PHYSICS 420 SPRING 2006 Dennis Papadopoulos

LECTURE 21

THE HYDROGEN AND HYDROGENIC ATOMS

The Hydrogen Atom

• 1 Application of the Schr ödinger Equation to the Hydrogen Atom • 2 • 3 Quantum Numbers • 4. Energy Levels and Electron Probabilities • 5 Magnetic Effects on Atomic Spectra – The Normal Zeeman Effect • 6 Solution of the Schr ödinger Equation for Hydrogen Intrinsic Spin

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 Schr ödinger Equation to the Hydrogen Atom

• The approximation of the potential energy of the electron proton system is electrostatic: • Rewrite the three-dimensional time-independent Schr ödinger Equation.

For Hydrogen-like atoms (He + • Replace

e

2 with

Ze

2 (

Z

or Li ++ ) is the atomic number).

Application of the Schr ödinger 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 Schr ödinger equation.

  1/   (1/

m e m i

)

Fig. 8-5, p.266

Application of the Schr ödinger 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 Schr ödinger Equation

• Eqs (8.11) to eqs (8.15) yield ----

Radial equation

----

Angular equation

2

d g d

 2 g : e im l    2

m g l

• SE has been separated into three ordinary second-order differential equations each containing only one variable.

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

to be zero.

each of the two expressions in parentheses 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 n

ℓ • 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

harmonics

Y

and the spherical 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

d τ

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

.

  2 4  2

r dr

Isotropic States only l=0

Fig. 8-9, p.282

Average vs. most probable distance   2 4 

r

2 1 

r f

 0     0  0    (3 / 2)

a o

Fig. 8-10a, p.283

Fig. 8-10, p.283

Probability Distribution Functions



R

(

r

) and

P

(

r

) for the lowest-lying states of the hydrogen atom.

Table 8-5, p.280

Fig. 8-11b, p.285

l=2

Fig. 8-11c, p.285

Probability Densities Symmetric about z-axis

Fig. 8-12, p.286

3: Quantum Numbers

The three quantum numbers: –

n

Principal quantum number – ℓ –

m

ℓ Orbital angular momentum quantum number Magnetic quantum number The boundary conditions: – –

n

= 1, 2, 3, 4, . . . – ℓ = 0, 1, 2, 3, . . . ,

n

− 1

m

ℓ = −ℓ, −ℓ + 1, . . . , 0, 1, . . . , ℓ − 1, ℓ The restrictions for quantum numbers: –

n

> 0 – ℓ <

n

–

|m

ℓ | ≤ ℓ Integer Integer Integer

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

mv

orbital

r

. • ℓ is related to

L

by .

• In an ℓ = 0 state, .

with

L

= It disagrees with Bohr’s semiclassical “planetary” model of electrons orbiting a nucleus

L

=

n ħ

.

n

 1 , 2 , 3 ,  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.

l  0,1, 2, K

n

 1

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

E

  13.6

Z

2

n

2     54.4

eV

• 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 2

E

  1 2

Z

 

r

Hund’s Rule Unpaired spins

Fig. 9-15, p.321

Table 9-2a, p.322

Table 9-2b, p.323

Fig. 9-16, p.324