Principles - Methods of Molecular Quantum Mechanics - page 18

described by the progressive wave in complex form:

Y ¼ A exp ð i aÞ¼ A exp ½ 2 p i ð kx n t Þ ð 1 : 83 Þ

where A is the amplitude and the phase of a monochromatic plane wave

of wavenumber k and frequency

a

, which propagates along x. Taking into
account the relations of De Broglie and Planck connecting wave-like and
particle-like behaviour:

n

p
h ;

E
h

k ¼ n ¼ ð 1 : 84 Þ

the phase of a matter wave could be written as

1
h ð px Et Þ

a ¼ ð 1 : 85 Þ

giving the wave equation for the matter particle in the form

i
h ð px Et Þ

Y ¼ A exp ¼ Yð x ; t Þ ð 1 :

86 Þ

which defines as a function of x and t at constant values of p and E. Then,

taking the derivatives of

Y

Y with respect to x and t, we obtain respectively

@Y
@ x ¼

h @
@ x

i
h p Y

whence ) p ¼ i ð 1 : 87 Þ

which is our first postulate (1.61) for the x-component of the linear
momentum, and

@Y
@ t ¼

i
h E Y

h @
@ t

whence ) E ¼ H ¼ i ð 1 : 88 Þ

the time-dependent Schroedinger Equation (1.76) giving the time evolution
of the state function

. Hence, the two correspondences (1.87) and (1.88),
connecting linear momentum and total energy to the first derivatives of
the function

Y

, necessarily implythe fundamental relationsoccurring inour
axiomatic proposition of quantum mechanics.
As a consequence of our probabilistic description, in doing experiments
in atomic physics we usually obtain a distribution of the observable
eigenvalues, unless the state function

Y

Y coincides at time t with the

eigenfunction w k of the corresponding quantum mechanical operator, in

which case we have a 100% probability of observing for A the value A k .
Such probability distributions fluctuate in time for all observables but the
energy, where we have a distribution of eigenvalues (the possible values of