Homework #2 — PHYS 622 — Fall 2016
Due on Friday, September 23, 2016 online
Professor Victor Yakovenko
Office: 2115 Physics Building
Web page: http://physics.umd.edu/~yakovenk/teaching/
Textbook: Sakurai and Napolitano, Modern Quantum Mechanics,
2nd edition, Addison Wesley Pearson, 2011, ISBN 9780805382914
Total score is 55 points.
Ch. 1 Fundamental Concepts
1. Problem 1.17, 5 points. Degeneracy for non-commuting operators.
2. Problem 1.18, 10 points. The states with minimal uncertainty.
(a) Do as directed in the book.
(b) Prove that the minimal uncertainty, i.e. the equality in Eq. (1.4.53),
1
h(∆A)2 i h(∆B)2 i = |h[A, B]i|2
4
(1)
is achieved for such a state |αi that
∆A|αi = −iq ∆B|αi,
(2)
where i is imaginary, and q is a real coefficient.
Hint: Follow the proof on page 35. The Schwartz equality in Eqs. (1.4.54),
(1.4.55), and (1.4.59) is achieved when the two vectors are parallel, i.e. when
∆A|αi = c∆B|αi with some (complex) coefficient c. The last term in Eqs. (1.4.62)
and (1.4.63) vanishes when c is imaginary.
(c) Let us apply Eq. (2) to the coordinate x and momentum p operators acting on
the wave function ψ(x0 ) = hx0 |αi in coordinate representation:
!
d
(x0 − hxi)ψ(x0 ) = −iq −ih̄ 0 − hpi ψ(x0 ).
dx
(3)
By solving the differential equation (3), calculate the minimal-uncertainty wave
function ψ(x) and show that it is a Gaussian wave packet. What is the dimensionality of q in this case?
3. 15 points. Uncertainty relation for classical signals.
The uncertainty relation is usually associated with quantum mechanics (it contains h̄).
However, it can be rewritten without h̄ as an uncertainty relation between the wave
number k = p/h̄ and coordinate:
h(∆k)2 i h(∆x)2 i ≥ 1/4,
(4)
2
Homework #2, PHYS 622, Fall 2016, Prof. Yakovenko
or frequency and time:
h(∆ω)2 i h(∆t)2 i ≥ 1/4.
(5)
Eqs. (4) and (5) are well known in such classical disciplines as optics, electrical engineering, data transmission, etc. They are nothing but identities relating the width of
a function and the width of its Fourier transform.
Show that computer circuits and music record players obey the uncertainty relation
(5). Let us denote the voltage as a function of time t somewhere in the electric circuit
by f (t). For generality, let us take f (t) to be a complex function of the real time t.
Let us assume that f (t) has the shape of a normalized pulse vanishing at t → ±∞
sufficiently fast, so that
Z
∞
|f (t)|2 dt = 1.
(6)
−∞
Arrival of the pulse signifies arrival of a bit of information. However, since f (t) has
some width in time, there is uncertainty in the arrival time. The center of the pulse is
hti =
Z ∞
t |f (t)|2 dt,
(7)
−∞
and the uncertainty of the arrival time is
2
h(∆t) i =
Z ∞
(t − hti)2 |f (t)|2 dt.
(8)
−∞
The Fourier transform of the signal is
1 Z ∞ iωt
e f (t) dt.
h(ω) = √
2π −∞
(9)
The mean and the variance of frequency are defined as follows
hωi =
Z ∞
2
ω |h(ω)| dω,
2
h(∆ω) i =
−∞
Z ∞
(ω − hωi)2 |h(ω)|2 dω.
(10)
−∞
With the definitions (8)–(10), prove Eq. (5).
Directions:
(a) First, let us shift the time argument to t0 = t − hti in the function f (t), so that
ht0 i = 0. This causes a phase shift h(ω) → h(ω)eiωhti in Eq. (9), but this phase
shift does not affect Eq. (10).
Similarly, let us shift the frequency argument to ω 0 = ω − hωi in h(ω) in Eq. (9)
so that hω 0 i = 0. This causes a phase shift f (t) → f (t)eihωit , but this phase shift
does not affect Eq. (8).
Thus, without loss of generality, you may consider the case where the means
vanish ht0 i = 0 and hω 0 i = 0, and drop the primes in notation.
Homework #2, PHYS 622, Fall 2016, Prof. Yakovenko
3
(b) Show that
h(∆ω)2 i =
Z ∞
ωh∗ (ω) ωh(ω) dω =
−∞
Z ∞
−∞
df ∗ (t) df (t)
dt.
dt
dt
(11)
The following expression for the delta-function may be useful, see Eq. (1.7.31)
Z ∞
−∞
dk ik(x−x0 )
e
= δ(x − x0 ).
2π
(12)
(c) Now, using the Schwartz inequality, show that
2
2
h(∆t) ih(∆ω) i =
Z ∞
∗
tf (t) tf (t) dt
−∞
Z ∞
−∞
2
Z ∞
df (t) df ∗ (t) df (t)
tf ∗ (t)
dt ≥ dt .
dt
dt
dt
−∞
(13)
(d) Integrating by parts in the integral
Z ∞
tf ∗ (t)
−∞
df (t)
dt,
dt
(14)
show that the real part is
<
Z ∞
tf ∗ (t)
−∞
so
Z
∞
tf ∗ (t)
−∞
df (t)
1
dt = − ,
dt
2
(15)
2
df (t) 1
dt ≥ ,
dt
4
(16)
which finishes the proof of Eq. (5).
Eq. (5) demonstrates that, in order to reproduce short-time variations of a signal with
high fidelity, the system must have sufficiently wide bandwidth in frequency. That is
why a fiber optics data network, utilizing electromagnetic waves in the frequency range
of visible light, can transmit digital data much faster than a copper wire network, whose
range is limited to much lower frequencies.
4. 25 points. Uncertainty relation in 3D.
In three dimensions (3D), the coordinate x = (x, y, z) and momentum p = (px , py , pz )
are vectors, and their squares (scalars) are denoted as r2 = x2 and p2 = p2 .
(a) Generalize the uncertainty relation from 1D to 3D as
h(x − hxi)2 i h(p − hpi)2 i ≥ ?
and obtain the right-hand side.
(17)
4
Homework #2, PHYS 622, Fall 2016, Prof. Yakovenko
(b) The wave function of the ground state of the hydrogen atom is spherically symmetrical and has the form
ψ0 (r) = c e−r/a ,
(18)
where c is a normalization coefficient, and a is the Bohr radius.
Obtain the coefficient c such that the wave function (18) is normalized:
Z
d3 x0 ψ02 (r) = 1.
(19)
Hint: Use spherical coordinates d3 x0 . . . = 4πr2 dr . . ., beware that (ey )2 = e2y
2
not ey , and prove the following formula by integration by parts
R
Z ∞
R
dy y n e−y/b = n! bn+1 .
(20)
0
(c) For the hydrogen wave function, hxi = 0 by spherical symmetry. Calculate the
coordinate uncertainty
h(x − hxi)2 i =
Z
d3 x0 r2 ψ02 (r).
(21)
Check that it has a correct dimension proportional to a2 .
(d) Using Eq. (1.7.51b), calculate the hydrogen wave function in the momentum representation
Z
1
0 0
d3 x0 ψ0 (r) e−ip ·x /h̄ ,
(22)
φ0 (p) =
3/2
(2πh̄)
Hint: Use spherical coordinates p0 ·x0 = pr cos θ and d3 x . . . = 2π(sin θ dθ)r2 dr . . .,
and integrate over θ first.
R
R
(e) Obviously, hpi = 0 by spherical symmetry. Calculate the momentum uncertainty
h(p − hpi)2 i =
Z
d3 p0 p2 φ20 (p).
(23)
Check that it has a correct dimension.
(f ) Using your results from Eqs. (21) and (23), verify that the uncertainty relation
(17) is satisfied for the hydrogen wave function.
September 16, 2016