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248 7 QUANTUM THEORY
to proceed to eqn 7D.20a, invert this expression
T = ((κ2 + k2)2(eκW − e−κW)2 + 16κ2k2
16k2κ2
)
−1
= ((κ2 + k2)2(eκW − e−κW)2
16κ2k2
+ 1)
−1
Now express (k2 + κ2)/k2κ2 in terms of a ratio of energies, ε = E/V0. k and κ
are de�ned by [7D.17–269] and [7D.18–269], respectively.�e factors in ħ and
m cancel out, leaving κ ∝ (V0 − E) and k ∝ E, which gives
(κ2 + k2)2
k2κ2
= (E + V0 − E)2
E(V0 − E)
= V 20
E(V0 − E)
= 1
ε(1 − ε)
making the transmission probability
T = ((eκW − e−κW)2
16ε(1 − ε)
+ 1)
−1
When κW ≫ 1 the negative term inside the parentheses is negligible compared
to the positive term, and the 1 is negligible compared to the exponential term,
such that
T ≈ ( e2κW
16ε(1 − ε)
)
−1
= 16ε(1 − ε)e−2κW
P7D.14 �e probability of the particle being inside the barrier is the integral of the
probability density, ∣ψ∣2, within the barrier which extends from x = 0 to x =∞
P = ∫
∞
0
(Ne−κx)2 dx = N2 ∫
∞
0
e−2κxdx = N2/2κ
�e average penetration depth is interpreted as the expectation values of x,
computed inside the barrier. �e required integral is of the form of Integral
G.2 with k = 2κ.
⟨x⟩ = ∫
∞
0
x(Ne−κx)2 dx = N2 ∫
∞
0
xe−2κxdx = N2/(2κ)2
7E Vibrational motion
Answers to discussion questions
D7E.2 For the harmonic oscillator the spacing of the energy levels is constant.�ere-
fore, relative to the energy of the oscillator, the spacing becomes progressively
smaller as the quantum number increases. In the limit of very high quantum
numbers this spacing becomes negligible compared to the total energy, and
e�ectively the energy can take any value, as in the classical case.