Elements of Quantum Theory
85
2. The discreteness of the energy levels is significant only for small m and l.
For example, if m ≈ 10
–3
kg and l ≈ 0.1 m, the separation between the
energy levels is of the order of 10
–62
J which is quite negligible. On the
other hand, for an electron in an atom, m ≈ 10
–30
kg and l ≈ 10
–10
m, so that
∆E n ~ (60n) eV and the discreteness becomes important.
3. It is seen that the states with n ≥ 2 have nodes inside the box. Since
probability density is given by |φ n (x)|
2
, this means that there are some
regions where the particle will not be found, which is totally incompatible
with the classical ideas of trajectories.
4. If the potential in the region x < 0 and x > l, is not infinite but finite, the
wave function can penetrate into this region. Therefore, it is not forced to
be zero at x = 0 or x = l. Hence, the wave function varies more gently
inside the region 0 ≤ x ≤ l, and the energies, which are related to the
second derivative of the wave function are lower than in the case where
the potential is infinite for x < 0 and x > l.
5. It is observed that
'
0
( ) , ( )
l
n
n
x
x dx
φ
φ
∫
= 0, for n ≠ n′
(3.103)
0
( ) ( )
l
n
n
x
x dx
φ
φ
∫
= 1
(3.104)
which can together be written as
'
0
( )
( )
l
n
n
x
x dx
φ
φ
∫
= δ n, n ′
(3.105)
where the Kronecker delta δ n,n ′ is 1 for n = n′ and zero otherwise. Thus,
these states are orthonormal. It can also be shown that any general state
of a particle in the box can be written as a linear combination of the
energy eigenstates, i.e.
ψ (x) =
1
( )
n n
n
a
x
∞
=
φ
∑
2
1
| |
n
n
a
∞
=
∑
= 1
(3.106)
which means that the eigenstates φ n (x) are complete. The orthonormality
and completeness are important properties associated with the eigenstates
of any physical observable (see Sec. 3.4).