Chemistry Reference
In-Depth Information
McMurry, S. M. (1993)
Quantum Mechanics
, Second Edition, Addison-Wesley,
London.
Matthews, P. T. (1996)
Introduction to QuantumMechanics
, McGraw-Hill Education,
New York.
Schiff, L. I. (1968)
Quantum Mechanics
, Third Edition, McGraw-Hill, Tokyo.
Problems
1.1 Show that in a quantum well of depth
V
0
and width
a
the energies of
states of odd parity are given by
, where
k
2
2
−
k
cot
(
ka
/
2
)
=
κ
=
2
mE
/
2
.
1.2 Normalise the wavefunctions,
2
and
κ
=
2
m
(
V
0
−
E
)/
ψ
(
x
)
=
a
n
sin
(
n
π
x
/
L
)
, of the infinite
n
square well, for which
V
otherwise.
Show that the wavefunctions are orthogonal to each other, that is,
L
0
ψ
n
(
(
x
)
=
0, for 0
<
x
<
L
,and
=∞
x
)ψ
(
x
)
d
x
=
δ
m
mn
1.3 A trial function,
f
(
x
)
, differs from the ground state wavefunction,
ψ
(
x
)
, by a small amount, which we write as
1
f
(
x
)
=
ψ
(
x
)
+
ε
u
(
x
)
1
where
, the
variational estimate of the ground state energy
E
1
, differs from
E
1
only by a term of order
ψ
(
x
)
and
u
(
x
)
are normalised, and
ε
1. Show that
E
1
2
, and find this term. [This shows that the
relative errors in the calculated variational energy can be considerably
smaller than the error in the trial function used.]
1.4 Consider an infinite square well between
ε
2.
a Use the variational method to estimate the ground state energy
in this well assuming
f
−
L
/
2 and
+
L
/
n
x
n
, where
n
is an even
(
x
)
=
(
L
/
2
)
−
integer,
2. Comment why the function becomes an increasingly
unsuitable starting function with increasing
n
.
≥
3
to
b Justify the choice of the cubic function
g
(
x
)
=
(
2
x
/
L
)
−
(
2
x
/
L
)
estimate the energy of the first excited state. Use
g
(
x
)
to estimate
E
2
and compare your result with the exact solution.
c Suggest a suitable polynomial form for the variational function
which might be chosen to estimate the energy of the second and
higher excited states.
1.5 Consider a particle moving in the one-dimensional harmonic
oscillator potential,
V
1
2
kx
2
. By using the trial function,
f
(
x
)
=
(
x
)
=
x
2
exp
, estimate the ground state energy of the harmonic oscilla-
tor. We can use
g
(
−
α
)
x
2
as a trial function to estimate the
lowest state of odd parity, that is, the first excited state. Estimate this
energy.
(
x
)
=
x
exp
(
−
β
)
