50
3.
WAVE-LENGTH OF THE ELECTRON
1926.1n simplied, non-relativistic form, for a particle of mass
"
—
—
m}}, moving in the x direction, this equation is
d—‘äp
87r2mo(é"— U)
,——
——————
=
0,
3—1
dx2
+
112
(
)
where 71 is Planck’s constant,
is the total and
U is the
potential energy of the particle. The square
of the absolute value
'
of the
“
wave-amplitude,” |u,b |2, is equal to the probability per unit
length of nding the particle at a given point.
'
For a free particle travelling with a velocity @ in the x direction,
the potential energy, U, is a constant and may
be set equal to zero
and
is equal to the kinetic energy m,,v2/2. Equation 3—1 then
becomes
d2gb
<27rm00>2
__
__
=
0
_
,
+
w
,
<3 2>
which has a solution of the form.
'
_
2
=
A sm
lx
,
(3—3)
.
)\
provided the wave—length )\ is given by
,
)\ = _
_
mov
,
(3 4)
as may be seen by twice differentiating (3—3) and substituting in
(3-2). This equation, (3—4), is the non—relativistic form of
de-Broglie’s equation. -Here, >\ is the wave-length of the
in centimeters, ]; = 6.54"X 10“27 erg—seèonds, mo = 9.03 X 10”28
grams (the rest mass of the electron), and 0 is the Velocity of the
electronin centimeters per second.
This equation applies With
-
good accuracy to free electrons accelerated by not more than a few
thousand volts. The aCcelerating Voltage, V, times the electronic
charge,
is equal to the kinetic energy
of the electron.
Then
'
_
.
'
>\2V=_ 150,
g
‘
Ï
(3—5)
where >\ is in angstroms and V is involts.
'
'
_
,