Page 134 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

4.6 Three-dimensional harmonic oscillator 125

4.5 Heisenberg uncertainty relation
Using the results of Section 4.4, we may easily verify for the harmonic
oscillator the Heisenberg uncertainty relation as discussed in Section 3.11.
Speci®cally, we wish to show for the harmonic oscillator that
1
ÄxÄp > "
2
where
2 2
(Äx) ˆh(x ÿhxi) i
2 2
(Äp) ˆh(^ p ÿhpi) i
The expectation values of x and of ^ p for a harmonic oscillator in eigenstate
jni are just the matrix elements hnjxjni and hnj^ pjni, respectively. These matrix
elements are given in equations (4.45c) and (4.46c). We see that both vanish,
2
2
2
so that (Äx) reduces to the expectation value of x or hnjx jni and (Äp) 2
2
2
reduces to the expectation value of ^ p or hnj^ p jni. These matrix elements are
given in equations (4.48b) and (4.49b). Therefore, we have
1=2
"
1 1=2
Äx ˆ (n ‡ )
mù 2
1 1=2
Äp ˆ (m"ù) 1=2 (n ‡ )
2
and the product ÄxÄp is
1
ÄxÄp ˆ (n ‡ )"
2
For the ground state (n ˆ 0), we see that the product ÄxÄp equals the
minimum allowed value "=2. This result is consistent with the form (equation
(3.85)) of the state function for minimum uncertainty. When the ground-state
harmonic-oscillator values of kxl, kpl, and ë are substituted into equation
(3.85), the ground-state eigenvector j0i in equation (4.31) is obtained. For
excited states of the harmonic oscillator, the product ÄxÄp is greater than the
minimum allowed value.

4.6 Three-dimensional harmonic oscillator
The harmonic oscillator may be generalized to three dimensions, in which case
the particle is displaced from the origin in a general direction in cartesian
space. The force constant is not necessarily the same in each of the three
dimensions, so that the potential energy is
2 2
2
2
2 2
2
2
2
1
1
1
1
V ˆ k x x ‡ k y y ‡ k z z ˆ m(ù x ‡ ù y ‡ ù z )
2 2 2 2 x y z