introduction-weak-interaction-volume-one

(r~ t)
_ f(r) ejEtm
(1 .5 .5 )
We now apply the function (1 .3.9) to (1 .5 .5) to obtai n
-Tt- '
E T - (1 .5 .6 )
Thus E is an eigenvalue5 of the energy operator, and 1)1 is an eigenfunction of
it .
An energy eigenfunction is said to represent a 'stationary state' of a
particle, since is constant in time . Similarly, E is an eigenvalue i n
► ^`r
(1 .5 .4) . Thus we see that only an eigenvalue E is a possible energy for the
particle, and hence we have quantized the SchrNdinger equation (1 .3.5) .
As an example of the process of quantization, we
shall now quantize
the harmonic oscillator6. The potential energy of a particle of mass m which
has been displaced x units from its equilibrium position is given by (26 )
V(x) = 271 2 m v.02 x2 . (1 .5 .7 )
Substituting with this in the one-dimensional SchrNdinger equation (1 .3 .5) we hav e
r, U -2 + m (W - 2 7T 2 my 2 x2 ) O . 1 .5
x n d 0 (
.8
)
We now introduce the variables (27 )
X = 8 712 mW/h2
and
oC = 4 7T 2m v0 /h ,
so that (1 .5 .8) become s
2
2 + (\ - c 2 x 2 ) 0 .
~ x
(1 .5 .9 )
(1 .5 .10 )
(1 .5 .11 )
Following Sommerfeld : Wave,:echanics, Dutton, 1929, p .11, we now use th e
polynomial method for finding a solution for 1V. First we find the asymptoti c
solution for 1f1 . Since for large 4x 4 , )\ is regligiLle compared with p2 x 2 ,
the asymptotic wave equation becomes
2 2 x2
a x
whose only physically meaningful solution i s
2r(x) = e -('x/2) x
Now we let
(1 .5 .12 )
(1 .5 .13)
( x ) = e(02) x2 f (x) (1 .5 .14 )
for finite x . Obtaining a power series, differentiating, and solving for ,
we obtain 1