7. The Hydrogen Atom in Wave Mechanics In this chapter we shall discuss : The Schr¨odinger equation in spherical coordinates • Spherical harmonics • Radial probability densities • The hydrogen atom wavefunctions • Angular momentum • Intrinsic spin, Zeeman effect, Stern-Gerlach experiment • Energy levels and spectroscopic notation, fine structure • Elements of Nuclear Engineering and Radiological Sciences I NERS 311: Slide #1 The Schr¨odinger equation in spherical coordinates ... In 3D Cartesian (x,y,z) coordinates, the time-independent Schr¨odinger equation for a single particle bound by a potential, V (~x), is: ~2 2u(~x)+ V (~x)u(~x)= Eu(~x) , (7.1) −2m∇ where ∂2 ∂2 ∂2 2 = + + . (7.2) ∇ ∂x2 ∂y2 ∂z2 If the potential specialized to V (~x) = V (r), where r is the distance to the origin, then, it is natural to use the spherical-polar coordinate system (r,θ,φ). Elements of Nuclear Engineering and Radiological Sciences I NERS 311: Slide #2 ... The Schr¨odinger equation in spherical coordinates ... where: r is the distance from origin to the particle location θ is the polar coordinate φ is the azimuthal coordinate Connection between Cartesian and spherical-polar: x r sin θ cos φ, y r sin θ sin φ, z r cos θ (7.3) → → → With dV =d~x =dx dy dz = r2 dr sin θ dθ dφ, (volume element in both systems) 2π π ∞ ∞ ∞ ∞ dx dy dzf(x,y,z) dφ sin θ dθ r2drf(r,θ,φ) (7.4) −→ 0 0 0 Z−∞ Z−∞ Z−∞ Z Z Z The Laplacian operator of (7.2) becomes, in spherical-polar coordinates: 2 2 = L , where (7.5) ∇ R− r2 ∂2 2 ∂ = + , and (7.6) R ∂r2 r ∂r 1 ∂ ∂ 1 ∂2 2 = sin θ + (7.7) L sin θ ∂θ ∂θ sin2 θ ∂φ2 Elements of Nuclear Engineering and Radiological Sciences I NERS 311: Slide #3 ... The Schr¨odinger equation in spherical coordinates ... Since V (~x) = V (r), a separation of variables occurs, and the most general solution of (7.1) in spherical-polar coordinates is given by: u (r,θ,φ) = R (r)Y (θ,φ) , (7.8) nlml nl l,ml where ∂2 2 ∂ R (r)= + R (r)= E R (r) (7.9) R nl ∂r2 r ∂r nl nl nl and 2 2 1 ∂ ∂ 1 ∂ Yl,m (θ,φ)= sin θ + Yl,m (θ,φ)= l(l + 1)Yl,m (θ,φ) L l sin θ ∂θ ∂θ sin2 θ ∂φ2 l l (7.10) The “radial” equation, (7.9) determines the energy of the system, since V (r) only involves the radial coordinate. The radial wavefunctions and the quantized energies are obtained by solving (7.10). Note that the radial eigenfunctions functions and energies may depend on two quantum numbers, n and l. We shall see several examples in due course. The angular equation, (7.10), is a generic solution that applies for all the potentials of the form, V (~x) = V (r). The Y (θ,φ)’s are the eigenfunctions of the angular part of the l,ml Laplacian operator, , with l(l + 1) being its eigenvalue. L Elements of Nuclear Engineering and Radiological Sciences I NERS 311: Slide #4 ... The Schr¨odinger equation in spherical coordinates We have 3 quantum numbers: name range of values n principal can be anything, depends on the specific of V (r) l orbital angular momentum positive integer or 0 m orbital magnetic quantum number integer, l m l l − ≤ l ≤ Their meanings are: n primary quantum number used to quantize E l quantized orbital angular momentum may also have a role in determining the quantized E ml measures the z-component of the angular momentum plays no role in determining E l if E depends on n and l, D = l 1=2l +1 − P Elements of Nuclear Engineering and Radiological Sciences I NERS 311: Slide #5 The Spherical Harmonics Y (θ,φ) ... l,ml As evident in (7.10), the Y (θ,φ)’s are the eigenfunctions of the angular part of the l,ml Laplacian, 2 Yl,m (θ,φ)= l(l + 1)Yl,m (θ,φ) , L l l with eigenvalue l(l + 1). Low-order explicit forms: l m Y (θ,φ) l l,ml 1 1/2 00 (4π) 3 1/2 3 1/2 iφ 1 0, 1 ( ) cos θ, ( ) sin θe± ± 4π ∓ 8π 5 1/2 2 15 1/2 iφ 15 1/2 2 i2φ 2 0, 1, 2 ( ) (3cos θ 1), ( ) sin θ cos θe± , ( ) sin θe± ± ± 16π − ∓ 8π 32π . Elements of Nuclear Engineering and Radiological Sciences I NERS 311: Slide #6 ... The Spherical Harmonics Y (θ,φ) ... l,ml The general form is: 1/2 m 2l +1(l ml)! ml im φ Yl,m (θ,φ)=( 1) l − P (cos θ)e l (7.11) l − 4π (l + m )! l l ml where the Pl ’s are the “associated Legendre polynomials”. (You may be familiar with ml the usual Legendre polynomials, the Pl’s. They are a special case of the Pl ’s obtained by setting ml =0. The most useful identities and recursion relationships for the generating all of the associ- ated Legendre polynomials are: l+1 l Pl+1 (cos θ)= (2l + 1)sin θPl (cos θ) (7.12) l − l 2 l/2 Pl (cos θ)=( 1) (2l 1)!!(1 cos θ) (7.13) l − − l − Pl+1(cos θ) = cos θ(2l + 1)Pl (cos θ) (7.14) ml ml ml (l ml + 1)Pl+1(cos θ)=(2l + 1)cos θPl (cos θ) (l + ml)Pl 1(cos θ) (7.15) − m m +1 − − m 1 2m cos θP l (cos θ)= sin θ P l (cos θ)+(l + m )(l m + 1)P l− (cos θ) l l − l l − l l h (7.16)i Elements of Nuclear Engineering and Radiological Sciences I NERS 311: Slide #7 ... The Spherical Harmonics Y (θ,φ) ... l,ml ml The relationship between the Pl ’s for negative ml is: m (l m )! m − l ml l l Pl (cos θ)=( 1) − Pl (cos θ) . (7.17) − (l + ml)! The complex conjugate of Y (θ,φ) is given by: l,ml Yl,m∗ (θ,φ)= Yl, m (θ,φ) (7.18) l − l The statement of orthonormality is: 2π π δl,l δm ,m = l, ml l′, m′ = dφ sin θdθYl,m∗ (θ,φ)Yl ,m (θ,φ) (7.19) ′ l l′ h | l′i l ′ l′ ′ Z0 Z0 ′ Recalling that the parity operator has the effect: Π[(θ,φ)]=(π θ,φ + π) , (7.20) − it follows that l Π[Yl,m (θ,φ)]=( 1) Yl,m (θ,φ) , (7.21) l − l Elements of Nuclear Engineering and Radiological Sciences I NERS 311: Slide #8 ... The Spherical Harmonics Y (θ,φ) l,ml Finally, the ml’s are the quantum number associated with the azimuthal component (equivalently, the z component of the angular momentum). From the general form (7.11), ∂ zYl,m (θ,φ)= i Yl,m (θ,φ)= m Yl,m (θ,φ) . (7.22) L l − ∂φ l l l Elements of Nuclear Engineering and Radiological Sciences I NERS 311: Slide #9 Radial wave functions ... As we have seen, for potentials of the form, V (~x)= V (r), the angular part of the of the Laplacian in the Schr¨odinger equation, the , given in (7.7), is completely general. L In fact, it was discovered, and its solutions explored (eventually to become known as spher- ical harmonics) in a paper written by Laplace in 1783, well before Quantum Mechanics, by Pierre-Simon Laplace (1749—1827). It even predated Electromagnetism and Maxwell’s equations. Laplace was an astronomer, as well as a Mathematician, and his application was in the classical mechanics of celestial bodies. The applications are many! From Wikipedia: “Spherical harmonics are important in many theoretical and practical applications, particularly in the computation of atomic orbital electron configurations, representation of gravitational fields, geoids, and the magnetic fields of planetary bodies and stars, and characterization of the cosmic microwave back- ground radiation. In 3D computer graphics, spherical harmonics play a special role in a wide variety of topics including indirect lighting (ambient occlusion, global illumination, precomputed radiance transfer, etc.) and modeling of 3D shapes. You will encounter them in many applications in your technical career. Elements of Nuclear Engineering and Radiological Sciences I NERS 311: Slide #10 Radial wave functions Returning to quantum mechanics, we substitute the Y (θ,φ)’s into the Schr¨odinger l,ml equation ~2 2 L Rnl(r)Yl,m (θ,φ)+ V (r)Rnl(r)Yl,m (θ,φ)= EnlRnl(r)Yl,m (θ,φ)(7.23) −2m R− r2 l l l 2 operate using on Yl,m (θ,φ), and divide by Yl,m (θ,φ) to obtain: L l l ~2 ∂2 2 ∂ ~2l(l + 1) + R (r)+ V (r)+ R (r)= E R (r) , (7.24) −2m ∂r2 r ∂r nl 2mr2 nl nl nl that is known as the “radial wave equation”. It is a second-order differential equation in the variable r, used to quantize the energies. Note the role that the angular momentum part plays. It appears as a repulsive force that tends to push the particle away from the origin. This is the effect, quantum-mechanically speaking, of the effect of the centrifugal force. We can anticipate that this will cause the probability density to be pushed away from the origin, with increasing effect as l increases, i.e. as angular momentum increases. Elements of Nuclear Engineering and Radiological Sciences I NERS 311: Slide #11 The l =0 radial wave functions, the “s-states” ..