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Chapter 42

Atomic Physics (cont.)

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Outline

The quantum model of the hydrogen atom

The wave functions for hydrogen

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The quantum model of the hydrogen atom Difficulties with Bohr theory:

Many of the lines in the Balmer and other series of hydrogen are not single lines at all.

Splitting of spectral lines when atoms are placed in a strong magnetic filed.

Need a full quantum model involving the Schrödinger equation to describe the hydrogen atom.

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The quantum model of the hydrogen atom

General strategy: Solving the Schrödinger equation for the hydrogen atom

The three-dimensional time-independent Schrödinger equation:

Formal procedure for solving the problem: The potential energy function for the hydrogen

atom:

Substitute U(r) into the Schrödinger equation and find the appropriate solutions to the equation satisfied by appropriate boundary conditions.

r

ekrU e

2

EU

zyxm

2

2

2

2

2

22

2

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Results for a Hydrogen atom Three different quantum numbers for

each allowed state of the hydrogen atom ( ): Principal quantum number, n: The

energies of the allowed states for the hydrogen atom depend only on n,

Orbital quantum number, l Orbital magnetic quantum number, ml

,...3,2,1 606.131

2 220

2

neV

nna

ekE en

lmln ,,

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Relationship among the three quantum numbers

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Atomic shell and subshell notations

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Example: The n=2 level of Hydrogen

For a hydrogen atom, determine the number of allowed states corresponding to the principal quantum number n = 2, and calculate the energies of these states.

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The wave functions for Hydrogen Wave function in the ground state 1s:

Radial probability density function P(r) = 4r2||2.

For the hydrogen atom in its ground state:

0/

30

1

1 ars e

ar

0/230

2

1

4 ars e

a

rrP

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Hydrogen atom in its ground state

The charge of the electron is extended throughout a diffuse region of space - Electron cloud

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Example: Probabilities for the electron in hydrogen

Calculate the probability that the electron in the ground state of hydrogen will be found outside the first Bohr radius, a0.

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Example: The ground state of hydrogen

Calculate the most probable value of r for an electron in the ground state of the hydrogen atom.

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Homework

Chapter 42, P. 1393, Problems: #19, 20.