Physics 89 (Mathematical Methods)
Problem Set #4
Due by October 5, 2021
1
Pauli matrices
The three matrices:
0 1
A=
,
1 0
0 −i
B=
,
i 0
1 0
C=
0 −1
are known as Pauli spin matrices. They play an important role in the quantum mechanical theory
of particles with spin.1
(a) Calculate the following matrices:
AB =?,
BA =?,
AC =?,
CA =?,
BC =?,
CB =?
Is AB equal to BA or not?
(b) Express your answers to part (a) in terms of the matrices ±iA, ±iB, and ±iC.
(c) Calculate
A2 =?,
2
B 2 =?,
C 2 =?,
Polynomial approximation
We wish to approximate a function y = f (x) by a polynomial
y ≈ α + βx + γx2
such that the approximation will be exact at three points: x = 1, 2, 3. Find α, β, γ in terms of
f (1), f (2), f (3), by writing three linear equations for α, β, γ. Write the equations in matrix form,
and perform row-operations to obtain the solution.
3
Masses connected by springs
K
m
K
K
m
- x1
m
- x2
- F (t)
m
- x3
= f cos ωt
- x4
Four equal masses m are connected in a row with three identical springs between them. The
spring constant of each of the springs is K. An oscillating force F (t) = f cos ωt is applied to
1
The standard notation of Quantum Mechanics is σx = A, σy = B, and σz = C.
1
the right-most mass. Let x1 , . . . , x4 be the displacements (from rest position) of the masses. The
equations of motion that follow from Newton’s second law are:
mẍ1
mẍ2
mẍ3
mẍ4
=
=
=
=
K(x2 − x1 ) ,
K(x3 − x2 ) − K(x2 − x1 ) ,
K(x4 − x3 ) − K(x3 − x2 ) ,
−K(x4 − x3 ) + f cos ωt .
Assuming each mass oscillates with the same frequency ω, we can guess a solution of the form
x1 = A1 cos ωt ,
x2 = A2 cos ωt ,
x3 = A3 cos ωt ,
x4 = A4 cos ωt ,
where A1 , . . . , A4 are unknown constants.
1. Substitute the above expressions into the equations of motion, cancel common terms, and
rearrange the equations to obtain a set of 4 linear equations for the 4 unknowns A1 , A2 , A3 ,
and A4 . Express the equations in matrix form.
(You may use the shorthands λ ≡ mω 2 /K and β ≡ f /K.)
2. Solve the equations for the case K = 13 mω 2 .
4
Determinants
Evaluate the following determinants using your favorite method (please show your work):
1 2
2 1
,
1 2 3
2 3 1
3 1 2
,
1
2
3
4
2
3
4
1
3
4
1
2
4
1
2
3
,
?? Extra problem ot for credit
can you derive, conjecture, or find online a general formula for
1 2 3 4 ···
n
2 3 4 5 ···
1
.. .. .. .. . .
..
.
. . . .
.
n 1 2 3 · · · (n − 1)
=?
(The three determinants in the first part of the problem are a special case of that general formula,
for n = 2, 3, 4.)
2