1
Ph 12b
Homework Assignment No. 5
Due: 5pm, Thursday, 18 February 2010
1. Minimal uncertainty I: particle in one dimension
(10 points).
If we measure the Hermitian operator
ˆ
A
in the state vector

ψ
a
, the
variance of the measurement outcomes is
(Δ
A
)
2
=
v
v
v
v
A
ψ

p
ˆ
A
A
ˆ
A
a
P
2

ψ
a
v
v
v
v
2
,
where
A
ˆ
A
a
denotes
A
ψ

ˆ
A

ψ
a
; the standard deviation Δ
A
is also called
the “uncertainty” of the observable
ˆ
A
in the state

ψ
a
. The
uncertainty
principle
, derived in class, is an inequality relating the product of the
uncertainties for two Hermitian operators
ˆ
A
and
ˆ
B
to the expectation
value of their commutator:
Δ
A
Δ
B
≥
1
2
v
v
v
A
ψ

b
ˆ
A,
ˆ
B
B

ψ
a
v
v
v
.
(1)
The online lecture notes for the Feb. 4 lecture include a discussion of
when eq.(1) is satis±ed as an equality — we have
Δ
A
Δ
B
=
1
2
v
v
v
A
ψ

b
ˆ
A,
ˆ
B
B

ψ
a
v
v
v
.
(2)
if and only if there exists a real number
γ
and a complex number
λ
such that
p
ˆ
A

iγ
ˆ
B

λ
P

ψ
a
= 0
.
For a particle moving in one dimension, with position operator ˆ
x
and
wavenumber operator
ˆ
k
=

i
d
dx
, eq.(2) becomes
Δ
x
Δ
k
= 1
/
2;
we say that a wavefunction satisfying this condition has
minimal un
certainty
.
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 Spring '09
 DUDKO
 mechanics, Work, Meter, Uncertainty Principle, Particle, Hilbert space, minimal uncertainty

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