Chap 4. Quantum Mechanics In Three Dimensions Schrodinger Equation in Spherical Coordinates The Hydrogen Atom Angular Momentum Spin
4.1. Schrodinger Equation in Spherical Coordinates Separation of Variables The Angular Equation The Radial Equation Read Prob 4.1
Orthogonal Curvilinear Coordinates Ref: G.Arfken, “Mathematical Methods for Physicists”, 3rd ed., Chap. 2. B.Schutz, “Geometrical Methods of Mathematical Physics”, p.148. Spherical coordinates :
4.1.1. Separation of Variables V V(r) Spherical coordinates : Ansatz :
Set dimensionless constant Or
Mnemonics Do Prob 4.2
4.1.2. The Angular Equation Ansatz : Set
Azimuthal Solutions single-valued, i.e.,
Legendre Polynomials Setting m 0 gives Frobenius method shows that convergence requires See Arfken (3rd ed) Ex 8.5.5 The corresponding solutions are called the Legendre polynomials, which can also be defined by the Rodrigues formula :
1st few Legendre Polynomials Normalization: Pl (1) = 1
Associated Legendre Functions Solutions to the m 0 case : are called associated Legendre functions defined by Griffiths: where Arfken, Mathenmatica: Thus while m takes on 2l + 1 values : Note : Another independent solution exists but is not physically acceptable ( see Prob. 4.4 ).
1st few Associated Legendre Polynomials
Normalization Spherical Harmonics Normalization : Griffiths : where Arfken, Mathenmatica: Note : Orthonormality :
1st few Spherical Harmonics Do Prob 4.3 Read Prob 4.4, 4.6
4.1.3. The Radial Equation Set Effective potential Centrifugal term Normalization :
Example 4.1. Infinite Spherical Well Let Find the wave functions and the allowed energies. Ans : jl = spherical Bessel function nl = spherical Neumann function
Spherical Bessel & Neumann Functions jl (0) is finite nl (0) Let n l be the nth zero of jl . (2l+1)-fold degeneracy in m.
1st Few Spherical Bessel & Neumann Functions Do Prob 4.9
Bessel & Neumann Functions The Bessel & Neumann functions are solutions to the radial part of the Helmholtz equation in cylindrical coordinates : Bessel Neumann functions Modified Bessel functions ( for 2 < 0 )
Spherical Bessel & Neumann Functions The spherical Bessel & Neumann functions are solutions to the radial part of the Helmholtz equation in spherical coordinates : Spherical Bessel Neumann functions
Asymptotic Forms for x 0 for x
The Radial Wave Function The Spectrum of Hydrogen 4.2. The Hydrogen Atom The Radial Wave Function The Spectrum of Hydrogen
Bohr’s Model Circular orbit : Quantization of angular momentum : Bohr radius
4.2.1. The Radial Wave Function Bound States ( E < 0 ) : Set Set
Asymptotic Behavior : u finite everywhere 0 : Set
Factor-Out Asympototic Behavior
Frobenius Method
Series Termination j : ( unacceptable for large ) Series must terminate :
Eigenenergies Let = principal quantum number n 1, 2, 3, ... Bohr radius :
Eigenfunctions Eigenfunction belonging to eigenenergy is where with
Ground State n 1, l 0. Normalization :
1st Excited States n 2, l 0, 1. m 1, 0, 1 Normalization : see Prob. 4.11 Degeneracy of nth excited state :
Associated Laguerre Polynomials qth Laguerre polynomial ; Used by Griffiths. Used by Arfken & Mathematica. 1/n! of Griffiths’ value. Associated Laguerre polynomial Used by Arfken & Mathematica. 1/(n+p)! Griffiths’ value. Differential eqs. :
1st Few Laguerre & Associated LaguerrePolynomials Ln : Arfken & Mathematica convention Griffiths’ / n! Lna : Arfken & Mathematica convention Griffiths’ / (n+a)!
Orthogonal Polynomials Ref: M.Abramowitz, I.A.Stegun, “Handbook of Mathematical Functions”, Chap 22. Orthogonality: w weight function Differential eq.: Recurrence relations: Rodrigues’ formula: fn (a,b) en w g Standard An Pn (1,1) ()n n! 2n 1 x2 1 Pn(1) = 1 2 / (2n+1) Ln ( 0, ) e x x Lnp ()p e x xp (p+n)! / n! Hn ( , ) ()n exp(x2/2) en = (1)n n! 2
Hydrogen Wave Functions Griffiths convention Arfken convention : [3rd ed., eq(13.60)] Orthonormality : Arfken
First Few Rnl (r)
Rnl Plots (n 0) has n1 nodes (n, n1) has no node Note: Griffiths’ R31 plot is wrong.
Density Plots of 4 l 0 (400) (410) (430) (420) (n 0 m) has n1 nodes White = Off-scale (400) (410) (n 0 m) has n1 nodes (n, n1, m) has no node (430) (420) White = Off-scale
Surfaces of constant | 3 l m | (300) (322) (320) (310) Warning : These are plots of | |, NOT | |2 . (321) Do Prob 4.13, 4.15.
4.2.2. The Spectrum of Hydrogen H under perturbation transition between “stationary” levels: energy absorbed : to higher excited state energy released : to lower state H emitting light ( Ei > Ef ) : Planck’s formula : Rydberg formula where Rydberg constant
H Spectrum nf 1 nf 2 nf 3 Series Lyman Balmer Paschen Radiation UV Visible IR
Eigenvalues Eigenfunctions 4.3. Angular Momentum CM : QM : Eigenvalues Eigenfunctions
Commutator Manipulation distributive Similarly
[ Li , Lj ] Cyclic permutation :
Uncertainty Principle Only one component of L is determinate.
[ L2, L ] Similarly for i x, y, z i.e. L2 & Lz share the same eigenfunction :
4.3.1. Eigenvalues Ladder operators : Let L f is an eigenfunction of L2. L f is an eigenfunction of Lz. L raiseslowers eigenvalue of Lz by .
Lz finite max max(). Let Also Now Lz finite min min(). Let Also
Since Let N = integer max must be integer or half integer or Let where
Diagram Representation of L L can’t be represented by a vector fixed in space since only ONE of it components can be determinate.
4.3.2. Eigenfunctions Gradient in spherical coordinates :
Do Prob. 4.21 Read Prob 4.18, 4.19, 4.20 Do Prob 4.24
Spin 1/2 Electrons in a Magnetic Field Addition of Angular Momenta
Spin Spin is an intrinsic angular momentum satisfying with
4.4.1. Spin 1/2 2-D state space spanned by (spin up ) (spin down ) In matrix form (spinors) : General state : Operators are 22 matrices.
Pauli Matrices Pauli matrices
Spin Measurements Let particle be in normalized state : with Measuring Sz then has a probability | a |2 of getting /2, and probability | b |2 of getting /2, Characteristic equation is Eigenvalues : Eigenvectors :
Writing in terms of (x) : Measuring Sx then has a probability | |2 of getting /2, and probability | |2 of getting /2, Read last paragraph on p.176. Do Prob 4.26, 27 Read Prob 4.30
4.4.2. Electrons in a Magnetic Field Ampere’s law : Current loop magnetic moment . Likewise charge particle with angular momentum. gyromagnetic ratio QM : Spin is an angular momentum : experiences a torque when placed in a magnetic field B : tends to align with B, i.e., // B is the ground state with 0 QM : L has no fixed direction can’t be aligned to B Larmor precession
Example 4.3. Larmor Precession Consider a spin ½ particle at rest in uniform The Hamiltonian is + H E+ B0 / 2 E + B0 / 2 Sz / 2 / 2 H & S share the same eigenstates : Time evolution of is
if a, b are real Sinilarly
Set i.e., S is tilted a constant angle from the z-axis, and precesses with the Larmor frequency ( same as the classical law )
Example 4.4. The Stern-Gerlach Experiment Force on in inhomogeneous B : Consider a particle moving in the y-direction in a field Note : The x term is to make sure Due to precession about B0 , Sx oscillates rapidly & averages to zero. Net force is incident beam splits into two. In contrast, CM expects a continuous spread-out.
Alternative Description In frame moving with particle : x component dropped for convenience Let for t 0 for 0 t T where for t > T
for t > T spin up particles move upward spin up particles move downward S-G apparatus can be used to prepare particles in particular spin state.
4.4.3. Addition of Angular Momenta The angular momentum of a system in state can be found by writing where is proportional to the probability of measuring If the system has two types of angular momenta j1 and j2 , its state can be written as where The total angular momentum of the system is therefore described by the quantities
Since either set of basis is complete, we have The transformation coefficients are called Clebsch-Gordan coeffiecients (CGCs). The problem is equivalent to writing the direct product space as a direct sum of irreducible spaces
4.4.3. Addition of Angular Momenta Rules for adding two angular momenta : j1 j2 j1j2 J1J2...Jn State | j1 m1 | j2 m2 | j1 m1 | j2 m2 | J1 M1 ...| Jn Mn Possible values of S are j1 + j2 , j1 + j2 1, ..., | j1 j2 | + 1, | j1 j2 | Only states with M = m1 + m2 are related. Linear transformation between | j1 m1 | j2 m2 and | Jk Mk can be obtained by applying the lowering operator to the relation between the “top” states. Coefficients of these linear transformation are called the Clebsch-Gordan ( C-G ) coefficients.
Example: Two Spin ½ Particles s1 = ½ , s2 = ½ s1 + s2 = 1, s1 s2 = 0 Possible total S 1, 0 s1 s2 s1s2 S1 S2 States | | | | 1 1 | 1 0 | 1 1 | 0 0 | | | |
“Top” state for S = 1 : “Top” state for S = 0 : must be orthogonal to | 10 . Normalization then gives
J1 J2 ... M = m1 + m2 m1 , m2 M
Clebsch-Gordan ( C-G ) coefficients Shaded column gives Shaded row gives Sum of the squares of each row or column is 1. Do Prob 4.36