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Chapter 2 Angular Momentum, Hydrogen Atom, and Helium Atom

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Chapter 2 Angular Momentum, Hydrogen Atom, and Helium Atom Chapter 2 Angular Momentum, Hydrogen Atom, and Helium Atom Contents 2.1 Angular momenta and their addition . .24 2.2 Hydrogenlike atoms . .38 2.3 Pauli principle, Hund's rules, and periodical table . 42 2.4 Ground state of helium atom: 1st approximation . .48 Basic Questions A. What are the eigenvalues of angular momentum operator? B. What are the quantum numbers of a state of the single electron in hydrogen atom? C. What is total electron spin of ground-state helium atom, and the spin eigenstate? 23 24CHAPTER 2. ANGULAR MOMENTUM, HYDROGEN ATOM, AND HELIUM ATOM 2.1 Angular momentum and addition of two an- gular momenta 2.1.1 Schr¨odinger Equation in 3D Consider the Hamiltonian of a particle of mass m in a central potential V (r) h¯2 H^ = 2 + V (r) : −2mr Since V (r) depends on r only, it is natural to express 2 in terms of spherical coordinates (r; θ; ') as r 2 2 1 @ 2 @ 1 @ @ 1 @ = r + sin θ + 2 : r r2 @r @r ! r2 sin θ @θ @θ ! r2 sin θ @'2 Operators in Spherical Coordinates A. Laplacian operator By coordinate transformation between Cartesian (x; y; z) and spherical coordi- nates (r; θ; ') x = r sin θ cos '; r = x2 + y2 + z2 q x2 + y2 y = r sin θ sin '; tan2 θ = z2 y z = r cos θ; tan ' = x we have @r @r @r = sin θ cos '; = sin θ sin '; = cos θ @x @y @z @θ 1 @θ 1 @θ 1 = cos θ cos '; = cos θ sin '; = sin θ @x r @y r @z −r @' 1 sin ' @' 1 cos ' @' = ; = ; = 0 : @x −r sin θ @y −r sin θ @z Using the derivative rule @ @r @ @θ @ @' @ = + + @x @x @r @x @θ @x @' 2.1. ANGULAR MOMENTUM AND ADDITION OF TWO ANGULAR MOMENTA25 @ 1 @ 1 sin ' @ = sin θ cos ' + cos θ cos ' @r r @θ − r sin θ @' @ @ 1 @ 1 cos ' @ = sin θ sin ' + cos θ sin ' @y @r r @θ − r sin θ @' @ @ 1 @ = cos θ sin θ @z @r − r @θ Therefore @2 @2 @2 @ 1 @ 1 sin ' @ + + = sin θ cos ' + cos θ cos ' @x2 @y2 @z2 @r r @θ − r sin θ @' ! @ 1 @ 1 sin ' @ sin θ cos ' + cos θ cos ' × @r r @θ − r sin θ @'! @ 1 @ 1 cos ' @ + sin θ sin ' + cos θ sin ' @r r @θ − r sin θ @' ! @ 1 @ 1 cos ' @ sin θ sin ' + cos θ sin ' × @r r @θ − r sin θ @' ! @ 1 @ @ 1 @ + cos θ sin θ cos θ sin θ @r − r @θ ! × @r − r @θ ! and this is equal to, after some considerable algebra @2 1 @2 1 @2 2 @ cot θ @ 2 = + + + + r @r2 r2 @θ2 r2 sin2 θ @'2 r @r r2 @θ 2 1 @ 2 @ 1 @ @ 1 @ = r + sin θ + 2 r2 @r @r ! r2 sin θ @θ @θ ! r2 sin θ @'2 1 @ @ 1 = r2 + A^ (θ; ') r2 @r @r ! r2 with definition 2 ^ 1 @ @ 1 @ A (θ; ') sin θ + 2 : ≡ sin θ @θ @θ ! sin θ @'2 B. Angular momentum operators @ @ Consider first the z-component, using the above formulas for @x and @y @ @ L^z = ih¯ x y − @y − @x! @ 1 @ 1 cos ' @ = ( ih¯) r sin θ cos ' sin θ sin ' + cos θ sin ' − @r r @θ − r sin θ @' ! 26CHAPTER 2. ANGULAR MOMENTUM, HYDROGEN ATOM, AND HELIUM ATOM @ 1 @ 1 sin ' @ ( ih¯) r sin θ sin ' sin θ cos ' + cos θ cos ' − − @r r @θ − r sin θ @' ! @ = ih¯ − @' similarly we can derive the x- and y-components as @ @ L^x = ih¯ sin ' cot θ cos ' − − @θ − @' ! @ @ L^y = ih¯ + cos ' cot θ sin ' : − @θ − @' ! >From the definition of the raising and lowering operators L^ L^ iL^ ≡ x y it is straightforward to obtain i' @ @ L^ = ih¯e i cot θ : − @θ − @' ! C. Angular momentum square In order to obtain the square of angular momentum operator in the spherical 2 coordinates, consider L^x 2 ^ 2 Lx @ @ 2 = sin ' + cot θ cos ' h¯ @θ @' ! − @ @ @ @ @ @ = sin ' sin ' + sin ' cot θ cos ' + cot θ cos ' sin ' @θ @θ @θ @' @' @θ @ @ + cot θ cos ' cot θ cos ' @' @' @2 @ @ @ = sin2 ' + sin ' cos ' cot θ + cot θ cos2 ' @θ2 @θ @' @θ @2 @ @2 + cot θ cos ' sin ' cot2 θ cos ' sin ' + cot2 θ cos2 ' @'@θ − @' @'2 2 ^ 2 Ly @ @ 2 = cos ' cot θ sin ' h¯ @θ − @'! − @2 @ @ @ = cos2 ' sin ' cos ' cot θ + cot θ sin2 ' @θ2 − @θ @' @θ @2 @ @2 cot θ cos ' sin ' + cot2 θ sin ' cos ' + cot2 θ sin2 ' − @'@θ @' @'2 2.1. ANGULAR MOMENTUM AND ADDITION OF TWO ANGULAR MOMENTA27 now L^2 L^2 + L^2 + L^2 @2 @ @2 = x y z = + cot θ + cot2 θ + 1 h¯2 h¯2 @θ2 @θ @'2 − − 1 @ @ 1 @2 = sin θ + = A^ (θ; ') sin θ @θ @θ sin2 θ @'2 1 @ @ 1 L^2 = sin θ z sin θ @θ @θ − sin2 θ h¯2 where A^ (θ; φ) is as defined before in the Laplacian 2. The Schr¨odinger eq. becomes r 2 2 h¯ 1 @ 2 @ 1 @ @ 1 @ r + sin θ + 2 + V (r) Ψ (r; θ; ') (−2m "r2 @r @r ! r2 sin θ @θ @θ ! r2 sin θ @'2 # ) = EΨ (r; θ; ') or h¯2 1 @ @ L^2 r2 + V (r) + Ψ (r; θ; ') = EΨ (r; θ; ') : ("−2m r2 @r @r ! # 2mr2 ) This eq. is separable. Let Ψ (r; θ; ') = R (r) Y (θ; ') where Y (θ; ') is assumed to be the eigenstate of L^2 with eigenvalue λ L^2Y (θ; ') = λY (θ; ') we have the equation for the radial part of wavefunction h¯2 1 @ @ λ r2 + V (r) + R (r) = ER (r) : ("−2m r2 @r @r ! # 2mr2 ) The solution of this equation depends on the given potential V (r). But the solution for the angular part of wavefunction Y (θ; ') is universal and can be discussed in general. 2.1.2 Operators and their algebra Classically, angular momentum is defined as L = r p × 28CHAPTER 2. ANGULAR MOMENTUM, HYDROGEN ATOM, AND HELIUM ATOM or in component form L = yp zp ; L = zp xp ; L = xp yp x z − y y x − z z y − x in QM, they all become operator L^ = y^p^ z^p^ ; L^ = z^p^ x^p^ ; L^ = x^p^ y^p^ : x z − y y x − z z y − x Their commutation relations are given by L^ ; L^ = [y^p^ z^p^ ; z^p^ x^p^ ] x y z − y x − z h i = [y^p^z; z^p^x] + [z^p^y; x^p^z] = y^[p^z; z^] p^x + [z^; p^z] p^yx^ = ih¯y^p^ + ih¯p^ x^ − x y = ih¯L^z L^y; L^z = ih¯L^x h i L^z; L^x = ih¯L^y h i we can memorize these relations by L^ L^ = ih¯L^ : × In spherical coordinates, these operators are expressed as @ @ L^x = ih¯ sin ' cot θ cos ' − − @θ − @'! @ @ L^y = ih¯ + cos ' cot θ sin ' − @θ − @' ! @ L^ = ih¯ z − @' and 2 ^2 2 1 @ @ 1 @ L = h¯ sin θ + 2 : − sin θ @θ @θ sin θ @'2 ! Hence the Laplacian operator 2 and Hamiltonian take the following simple form r 1 @ @ L^2 2 = r2 r r2 @r @r − h¯2r2 h¯2 1 @ @ 1 L^2 H^ = r2 + V (r) + −2m r2 @r @r 2m r2 2.1. ANGULAR MOMENTUM AND ADDITION OF TWO ANGULAR MOMENTA29 2 It is easy to see L^z commutes with L^ 2 L^z; L^ = 0 h i and L^2 commute with operators H^ , L^2; H^ = 0 : h i 2 2 (note: we also have L^x; L^ = L^y; L^ = 0) Therefore we can find a wavefunction h i h 2 i which is eigenfunctions to both H^ ; L^ ; and L^z. Furthermore, since they are separable, the eigenfunction of H^ can be written in general as Ψn:l;m (r; θ; ') = Rnl (r) Θl (θ) Φm (') = Rnl (r) Yl;m (θ; ') with @ L^ Φ (') = λ Φ (') i Φ (') = λ Φ (') z m m m ! − @' m m m 2 2 ^2 h¯ @ @ λm L Yl;m (θ; ') = λl;mYl;m (θ; ') sin θ + 2 Θl (θ) = λl;mΘl (θ) ! −sin θ @θ @θ sin θ ! H^ Ψn:l;m (r; θ; ') = λn;l;mΨn:l;m (r; θ; ') 2 h¯ 1 @ 2 @ 1 λl;m r + V (r) + Rnl (r) = λn;l;mRnl (r) ! −2µ r2 @r @r 2µ r2 ! In Dirac notation Φ (') m m ! j i Y (θ; ') l; m l;m ! j i Ψ (r; θ; ') n; l; m : n:l;m ! j i Instead of dealing with L^x; L^y, and L^z, one can define angular raising and lowering + operators L^ and L^− as, i' @ @ L^ L^x iL^y = ihe¯ i cot θ ≡ − @θ − @' ! + and we have the equivalent set L^x; L^y, and L^z or L^−; L^ and L^z. The algebra for this 2nd set is more convenient. It is easy to show the commutations between three operators Lz; L are given by + L^ ; L^ = h¯L^; L^ ; L^− = 2¯hL^ : z z h i h i 30CHAPTER 2. ANGULAR MOMENTUM, HYDROGEN ATOM, AND HELIUM ATOM Note: Compare this algebra with harmonic oscillator a;^ a^y = 1. These algebra will determine the eigenfunctions and eigenvalues of L^2. Ith is alsoi easy to derive ^+ ^ ^2 ^2 ^ L L− = L Lz + h¯Lz; + 2 − 2 L^−L^ = L^ L^ h¯L^ ; − z − z and 2 2 + 2 + 2 1 + + L^ = L^ + h¯L^ + L^−L^ = L^ h¯L^ + L^ L^− = L^ + L^ L^− + L^−L^ : z z z − z z 2 ^ 1 Note: compare this with harmonic oscillator H = h!¯ a^ya^ + 2 .
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