LECTURE 21 THE HYDROGEN AND HYDROGENIC ATOMS PHYSICS 420 SPRING 2006 Dennis Papadopoulos

1Application of the Schrödinger Equation to the Hydrogen Atom 2Solution of the Schrödinger Equation for Hydrogen 3Quantum Numbers 4. Energy Levels and Electron Probabilities 5Magnetic Effects on Atomic Spectra – The Normal Zeeman Effect 6Intrinsic Spin The Hydrogen Atom The atom of modern physics can be symbolized only through a partial differential equation in an abstract space of many dimensions. All its qualities are inferential; no material properties can be directly attributed to it. An understanding of the atomic world in that primary sensuous fashion…is impossible. - Werner Heisenberg

1: Application of the Schrödinger Equation to the Hydrogen Atom The approximation of the potential energy of the electron- proton system is electrostatic: Rewrite the three-dimensional time-independent Schrödinger Equation. For Hydrogen-like atoms (He + or Li ++ ) Replace e 2 with Ze 2 (Z is the atomic number).

Application of the Schrödinger Equation The potential (central force) V(r) depends on the distance r between the proton and electron. Transform to spherical polar coordinates because of the radial symmetry. Insert the Coulomb potential into the transformed Schrödinger equation.

Fig. 8-5, p.266

Application of the Schrödinger Equation The wave function ψ is a function of r, θ,. Equation is separable. Solution may be a product of three functions. We can separate the SE into three separate differential equations, each depending on one coordinate: r, θ, or.

Solution of the Schrödinger Equation Eqs (8.11) to eqs (8.15) yield SE has been separated into three ordinary second-order differential equations each containing only one variable. ----Radial equation ----Angular equation

Solution of the Radial Equation The radial equation is called the associated Laguerre equation and the solutions R that satisfy the appropriate boundary conditions are called associated Laguerre functions. Assume the ground state has ℓ = 0 and this requires m ℓ = 0. The radial equation becomes or

Solution of the Radial Equation Try a solution Take derivatives of R and insert them into the SE equation. To satisfy it for any r each of the two expressions in parentheses to be zero. Set the second parentheses equal to zero and solve for a 0. Set the first parentheses equal to zero and solve for E. Both equal to the Bohr result.

Quantum Numbers The appropriate boundary conditions to the radial and angular equations leads to the following restrictions on the quantum numbers ℓ and m ℓ : –ℓ = 0, 1, 2, 3,... –m ℓ = −ℓ, −ℓ + 1,..., −2, −1, 0, 1, 2,. ℓ., ℓ − 1, ℓ –|m ℓ | ≤ ℓ and ℓ < 0. The predicted energy level is

Hydrogen Atom Radial Wave Functions First few radial wave functions R nℓ Subscripts on R specify the values of n and ℓ.

Solution of the Angular and Azimuthal Equations The solutions for azimuthal equation are. Solutions to the angular and azimuthal equations are linked because both have m ℓ. Group these solutions together into functions spherical harmonics

Solution of the Angular and Azimuthal Equations The radial wave function R and the spherical harmonics Y determine the probability density for the various quantum states. The total wave function depends on n, ℓ, and m ℓ. The wave function becomes

Normalized Spherical Harmonics

Table 8-2, p.269

Table 8-3, p.269

Probability Distribution Functions We must use wave functions to calculate the probability distributions of the electrons. The “position” of the electron is spread over space and is not well defined. We may use the radial wave function R(r) to calculate radial probability distributions of the electron. The probability of finding the electron in a differential volume element dτ  is.

Probability Distribution Functions The differential volume element in spherical polar coordinates is Therefore, We are only interested in the radial dependence. The radial probability density is P(r) = r 2 |R(r)| 2 and it depends only on n and l.

Fig. 8-9, p.282 Isotropic States only l=0

Fig. 8-10a, p.283 Average vs. most probable distance

Fig. 8-10, p.283

R(r) and P(r) for the lowest-lying states of the hydrogen atom. Probability Distribution Functions

Table 8-5, p.280

Fig. 8-11b, p.285

Fig. 8-11c, p.285 l=2

Fig. 8-12, p.286 Probability Densities Symmetric about z-axis

3: Quantum Numbers The three quantum numbers: –nPrincipal quantum number –ℓOrbital angular momentum quantum number –m ℓ Magnetic quantum number The boundary conditions: –n = 1, 2, 3, 4,... Integer –ℓ = 0, 1, 2, 3,..., n − 1Integer –m ℓ = −ℓ, −ℓ + 1,..., 0, 1,..., ℓ − 1, ℓInteger The restrictions for quantum numbers: –n > 0 –ℓ < n –|m ℓ | ≤ ℓ

Principal Quantum Number n It results from the solution of R(r) in because R(r) includes the potential energy V(r). The result for this quantized energy is The negative means the energy E indicates that the electron and proton are bound together.

Orbital Angular Momentum Quantum Number ℓ It is associated with the R(r) and f(θ) parts of the wave function. Classically, the orbital angular momentum with L = mv orbital r. ℓ is related to L by. In an ℓ = 0 state,. It disagrees with Bohr’s semiclassical “planetary” model of electrons orbiting a nucleus L = nħ.

The allowed energy levels are quantized much like or particle in a box. Since the energy level decreases a the square of n, these levels get closer together as n gets larger.

Chemical properties of an atom are determined by the least tightly bound electrons. Factors: Occupancy of subshell Energy separation between the subshell and the next higher subshell.

Pauli principle and Minimum Energy Principle

He Z=2, n=1, l=0, m=0. Two electron with opposite spin Zero angular momentum High ionization energy 54.4 eV Inert

Li Z=3, n=1 full, go to n=2, L-shell Bigger atom, 4 times a o (~n 2 ) Nuclear charge partially screened by n=1 electrons Low ionization potential Energy of outer electrons

Fig. 9-15, p.321 Hund’s Rule Unpaired spins

Table 9-2a, p.322

Table 9-2b, p.323

Fig. 9-16, p.324