated as thick lines with solid arrow), is j(j + ) h = 5/4 h.

22 CHAPTER. HYDROGENIC ATOMS There are four possible z axis projections m j = 3/, /,+/,+3/, though oy the latter is shown. The spin vector is represented originating at the head of the angular momentum vector (but it does not precess around J; it precesses about the z axis)..3.3 Energy structure The Dirac wave functions are represented by the quantum numbers l, j, and m j, and m s. These wave functions are spinors, which are beyond the scope of this text. A heuristic method has been to write the wave function as ψ jmj (r), and compute it as the linear superposition of the Schrödinger stationary states, m l, and the spin wave functions with m s stationary states, ψ jmj (r) = m l m s α jmj m s R (r)y lml (θ,φ)χ ms. (.5) where the sum is over all m l and m s states for a given, and where αm jmj s are known as the Clebsch-Gordon coefficients, which we do not discuss further. The interaction energies due to relativistic and spin-orbit effects are on the order of 5 E n. Thus, we present the perturbation theory approach of adding low-order correction terms to the Schrödinger Hamiltonian, H = h ˆm +V (r,t) p4 8 ˆm 3 c + dv ˆm c r dr L S+ h 8 ˆm c V (r,t). (.5) The first two terms are the Schrödinger Hamiltonian. The third term is a relativistic energy correction of order v /c. The fourth term is the spin-orbit interaction energy for l electrons, which includes a relativistic correction to the electromagnetic potential. The final term is a correction called the Darwin term for l = electrons (to account for zwitterbewegung, a jittering precession of the electron spin due to interaction with the angular momentum of the photon field transporting the electromagnetic force between the nucleus and the electron). Thus, the three corrections obtained from the perturbation treatment are the relativistic energy correction, E, the spin-orbit coupling correction, E, and the Darwin effect correction, E 3. Dirac energies The resulting energy corrections are obtained by inserting Eq..5 into Eq..43. For example, the relativistic energy correction is given by E = ψ jmj (r) p4 8 ˆm 3 c ψ jm j (r). (.5)

23 .3. THE DIRAC ATOM 3 The correction is applied by adding E to Schrödinger energies, E n (Eq..44). The other correction terms to the energy are similarly computed. We have [ E = (Zα) 3 E n n 4 n ] l+/ E E n E E n = (Zα) n = + (Zα) n [ ] j(j +/) [ ] (j +/)(j +) l, j = l l, j = l+ (.53) E 3 E n = (Zα) n l =, where the total energy correction is E/E n = ( E + E + E 3 )/E n, which yields, E nj = E n (+ E/E n ) = ˆmc (Zα) ( [+ (Zα) n n n j+/ 3 )], (.54) 4 where α = e / hc is the fine structure constant. Note that the energies now depend upon both the n and the j states. The exact solution was obtained by Dirac using the so-called Dirac equation based upon full treatment of relativistic energies, electron spin, and interaction of the electron and nuclear magnetic moments as applied to spinors (two-by-two matrix wave equations). The treatment is well beyond the scope of this text. Dirac s result is [ [ ] ] / E nj = ˆmc Zα + n (j /) + [(j+/) (Zα) ] /, (.55) which is equivalent to Eq..54 to order (Zα). For an infinite mass nucleus, the reduced mass of the electron, ˆm is replaced with the rest mass, m e. Shifts and fine structure splittings Eq..54 clearly shows that, for level n, the Dirac energies are shifted from the Schrödinger energies and that these shifts depend on both n and j. Furthermore, since there are n possible j states for level n, Eq..54 shows that there are n unique energy states in level n. The magnitude of the shifts and splittings are on the order of α E n ( 5 E n ) and decrease with increasing n. The resulting energy splittings are referred to as fine structure. In Figure.4, the quantity E/E n from Eq..54 is schematically illustrated for n =,, and 3 for the hydrogen atom. We have invoked the spectroscopic notation, i.e., the n =, l = state is written p, etc., with the added convention

24 4 CHAPTER. HYDROGENIC ATOMS n = 3 6 E / En Quantum States ( l,j ) 3d 5/ (,5/) 3p 3/ (,3/) 3d 3/ (,3/) 3s / (,/) 3p / (,/) n = p / (,3/) s / (,/) p / (,/) n = 3.38 s / (,/) Figure.4: The hydrogen fine structure energy shifts and splittings, 6 E/E n, for the first three principle levels n =,, and 3. Note that there are n fine-structure states for level n. The quantum states are identified using spectroscopic notation. The energy differences between fine structure states are given relative to the Schrödinger states (left). of including the subscript j to denote the total angular momentum state. Thus, the n =, l =, j = / state is written p /, etc. The result of relativistic spin-orbit coupling is that the binding energy of the electron is increased (the energy is more negative) relative to the non-relativistic Schrödinger treatment. In the Dirac model, since there are j + possible projections for m j, the E nj are j + energy degenerate..3.4 Selection rules and Grotrian diagram The transition energies in the Dirac model are E n j nj = E nj n j = E nj E n j, (.56) where the energies are given by Eq..54 or.55. Given the fine structure multiplets, it is clear that the transition energies will cluster in multiplets as well. The closely spaced spectral lines that result from transitions between fine structure multiplets associated with states and n l are also simply called a multiplet. Selection rules As with the Schrödinger model, the non-vanishing terms of the dipole moment between stationary states provide the selection rules for permitted transitions. The dipole moment vanishes for all transitions except for those with l = ±

25 .3. THE DIRAC ATOM 5 and m l =, ±; these selection rules immediately apply. Since j = l ± /, it also immediately follows that j = ± is a selection rule. An additional rule is that there must be a parity change in the stationary states; this is known as LaPorte s rule. Parity and LaPorte s rule will be discussed further in The consequence of LaPorte s rule is that there is no spin flip ( s = ) for dipole transitions. Full treatment of the relative polarization of the electromagnetic field with respect to the z axis and the xy plane, yields the full set of Dirac model dipole selection rules (see.3 for details). Though l cannot be zero for either polarization; j can be zero for both polarizations. Without further elaboration, we present the dipole selection rules for the hydrogenic Dirac model in Table.4. Table.4: Dirac model dipole selection rules Transition Rule l ± j, ± m j, ± s Grotrian diagram: doublets and multiplets In Figure.5, the permitted transitions for hydrogen Lyα and Hα are illustrated in a Grotrian diagram of the Dirac model. There is oy a single s state with j = / (though its energy is shifted relative to the Schrödinger energy). There are three j states for n =, but the s / and p / states are energy degenerate (we will see that this degeneracy is broken when we introduce radiative corrections). According to the selection rules, a s / s / transition is forbidden, since it would require l =. Since the path for an electron in the s / state to transition to the s / state is forbidden, and s / is the oy n = state, the s / state is metastable. Both the p 3/ s / and the p / s / transitions are permitted with l = ± and with j = ± and, respectively. Thus, the fine structure Lyα spectral feature is a doublet. In fact, in the Dirac model, all Lyman series spectral features are doublets due to the transitions np / s / and np 3/ s /, where n = is Lyα, n = 3 is Lyβ, etc. The Lyman series doublets are { np/ s Lyman doublets / np 3/ s /. Similar discussion applies to the Hα transition, which can occur as any one of seven transitions. The 3d 5/ p / transition is forbidden, because it would require j = ±, even though l = ± would otherwise be allowed. Note that the transitions 3p 3/ s / and 3d 3/ p / are energy degenerate, as are the 3p / s / and 3s / p / transitions. Thus, the Dirac theory predicts that

26 6 CHAPTER. HYDROGENIC ATOMS ev s p d ev n =3 j = 5/ j = 3/ j = / s p 3/ / 3s p / / 3p s 3/ / 3p s / / 3d p 5/ 3/ 3d p 3/ 3/ 3d p 3/ / n = j = 3/ j = / p s 3/ / p s / /. n = j = / s p d Figure.5: The Grotrian diagram for the Dirac model of hydrogen for n =,, and 3. Excitation energy is shown on the left and binding energy is shown on the right. The energy levels are shifted relative to the Schrödinger model and exhibit fine structure splitting due to the j states. The dipole selection rules are l = ± and j =, ± for permitted transitions. The Lyα doublet (p 3/ s / and p / s / ) and Hα multiplet transitions are illustrated. Not illustrated is the permitted Lyβ doublet (3p 3/ s / and 3p / s / ). observed spectra will show five unique features in the Hα multiplet. In fact, all Balmer series spectral features are five-fold multiplets from seven transitions, Balmer multiplets np 3/ s / nd 3/ p / (energy degenerate) np / s / ns / p / (energy degenerate) nd 3/ p 3/ nd 5/ p 3/ ns / p 3/, where n = 3 is Hα, n = 4 is Hβ, etc. Note that fine structure energy splittings decrease with increasing n such that spectral feature multiplets from a high n to a much lower n (except to n = ) are dominated by the low n fine structure. The transitions between fine structure multiplets do not have equal probability of occurring. Thus, the observed spectral multiplets have different intensities (see.3.)..4 Radiative corrections Precise measurements of the hydrogen spectrum indicate that the Dirac transition energies do not match observations within one percent. As we discuss here,