[p2/(2m)]y
= Ey
= [hbar2k2/2m)]y
Yk(x,t) = e ikx -i hbar k2 t / (2m) (8-2)
and since pop|k> = hbar k |k>, from popYk = -i hbar dYk/dx = hbar k Yk , hbar k is reasonably regarded as the value of the momentum in |k>. Therefore cos(kx) and sin(kx) are linear combinations of the states that have definite momenta (of hbar k and - hbar k).
However, there is a problem: <k'|k> = ò dx ei(k-k')x = 2 pd(k-k') Although the states of different moment are orthogonal, the normalization integral is 2pd(0) which cannot be adjusted to be 1! There is no normalized state which corresponds to a free particle of definite momentum. This fact is not a mathematical anomaly, it is a simple consequence of the Uncertainty Principle.
The actual physical states can be written, however, as sums over the definite-momentum states in the usual fashion. Since there are now a continuous distribution of states, instead of discrete numbers of states, the sum will have to become an integral. Hence a general wave function will be
Y(x,t) = (2p)-1/2 ò -¥ ¥ dk f(k) exp{i[kx - hbar k2 t / (2m)]} (8-3)
We can use (8-1) to get the coefficients f(k) from the values of Y(x,t) at any set time, say t=0:
òdx e -ik'x
Y(x,0)
= (2p)-1/2
ò dx dk f(k)
ei(k-k')x
= (2p)1/2
ò dk f(k)
d(k-k')
f(k') = (2p)-1/2 òdx e -ik'x Y(x,0) (8-4)
The wave function (8-3) is called a "wave packet," and the most useful cases are where f(k) involves a narrow range of values of k.
Last Revised 05/03/02 |