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Quantum Mechanics Lecture 24 Dr. Mauro Ferreira E-mail: [email protected] Room 2.49, Lloyd Institute Eigenfunctions We need to rewrite Lx, Ly and Lz in spherical coordinates: !ˆ ! ˆ ! L = !r × ∇ i ! ˆ ! ⇒ L= i ! where ∂ 1 ∂ 1 ∂ ! ∇ = r̂ + θ̂ + φ̂ ∂r r ∂θ r sin θ ∂φ ∂ 1 ∂ φ̂ − θ̂ ∂θ sin θ ∂φ " Rewriting the unit vectors θ̂ and φ̂ in spherical coordinates, we have θ̂ = (cos θ cos φ) î + (cos θ sin φ) ĵ − (sin θ) k̂ φ̂ = − (sin φ) î + (cos φ) ĵ ! ! ∂ ∂ L̂x = − sin φ − cos φ cot θ i ∂θ ∂φ ! " ! ∂ ∂ L̂y = cos φ − sin φ cot θ i ∂θ ∂φ ! ∂ L̂z = i ∂φ " The raising and lowering operators become L̂± = ±! e ±iφ L̂+ L̂− = −!2 ! ! ∂ ∂ ± i cot θ ∂θ ∂φ " 2 ∂2 ∂ ∂ ∂ 2 + cot θ + cot θ 2 + i ∂θ2 ∂θ ∂φ ∂φ " Finally, the operator L̂2 is written as L̂ = −! 2 2 ! 1 ∂ sin θ ∂θ " ∂ sin θ ∂θ # 1 ∂ + sin2 θ ∂φ2 2 $ The eigenvalue equations for L̂2 and L̂z become 2 L̂ f!m = −! L̂z f!m 2 ! 1 ∂ sin θ ∂θ " ∂ sin θ ∂θ ! ∂ m = f! = ! m f!m i ∂φ # 1 ∂ + sin2 θ ∂φ2 2 $ f!m = !2 $($ + 1) f!m Remember lecture 18 ? In general V (x, y, z) != V (x) + V (y) + V (z) Typically V (!r) = V (r) (central potentials) (Laplacian in spherical coordinates) ∇2 = 1 ∂ r2 ∂r ! r2 ∂ ∂r " + 1 ∂ r2 sin θ ∂θ ! sin θ ∂ ∂θ " 1 r2 sin2 θ + ! 2 ∂ ∂φ2 " (Schroedinger equation in spherical coordinates) ! ! 1 ∂ − 2m r2 ∂r 2 " r 2 ∂ψ ∂r # 1 ∂ + 2 r sin θ ∂θ " ∂ψ sin θ ∂θ # 1 + 2 r sin2 θ " 2 ∂ ψ ∂φ2 #$ + V ψ = Eψ attempted solution: ψ(r, θ, φ) = R(r) Y (θ, φ) − ! ! Y ∂ 2m r2 ∂r 2 " r2 ∂R ∂r # + R ∂ r2 sin θ ∂θ " sin θ ∂Y ∂θ # + R r2 sin2 θ 2 −2mr dividing by R Y an multiplying by !2 " 2 ∂ Y ∂φ2 #$ + V RY = ERY Lecture 18 ! 1 d R dr " r 2 dR 1 + Y dr ! # $ 2mr − [V (r) − E] !2 1 ∂ sin θ ∂θ 2 " ∂Y sin θ ∂θ # function of r only 1 ∂ Y + sin2 θ ∂φ2 2 $ =0 function of θ and Φ only Separating the variables... (Radial ODE) ! " 1 d 2mr2 2 dR r − [V (r) − E] = !(! + 1) 2 R dr dr ! 1 Y ! (Angular PDE) 1 ∂ sin θ ∂θ " ∂Y sin θ ∂θ # 1 ∂ Y + sin2 θ ∂φ2 2 $ separation constant = −$($ + 1) Lecture 18 Angular equation 1 Y ! 1 ∂ sin θ ∂θ " ∂Y sin θ ∂θ # 1 ∂ Y + sin2 θ ∂φ2 2 $ = −$($ + 1) multiplying by Y sin2 θ ∂ sin θ ∂θ ! ∂Y sin θ ∂θ " ∂2Y 2 + = −$($ + 1) sin θY ∂φ2 Separating the variables again: Y (θ, φ) = Θ(θ) Φ(φ) ! " # $% & 1 d dΘ 1 d2 Φ 2 sin θ sin θ + "(" + 1) sin θ + =0 Θ dθ dθ Φ dφ2 ! " #$ 1 d dΘ sin θ sin θ + "(" + 1) sin2 θ = m2 Θ dθ dθ 1 d2 Φ 2 = −m Φ dφ2 Lecture 18 2 L̂ f!m = −! L̂z f!m 1 Y ! 2 ! 1 ∂ sin θ ∂θ " ∂ sin θ ∂θ # 1 ∂ + sin2 θ ∂φ2 2 $ f!m = !2 $($ + 1) f!m ! ∂ m = f! = ! m f!m i ∂φ (Angular PDE) 1 ∂ sin θ ∂θ " ∂Y sin θ ∂θ # 1 ∂ Y + sin2 θ ∂φ2 2 $ = −$($ + 1) 1 d2 Φ 2 = −m Φ dφ2 A simple comparison indicates that the eigenfunctions f!m are given by the spherical harmonics Y!m (θ, φ). CONCLUSION: Spherical harmonics are eigenfunctions of the operators L̂2 and L̂z . When solving the Schroedinger equation by separation of variables we were inadvertently building simultaneous eigenfunctions of the three commuting operators Ĥ, L̂2 and L̂z . (Schroedinger equation in spherical coordinates) ! ! 1 ∂ − 2m r2 ∂r 2 " r 2 ∂ψ ∂r # 1 ∂ + 2 r sin θ ∂θ can be simplified to " ∂ψ sin θ ∂θ # 1 + 2 r sin2 θ " 2 ∂ ψ ∂φ2 #$ + V ψ = Eψ ! " # $ 1 2 ∂ 2 ∂ 2 −! r + L̂ ψ+V ψ =Eψ 2 2mr ∂r ∂r The only discrepancy is in the fact that the quantum numbers ! associated with eigenvalues of the angular momentum L̂2 may be integer or half-integer whereas they were found to be integers by solving the Schroedinger Eq. by separation of variables. Spin angular momentum Classically, a particle possess orbital and ! =L !o + L !s spin angular momenta, i.e., L In QM, in addition to their orbital angular momentum, ˆ ! elementary particles also carry an intrinsic momentum S called spin (in analogy with classical mechanics) Angular momentum commutation relations are still valid: [Ŝx , Ŝy ] = i! Ŝz ; [Ŝy , Ŝz ] = i! Ŝx ; [Ŝz , Ŝx ] = i! Ŝy Ŝ 2 |s, ms ! = !2 s(s + 1) |s, ms ! Ŝz |s, ms ! = ! ms |s, ms ! Ŝ± |s, ms ! = ! ! Defining Ŝ± ≡ Ŝx ± iŜy s(s + 1) − ms (ms ± 1) |s, ms ± 1! !s L The allowed values for the quantum numbers s and ms are 1 3 s = 0, , 1, , ... 2 2 and ms = −s, −s + 1, ..., s − 1, s Unlike the orbital degree of freedom, the spin of an elementary particle is fixed, that is, s is constant. One very special case is for s=1/2, in which case there are only two eigenstates: 1 1 and ms = − ms = + 2 2 1 1 1 1 | , + ! and | , − " 2 2 2 2 1 1 | , ↑" and | , ↓" 2 2 | ↑ " and | ↓ " Using these as a basis, the general state of a spin-1/2 particle can be written as a linear combination |χ ! = a | ↑ ! + b | ↓ !