LECTURE 21

THE HYDROGEN AND HYDROGENIC ATOMS

PHYSICS 420SPRING 2006Dennis Papadopoulos

• 1 Application of the Schrödinger Equation to the Hydrogen Atom

• 2 Solution of the Schrödinger Equation for Hydrogen• 3 Quantum Numbers• 4. Energy Levels and Electron Probabilities• 5 Magnetic Effects on Atomic Spectra – The Normal

Zeeman Effect• 6 Intrinsic Spin

The Hydrogen AtomThe 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 e2 with Ze2 (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.

1/ (1/ ) (1/ )e ireduced mass m m

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

l

22

2

im

( )( ) --------Azimuthal equation

g e

l

d gm g

d

:

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 mℓ = 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 a0.

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 mℓ:

– ℓ = 0, 1, 2, 3, . . .

– mℓ = −ℓ, −ℓ + 1, . . . , −2, −1, 0, 1, 2, . ℓ . , ℓ − 1, ℓ

– |mℓ| ≤ ℓ and ℓ < 0.

• The predicted energy level is

Hydrogen Atom Radial Wave Functions

• First few radial wave functions Rnℓ

• 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 mℓ.

• 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 mℓ. 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 dτis .

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) = r2|R(r)|2 and it depends only on n and l.

Fig. 8-9, p.282

2 2( ) 4P r dr r dr

Isotropic States only

l=0

Fig. 8-10a, p.283

2 2

0

0

0

( ) 4

1 ( )

( ) ( )

( ) (3 / 2) o

P r r

P r dr

f f r P r dr

r rP r dr a

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 NumbersThe three quantum numbers:

– n Principal quantum number– ℓ Orbital angular momentum quantum number

– mℓ Magnetic quantum number

The boundary conditions:– n = 1, 2, 3, 4, . . . Integer– ℓ = 0, 1, 2, 3, . . . , n − 1 Integer

– mℓ = −ℓ, −ℓ + 1, . . . , 0, 1, . . . , ℓ − 1, ℓ Integer

The restrictions for quantum numbers:– n > 0– ℓ < n

– |mℓ| ≤ 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 = mvorbitalr.

• 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.

0,1,2, 1n l K0,1,2, 1n l K

,3,2,1n ,3,2,1n 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

• Inert2

213.6 4 13.6 54.4

ZE x eV

n

• Li Z=3, n=1 full, go to n=2, L-shell

• Bigger atom, 4 times ao (~n2)

• Nuclear charge partially screened by n=1 electrons

• Low ionization potential

• Energy of outer electrons21

21.1 1.2

kZ eE

rZ

Fig. 9-15, p.321

Hund’s RuleUnpairedspins

Table 9-2a, p.322

Table 9-2b, p.323

Fig. 9-16, p.324