BioPHYSICAL chemistry

Using this expression for velocity from eqn 9.17 gives us the wavelength

as:

=6.1 × 10 −^12 m (9.22)

The ability to manipulate the wavelength of an electron by use of an

electric field has led to the development of electron microscopes that are

commonly used for biological samples, as discussed in Chapter 18.

Schrödinger’s equation

Independently, Werner Heisenberg and Erwin Schrödinger both developed

mathematical formulations that, despite the differences in their details, were

found to yield the same basic results. In the Schrödinger formulation, the

key concepts are wavefunctions and operators. Each object is characterized

by a wavefunction that describes the distribution of the object over space,

reflecting its wave-like nature. The determination of the wavefunction

for an object in a particular environment is done by use of a second-order

differential equation called Schrödinger’s equation, as described below.

Once the wavefunction has been determined, then the various properties

of the object, such as position and momentum, can be described.

Classically, the energy of a particle is given by the sum of the kinetic

and potential energies:

(9.23)

In quantum mechanics this expression is modified with the introduction

of operators, as provided in Table 9.1. Some of these operators involve

Zand i, which are defined as:

(9.24)

In some cases, the operator is simply a variable, for example the posi-

tion operator is the position variable x. In other cases, the operator is

a differential term that operates on the wavefunction. For example, the

momentum is substituted by the derivative with respect to position, ,

∂

∂x

Z==−

h

i

2

1

π

and

EmVx

p

m

=+=+() Vx()

1

22

2

2

9

=

×

××

−

−

.

(. )(.

6 626 10

2 9 109 10 1 609

34

31

Js

kg 10 −^19 CV)( 4 × 104 )

λ== =

h

p

h

m

h

(^92) meVe

184 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY