Description
1. Byron & Fuller, Chapter 4, problem 4.
2. Byron & Fuller, Chapter 4, problem 6.
3. Byron & Fuller, Chapter 4, problem 17.
4. Consider the three vectors
~v1 = ˆi + ˆj + ˆk
~v2 = ˆi + 2 ˆj + 3 ˆk
~v3 = ˆi + 2 ˆj + ˆk
Perform Gram-Schmidt orthonormalization on the set {~v1, ~v2, ~v3}, starting with ~v1 as
your first basis vector.
5. Spinors: Spin is often introduced in undergraduate physics courses simply as a 2-vector.
This can be a bit confusing. Let’s see why.
In problem 28 from chapter 3 you learned that the operator T ≡ˆ e
a ∂x
is the translation
operator, in that
Tˆ f(x) = f(x + a)
In quantum mechanics this is written as:
T ≡ˆ e
iapˆx/¯h
since ˆpx = −ih∂¯ x.
This is stated as “the momentum operator is the generator of
translations.” In a similar fashion one can show that the generator of infinitesimal1
rotations about the z axis is the operator Lˆ
z, the operator which gives the z component
of the angular momentum, so that to rotate something in quantum mechanics about
the z-axis by an infinitesimal angle ∆φ one can use the operator.
Rˆ
z = e
−iLˆz∆φ/¯h
1We have to be a little careful since while translation operators are Abelian, rotation operators are not.
What about spin?
By analogy, the operator to rotate a spin about an arbitrary axis defined by the unit
vector ˆn, is given by:
Rˆ
nˆ(∆φ) = exp
−iS · ˆ nˆ ∆φ,
h¯
!
= exp
−iσˆ · nˆ ∆φ
2
!
where we have set the spin operator S →ˆ h¯σ/ˆ 2, the spin-1/2 operator made from the
three Pauli matrices.
Note that we have implicitly assumed our basis to the along the
z-axis, with basis states:
| ↑i =
1
0
| ↓i =
0
1
(a) Show that
(ˆσ · nˆ)
k =
1 k is even
σˆ · nˆ k is odd
(b) From the above prove that for the spin-1/2 case
Rˆ
nˆ(∆φ) = cos ∆φ
2
− i sin
∆φ
2
nˆ · σˆ
(c) If we repeatedly rotate about the same axis ˆn, then we know that rotations simply
add, and we can write the general rotation matrix for an angle φ in the z-axis
basis:
cos φ
2 − i nz sin φ
2
(−i nx − ny) sin φ
2
(−i nx + ny) sin φ
2
cos φ
2 + i nz sin φ
2
!
(d) Using this matrix, what do you get if you rotate the state | ↑i:
i. by π/2 about the x-axis?
ii. by π about the x-axis?
iii. by 2π about the x-axis?
(e) Does the state | ↑i rotate as a vector?
6. Variational Calculations: Consider the one dimensional Schr¨odinger equation, already converted to dimensionless units:
Hψ(x) = (
−
d
2
dx2
−
1
1 + x
2
)
ψ(x) = E ψ(x)
with the boundary conditions ψ(−∞) = ψ(∞) = 0. We will assume a variational form
for the groundstate:
ψ(x) = s
2α3
π
1
x
2 + α2
where α is a constant that must be determined.
(a) Show that ψ(x; α) is normalized in the infinite interval.
(b) We wish to determine the value of α in a variational fashion so that:
I ≡ hψ|H|ψi
hψ|ψi
is a maximum. Evaluate an analytic expression for I(α).
(c) Either plot your function and find its minimum, or take the dervivative and
determine where it crosses zero. What is this value of α? Use this value of α to
find an estimate of the smallest eigenvalue.
(d) Determine the groundstate eigenvalue directly by a numerical solution of the problem, using the eigenvalue solver for the Schrodinger equation that you developed
in an earlier homework. Compare it to the value obtain from the variational
calculation.
This problem is a bit tedious. You may question the wisdom of doing a variational calculation, since it required numerical evaluations only slightly simpler
than writing an eigenvalue solver.
On the other hand, eigenvalue solvers become
much harder in two and three dimensions, whereas a good variational calculation
is often much simpler.