B 12 ¼
π
3ε 0 ħ
2
ea
ð Þ
2 :
ð9:76Þ
In order to relate these results to those of quantum mechanics, we may replace a
2
in the above expressions with a square of an absolute value of the matrix elements of
the position operator r. That is, representing | 1i and | 2i as the quantum states of the
ground and excited states of a two-level atom, we define h1| r| 2i as the matrix
element of the position operator. Relating |h1| r| 2i|
2 to a
2 , we get
A 12 ¼
ω 21
3 e
2
3πε 0 c 3 ħ
1
h r
j j2i
j
j
2 and B 12 ¼
πe
2
3ε 0 ħ
2
1
h jrj2i
j
j
2 :
ð9:77Þ
From (9.77), we have
B 12 ¼
πe
2
3ε 0 ħ
2
1
h jrj2i 1
h jrj2i
à ¼
πe
2
3ε 0 ħ
2
1
h jrj2i 2
h jr
{
j1
¼
πe
2
3ε 0 ħ
2
1
h jrj2i 2
h jrj1i,
ð9:78Þ
where with the second equality we used (1.116); the last equality comes from that r is
Hermitian. Meanwhile, we have
B 21 ¼
πe
2
3ε 0 ħ
2
2
h jrj1i
j
j
2 ¼
πe
2
3ε 0 ħ
2
2
h jrj1i 1
h jr
{
j2
¼
πe
2
3ε 0 ħ
2
2
h jrj1i 1
h jrj2i: ð9:79Þ
Hence, we recover (9.41).
9.5 Lasers
9.5.1 Brief Outlook
A concept of the two-level atoms proposed by Einstein is universal and independent
of materials and can be utilized for some particular purposes. Actually, in later years
many researchers tried to validate that concept and verified its validity. After basic
researches of 1950s and 1960s, fundamentals were put into practical use as various
optical devices. Typical examples are masers and lasers, abbreviations for “microwave amplification by stimulated emission of radiation” and “light amplification by
stimulated emission of radiation.” Of these, lasers are common and particularly
important nowadays. On the basis of universality of the concept, a lot of materials
including semiconductors and dyes are used in gaseous, liquid, and solid states.
354
9 Light Quanta: Radiation and Absorption
π
3ε 0 ħ
2
ea
ð Þ
2 :
ð9:76Þ
In order to relate these results to those of quantum mechanics, we may replace a
2
in the above expressions with a square of an absolute value of the matrix elements of
the position operator r. That is, representing | 1i and | 2i as the quantum states of the
ground and excited states of a two-level atom, we define h1| r| 2i as the matrix
element of the position operator. Relating |h1| r| 2i|
2 to a
2 , we get
A 12 ¼
ω 21
3 e
2
3πε 0 c 3 ħ
1
h r
j j2i
j
j
2 and B 12 ¼
πe
2
3ε 0 ħ
2
1
h jrj2i
j
j
2 :
ð9:77Þ
From (9.77), we have
B 12 ¼
πe
2
3ε 0 ħ
2
1
h jrj2i 1
h jrj2i
à ¼
πe
2
3ε 0 ħ
2
1
h jrj2i 2
h jr
{
j1
¼
πe
2
3ε 0 ħ
2
1
h jrj2i 2
h jrj1i,
ð9:78Þ
where with the second equality we used (1.116); the last equality comes from that r is
Hermitian. Meanwhile, we have
B 21 ¼
πe
2
3ε 0 ħ
2
2
h jrj1i
j
j
2 ¼
πe
2
3ε 0 ħ
2
2
h jrj1i 1
h jr
{
j2
¼
πe
2
3ε 0 ħ
2
2
h jrj1i 1
h jrj2i: ð9:79Þ
Hence, we recover (9.41).
9.5 Lasers
9.5.1 Brief Outlook
A concept of the two-level atoms proposed by Einstein is universal and independent
of materials and can be utilized for some particular purposes. Actually, in later years
many researchers tried to validate that concept and verified its validity. After basic
researches of 1950s and 1960s, fundamentals were put into practical use as various
optical devices. Typical examples are masers and lasers, abbreviations for “microwave amplification by stimulated emission of radiation” and “light amplification by
stimulated emission of radiation.” Of these, lasers are common and particularly
important nowadays. On the basis of universality of the concept, a lot of materials
including semiconductors and dyes are used in gaseous, liquid, and solid states.
354
9 Light Quanta: Radiation and Absorption