Matrix quantum mechanics​
​by Paul Abbott​
School of Physics, Mathematics and Computing
The University of Western Australia
​paul.c.abbott@uwa.edu.au
Ref: Computing quantum eigenvalues made easy Korsch and Glück (2002), Eur. J. Phys. 23 413-426
A fundamental problem in quantum mechanics is the calculation of the discrete spectrum of the Schrödinger operator. This question introduces a straightforward matrix method for computing the eigenvalues of operators. By expanding both the states of the system and the operators in a suitable basis set
ν〉
,
ν=0,1,…
, the problem is transformed into a matrix equation.
With a discrete matrix representation of the position,
x
μ,ν
=μ

x
ν
, and momentum operators,
p
μ,ν
=μ

p
ν
, in a convenient representation
ν〉
,
μ,ν=0,1,2,…
, one can use the (completeness of the) matrix representation to compute matrix elements of other operators, such as the potential
μ

V
(

x
)ν
, the kinetic energy
μ

T


p
ν
, or a general operator
μ




x
,

p
ν
.

Harmonic oscillator eigenstates

In terms of normalized eigenstates of the harmonic oscillator Hamiltonian,

H
0
=
2

p
2m
+
1
2
m
2
ω
0
2

x
,
(
1
)
the (infinite) matrix representation of the position operator,

x
, is
x=
x
0
2
0
1
0
0
⋯
1
0
2
0
⋯
0
2
0
3
⋯
0
0
3
0
⋯
⋮
⋮
⋮
⋮
⋱
,
(
2
)
where
{x}
μ,ν
≡
x
μ,ν
=μ

x
ν
, and the matrix representation of the momentum operator,

p
, is
p=

p
0
2
0
-
1
0
0
⋯
1
0
-
2
0
⋯
0
2
0
-
3
⋯
0
0
3
0
⋯
⋮
⋮
⋮
⋮
⋱
,
(
3
)
where
{p}
μ,ν
≡
p
μ,ν
=μ

p
ν
and
x
0
=
1/2
(ℏ/(m
ω
0
))
and
p
0
=
1/2
(ℏm
ω
0
)
.
Exercise 1
The harmonic oscillator eigenfunctions in coordinate space are
ψ
ν_
(x_)=
-1/2

π
a
ν
2
ν!
-
2
x
2
2
a


H
ν
x
a
;
Verify that the
ψ
ν
are orthonormal by computing
〈μ|ν〉:=
∞
∫
-∞
*
ψ
μ
(x)
ψ
ν
(x)x
for
μ,ν=0,1,…,4
. [3 Marks]

Assuming
that
a>0
assists in the computation of this and other integrals in this section.
Solution
The
ψ
ν
are orthonormal:
〈μ_ν_〉:=〈μν〉=Assuminga>0,
∞
∫
-∞
*
ψ
μ
(x)
ψ
ν
(x)x
Table[〈μν〉,{μ,0,4},{ν,0,4}]
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
Exercise 2
Visualize
2
ψ
ν
(x)
+
ℰ
ν
for the energy eigenvalues
ℰ
ν
=ν+1/2
for
ν=0,1,…,6
, together with the potential,
1
2
2
x
, on one plot for
a=1
. [3 Marks]
Solution
Exercise 3

Exercise 4

Exercise 5

Exercise 6


Raising and lowering operators

In the following we will use units in which
m=ℏ=
ω
0
=1
. The harmonic oscillator Hamiltonian becomes

H
0
=
1
2
2

p
+
1
2
2

x
.
(
4
)
Introducing matrix representations of the raising operator
†


,
†
a
=
0
0
0
0
⋯
1
0
0
0
⋯
0
2
0
0
⋯
0
0
3
0
⋯
⋮
⋮
⋮
⋱
⋱
,
(
5
)
and lowering operator

a
,
a=
0
1
0
0
⋯
0
0
2
0
⋯
0
0
0
3
⋯
0
0
0
0
⋱
⋮
⋮
⋮
⋮
⋱
,
(
6
)
the matrix representation of the position operator,

x
, is
x=
1
2
(
†
a
+a),
(
7
)
and the matrix representation of the momentum operator,

p
, is
p=

2
(
†
a
-a).
(
8
)
Exercise 7
Define
k×k
matrix representations of the raising and lowering operators. Verify that
†
a
.a
— the so-called number operator — is a diagonal matrix with eigenvalues
0,1,2,…,k-1
, for
k=10
. [4 Marks]
Solution
a/:
a
k_
:=SparseArray{i_,j_}/;ji+1
j-1
,{k,k}
a
10
//Normal
0
1
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
3
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0
0
0
0
7
0
0
0
0
0
0
0
0
0
0
2
2
0
0
0
0
0
0
0
0
0
0
3
0
0
0
0
0
0
0
0
0
0
†
(
a
10
)
.
a
10
//Normal
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
3
0
0
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0
0
0
0
7
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
0
0
9
Eigenvalues[%]//Sort
{0,1,2,3,4,5,6,7,8,9}

Energies of the harmonic and quartic oscillator

Exercise 8
Use equations (
8
) and (
8
) to define
k×k
matrix representations of the position operator,

x
, momentum operator,

p
, and the harmonic oscillator Hamiltonian,

H
0
=
1
2
2

p
+
1
2
2

x
. Determine the eigenvalues (energies) of the Hamiltonian matrix for
k=10
and comment. (Hint: if the
k×k
matrix representation of the position operator is denoted
x
k
then
2

x
↔
x
k
.
x
k
and similarly for

p
. What do you expect for the harmonic oscillator energies?) [4 Marks]
Solution

Eigenfunctions

and momentum space wavefunction,
Exercise 10
Solution
Compare these to the corresponding coordinate space wavefunctions:
Note that the wavefunctions have an inflection point where they intersect the potential curve.