Concise Physical Chemistry

c15 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

246 EARLY QUANTUM THEORY: A SUMMARY

Butα=mee^2 /
^2 ,so

E(r)=−

^2

2 me

(

mee


^2

2 )^2

=−

mee

2 
^2

4

−

1

2

mee


^2

4

=−

1

2

that is,E(r)=^12 hartree in atomic units because the constantsmee/
^2

4

are all defined

as 1 in the atomic system.

Problem 15.1

The dominant line in the emission spectrum of atomic hydrogen is a beautiful crim-

son line at a wavelength of 656.1 nm. The hertz (Hz) is the unit of frequency that

corresponds to one cycle of the wave motion per second. The speed of electromag-

netic radiation isc= 2. 998 × 108 ms−^1. Each unit of frequency represents one step

of wavelengthλ, so the speed of wave propagation is the number of steps per second

(frequency) times the length of each step, orc=νλ. Express the crimson hydrogen

line in terms of frequency in hertz and find its energy from the Planck equation. It

is sometimes convenient to express frequency in wave numbers, ̃ν=ν/c.Find ̃νfor

the crimson line in the hydrogen spectrum and use it to findλ. (This is a crosscheck

because you already knowλ).

Problem 15.2

Einstein’s most famous equation isE=mc^2 and the Planck equation isE=hν.

Since they both express the energy, combine them to obtain the de Broglie equation,

which expresses wave particle duality.

Problem 15.3 (Classical Mechanics)

A projectile of massmwas fired upward so that it traveled along a parabolic arc

with velocityv. Its initial velocity wasv 0. How high will it go? Neglect wind and air

resistance.

Problem 15.4

Expand determinant a so as to obtain a single number. Expand determinant b so as

to obtain a single number. The job is already becoming tedious, so expand b using

machine software—for example, Mathcad©C. Expand c by machine. Could you have

guessed the answer simply by knowing b and looking at c?

(a)

∣

∣

∣

∣

23

56

∣

∣

∣

∣ (b)

∣ ∣ ∣ ∣ ∣ ∣

123

456

789

∣ ∣ ∣ ∣ ∣ ∣

(c)

∣ ∣ ∣ ∣ ∣ ∣

12 3

45 6

789. 1

∣ ∣ ∣ ∣ ∣ ∣