Page 123 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics
P. 123
114 Harmonic oscillator
only one independent solution and the ground state is non-degenerate. If we
arbitrarily set á equal to zero so that ö 0 (î) is real, then the ground-state
eigenvector is
2
j0i ð ÿ1=4 ÿî =2 (4:31)
e
^
We next show that if the eigenvalue n of the number operator N is non-
degenerate, then the eigenvalue n 1 is also non-degenerate. We begin with
the assumption that there is only one eigenvector with the property that
^
Njni njni
and consider the eigenvector j(n 1)ii, which satis®es
^
Nj(n 1)ii (n 1)j(n 1)ii
If we operate on j(n 1)ii with the lowering operator ^ a, we obtain to within a
multiplicative constant c the unique eigenfunction jni,
^ aj(n 1)ii cjni
We next operate on this expression with the adjoint of ^ a to give
y ^ y
^ a ^ aj(n 1)ii Nj(n 1)ii (n 1)j(n 1)ii c^ a jni
from which it follows that
c
y
j(n 1)ii ^ a jni
n 1
Thus, all the eigenvectors j(n 1)ii corresponding to the eigenvalue n 1 are
proportional to ^ a jni and are, therefore, not independent since they are
y
proportional to each other. We conclude then that if the eigenvalue n is non-
degenerate, then the eigenvalue n 1 is non-degenerate.
Since we have shown that the ground state is non-degenerate, we see that the
next higher eigenvalue n 1 is also non-degenerate. But if the eigenvalue n 1
is non-degenerate, then the eigenvalue n 2 is non-degenerate. By iteration, all
^
of the eigenvalues n of N are non-degenerate. From equation (4.30) we observe
that all the energy levels E n of the harmonic oscillator are non-degenerate.
4.3 Eigenfunctions
Lowering and raising operations
^
From equations (4.27) and (4.28) and the conclusions that the eigenvalues of N
are non-degenerate and are positive integers, we see that ^ ajni and ^ a jni are
y
^
eigenfunctions of N with eigenvalues n ÿ 1 and n 1, respectively. Accor-
dingly, we may write
^ ajni c n jn ÿ 1i (4:32a)
