The probability, P, of finding the mass outside the classically allowed region
is then:
(11.26)
The value of yAis given by:
(11.27)
Substitution of ωyields:
(11.28)
For the ground state, 9 =0 and yA=1. Substitution of the wavefunction gives:
(11.29)
(11.30)
This integral is related to the error function, erf z:
(11.31)
For the case presented above, z=1 and P=0.079. Thus, there is a 7.9%
probability of finding the mass past the classic turning point on each side
or a total probability of 15.8% of the mass being in the forbidden region.
If chemical bonds are pictured as springs holding atoms, then there is a
considerable probability of the bonds having large bond distances.
Transitions
Unlike the particle in a box, the energy levels for the harmonic oscillator
are evenly spaced (Figure 11.4). The difference between adjacent levels is
proportional to the frequency and independent of the quantum number:
ΔE=( 9 + 1 +1/2)Zω−( 9 +1/2)Zω=Zω (11.32)
erf z ey y
z
=−/
−∞(^1) ∫
2
122
πdPeyey==//yy
∞
−−∞
∫∫11
12
1
12
122
απα
πddψ
(^0) απ 12
()^122
/
ye= −y/y
kk
mmk
A//
=+⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎛
⎝
21
12 14(^92)
Z
Z
⎜⎜⎜
⎞
⎠
⎟⎟ =+
14
21/
9yAE
kmk
km
A()
/
==⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ =
+
α2212 ω
214Z9Z/ kk
Z^2⎛^14
⎝
⎜⎜
⎞
⎠
⎟⎟
/Pxxxyyy
AA
==()() ()()
∞∞
∫∫ψψ*d* dψψα