BioPHYSICAL chemistry

that are quantized. The mathematical analysis for solving the equation is

provided in Derivation box 11.1. The reader may skip to the main text

following the box, where detailed description of the solutions is given.

224 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

Derivation box 11.1

Solving Schrödinger’s equation for the simple harmonic oscillator

To solve this equation, we first rewrite it by letting:

(db11.1)

Substitution of yfor xyields:

(db11.2)

(db11.3)

(db11.4)

(db11.5)

(db11.6)

The expression can then be written in the simpler form of:

(db11.7)

d

d

2

2

(^20)
y
ψεψ() (yyy+−) ()=
Multiplication of this equation by yields:
−

2

2

m 2

Z

α

−+=

Z^2

2

2

2

2

2

1

my 2

y

k

yyEy

α

ψαψψ

d

d

() ( ) () ()

y

x

xy

y

xy xx

=== =

αα

d

d

d

d

d

d

d

d

and

d

d

d

d

d

d

1 2

(^2) xxy=

1

2

2

α^2

d

d

or

Now, let:

ε

α

/

==

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ =

222 2 ⎛

22

m^212

E

mE

mk

Em

ZZ k

Z

Z ⎝⎝

⎜⎜

⎞

⎠

⎟⎟ =

12

2

/

E

Zω

1

4

2

4

22 1/

===

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

mk

mk mk

α

αα

Z

ZZ

then or

44

d

d

2

2

2

2

4

2

(^220)
y
y
m
Ey
mk
ψ yy
α
ψ
α
()+− =() ψ()
ZZ
d
d
2
2
2
2
2
2
2

2

2

2

y

y

mk

yy

m

ψ Ey

α

αψ

α

()−=( ) () −ψ(

ZZ

))