BioPHYSICAL chemistry

So the expectation value agrees with the de Broglie relationship as expected.

We can also calculate the expectation value of position X:

(10.29)

(10.30)

The particle has an average position of L/2; that is, the particle is on

average in the middle of the box.

Transitions

A transition between states is associated with

absorption or emission of energy. The particle is

normally present in the lowest-energy state. If

energy is introduced into the box then the particle

can make a transition to a higher-energy state. If

a particle changes from the n=2 to the n=1 state

(Figure 10.4), the change in energy is:

(10.31)

For an electron m=9.11 × 10 −^31 kg. Assuming that L=1 nm then

(10.32)

Using conservation of energy we can set the energy of the light emitted

equal to the energy change between states, giving:

(10.33)

(10.34)

If there is more than one electron in the box, then the transition is not the

same, but rather proceeds from the highest occupied level to the lowest

unoccupied level, with each level containing two electrons. For example,

if there are four electrons in the box, then the first two levels are filled

λ==

××

×

hc −−

ΔE

(. )( )

.

663 10 3 10

18 1

(^34) Js (^8) m s 1
00

− 19 =×1 1 10−^6 =^1100

J

. mnm

ΔEh

hc

==ν

λ

ΔE

(. )

(. )(

=

×

×

−

−

3 6 63 10

8 9 11 10 10

34 2

31

Js

kg −−

=×−

92

18 10^19

m

J

)

.

ΔEE E

h

mL

h

mL

=−= 21 ()−=

2

2

22

2

8 21 2

3

8

X= sin

⎛

⎝

⎜⎜

⎞

⎠

∫ ⎟⎟ =

2

2

2

L 0

x

nx

L

x

L

L

π

d

X==() () sin

⎛

⎝

⎜⎜

⎞

⎠

∫ψψ ∫ ⎟⎟

π

*dxx x x

L

nx

L

x

L L

2

0 0

ssin

nx

L

x

⎛π

⎝

⎜⎜

⎞

⎠

⎟⎟d

204 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

n  3

n  2

n  1

1 electron 4 electrons

Figure 10.4The
longest-wavelength
transitions for one
and four electrons in
a particle in a box.