Consequences of the Uncertainty Principle
The deBroglie relation
λ
=
h/p
and the uncertainty principle Δ
p
x
Δ
x
≥
¯
h/
2, which reﬂect
the wavelike nature of matter, imply that a
localized
particle (in an atom or a nucleus or
passing through a slit, etc.) cannot have zero kinetic energy.
Three ways to see this (there are more!):
1. From the uncertainty principle, if a particle is conﬁned to Δ
x
, the momentum will be
at least Δ
p
x
= ¯
h/
(2Δ
x
), where ¯
h
=
h/
2
π
.
2. If a particle with initial momentum
p
x
=
p
and
p
y
= 0 passes through a slit of width
d
, it will di±ract, which means it spreads out in the
y
direction. So localizing in the
y
direction makes
p
y
>
0. Estimate: The angle to the ﬁrst minimum is given by
λ
=
d
sin
θ
and sin
θ
≈
p
y
/p
. The deBroglie relation
λ
=
h/p
gives the uncertainty
principle result with
δy
≈
d
and
δp
y
≈
p
y
.
3. The wavelength of a particle cannot be much larger than the size of the region of
localization. From
p
=
h/λ
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 Spring '11
 Furnstahl
 Kinetic Energy, Momentum, Quantum Physics, Photon, Special Relativity, Uncertainty Principle

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