In quantum mechanics, the uncertainty principle
(also known as Heisenberg's uncertainty principle)
is any of a variety of mathematical inequalities
asserting a fundamental limit to the precision
with which certain pairs of physical properties
of a particle, known as complementary variables,
such as position x and momentum p, can be
known.
Introduced first in 1927, by the German physicist
Werner Heisenberg, it states that the more
precisely the position of some particle is
determined, the less precisely its momentum
can be known, and vice versa. The formal inequality
relating the standard deviation of position
σx and the standard deviation of momentum
σp was derived by Earle Hesse Kennard later
that year and by Hermann Weyl in 1928:
where ħ is the reduced Planck constant, h/(2π).
Historically, the uncertainty principle has
been confused with a somewhat similar effect
in physics, called the observer effect, which
notes that measurements of certain systems
cannot be made without affecting the systems,
that is, without changing something in a system.
Heisenberg utilized such an observer effect
at the quantum level (see below) as a physical
"explanation" of quantum uncertainty. It has
since become clearer, however, that the uncertainty
principle is inherent in the properties of
all wave-like systems, and that it arises
in quantum mechanics simply due to the matter
wave nature of all quantum objects. Thus,
the uncertainty principle actually states
a fundamental property of quantum systems
and is not a statement about the observational
success of current technology. It must be
emphasized that measurement does not mean
only a process in which a physicist-observer
takes part, but rather any interaction between
classical and quantum objects regardless of
any observer.Since the uncertainty principle
is such a basic result in quantum mechanics,
typical experiments in quantum mechanics routinely
observe aspects of it. Certain experiments,
however, may deliberately test a particular
form of the uncertainty principle as part
of their main research program. These include,
for example, tests of number–phase uncertainty
relations in superconducting or quantum optics
systems. Applications dependent on the uncertainty
principle for their operation include extremely
low-noise technology such as that required
in gravitational wave interferometers.
== Introduction ==
The uncertainty principle is not readily apparent
on the macroscopic scales of everyday experience.
So it is helpful to demonstrate how it applies
to more easily understood physical situations.
Two alternative frameworks for quantum physics
offer different explanations for the uncertainty
principle. The wave mechanics picture of the
uncertainty principle is more visually intuitive,
but the more abstract matrix mechanics picture
formulates it in a way that generalizes more
easily.
Mathematically, in wave mechanics, the uncertainty
relation between position and momentum arises
because the expressions of the wavefunction
in the two corresponding orthonormal bases
in Hilbert space are Fourier transforms of
one another (i.e., position and momentum are
conjugate variables). A nonzero function and
its Fourier transform cannot both be sharply
localized. A similar tradeoff between the
variances of Fourier conjugates arises in
all systems underlain by Fourier analysis,
for example in sound waves: A pure tone is
a sharp spike at a single frequency, while
its Fourier transform gives the shape of the
sound wave in the time domain, which is a
completely delocalized sine wave. In quantum
mechanics, the two key points are that the
position of the particle takes the form of
a matter wave, and momentum is its Fourier
conjugate, assured by the de Broglie relation
p = ħk, where k is the wavenumber.
In matrix mechanics, the mathematical formulation
of quantum mechanics, any pair of non-commuting
self-adjoint operators representing observables
are subject to similar uncertainty limits.
An eigenstate of an observable represents
the state of the wavefunction for a certain
measurement value (the eigenvalue). For example,
if a measurement of an observable A is performed,
then the system is in a particular eigenstate
Ψ of that observable. However, the particular
eigenstate of the observable A need not be
an eigenstate of another observable B: If
so, then it does not have a unique associated
measurement for it, as the system is not in
an eigenstate of that observable.
=== Wave mechanics interpretation ===
(Ref )
According to the de Broglie hypothesis, every
object in the universe is a wave, i.e., a
situation which gives rise to this phenomenon.
The position of the particle is described
by a wave function
Ψ
(
x
,
t
)
{\displaystyle \Psi (x,t)}
. The time-independent wave function of a
single-moded plane wave of wavenumber k0 or
momentum p0 is
ψ
(
x
)
∝
e
i
k
0
x
=
e
i
p
0
x
/
ℏ
.
{\displaystyle \psi (x)\propto e^{ik_{0}x}=e^{ip_{0}x/\hbar
}~.}
The Born rule states that this should be interpreted
as a probability density amplitude function
in the sense that the probability of finding
the particle between a and b is
P
⁡
[
a
≤
X
≤
b
]
=
∫
a
b
|
ψ
(
x
)
|
2
d
x
.
{\displaystyle \operatorname {P} [a\leq X\leq
b]=\int _{a}^{b}|\psi (x)|^{2}\,\mathrm {d}
x~.}
In the case of the single-moded plane wave,
|
ψ
(
x
)
|
2
{\displaystyle |\psi (x)|^{2}}
is a uniform distribution. In other words,
the particle position is extremely uncertain
in the sense that it could be essentially
anywhere along the wave packet.
On the other hand, consider a wave function
that is a sum of many waves, which we may
write this as
ψ
(
x
)
∝
∑
n
A
n
e
i
p
n
x
/
ℏ
,
{\displaystyle \psi (x)\propto \sum _{n}A_{n}e^{ip_{n}x/\hbar
}~,}
where An represents the relative contribution
of the mode pn to the overall total. The figures
to the right show how with the addition of
many plane waves, the wave packet can become
more localized. We may take this a step further
to the continuum limit, where the wave function
is an integral over all possible modes
ψ
(
x
)
=
1
2
π
ℏ
∫
−
∞
∞
φ
(
p
)
⋅
e
i
p
x
/
ℏ
d
p
,
{\displaystyle \psi (x)={\frac {1}{\sqrt {2\pi
\hbar }}}\int _{-\infty }^{\infty }\varphi
(p)\cdot e^{ipx/\hbar }\,dp~,}
with
φ
(
p
)
{\displaystyle \varphi (p)}
representing the amplitude of these modes
and is called the wave function in momentum
space. In mathematical terms, we say that
φ
(
p
)
{\displaystyle \varphi (p)}
is the Fourier transform of
ψ
(
x
)
{\displaystyle \psi (x)}
and that x and p are conjugate variables.
Adding together all of these plane waves comes
at a cost, namely the momentum has become
less precise, having become a mixture of waves
of many different momenta.
One way to quantify the precision of the position
and momentum is the standard deviation σ.
Since
|
ψ
(
x
)
|
2
{\displaystyle |\psi (x)|^{2}}
is a probability density function for position,
we calculate its standard deviation.
The precision of the position is improved,
i.e. reduced σx, by using many plane waves,
thereby weakening the precision of the momentum,
i.e. increased σp. Another way of stating
this is that σx and σp have an inverse relationship
or are at least bounded from below. This is
the uncertainty principle, the exact limit
of which is the Kennard bound. Click the show
button below to see a semi-formal derivation
of the Kennard inequality using wave mechanics.
=== Matrix mechanics interpretation ===
(Ref )
In matrix mechanics, observables such as position
and momentum are represented by self-adjoint
operators. When considering pairs of observables,
an important quantity is the commutator. For
a pair of operators Â and B̂, one defines
their commutator as
[
A
^
,
B
^
]
=
A
^
B
^
−
B
^
A
^
.
{\displaystyle [{\hat {A}},{\hat {B}}]={\hat
{A}}{\hat {B}}-{\hat {B}}{\hat {A}}.}
In the case of position and momentum, the
commutator is the canonical commutation relation
[
x
^
,
p
^
]
=
i
ℏ
.
{\displaystyle [{\hat {x}},{\hat {p}}]=i\hbar
.}
The physical meaning of the non-commutativity
can be understood by considering the effect
of the commutator on position and momentum
eigenstates. Let
|
ψ
⟩
{\displaystyle |\psi \rangle }
be a right eigenstate of position with a constant
eigenvalue x0. By definition, this means that
x
^
|
ψ
⟩
=
x
0
|
ψ
⟩
.
{\displaystyle {\hat {x}}|\psi \rangle =x_{0}|\psi
\rangle .}
Applying the commutator to
|
ψ
⟩
{\displaystyle |\psi \rangle }
yields
[
x
^
,
p
^
]
|
ψ
⟩
=
(
x
^
p
^
−
p
^
x
^
)
|
ψ
⟩
=
(
x
^
−
x
0
I
^
)
p
^
|
ψ
⟩
=
i
ℏ
|
ψ
⟩
,
{\displaystyle [{\hat {x}},{\hat {p}}]|\psi
\rangle =({\hat {x}}{\hat {p}}-{\hat {p}}{\hat
{x}})|\psi \rangle =({\hat {x}}-x_{0}{\hat
{I}}){\hat {p}}\,|\psi \rangle =i\hbar |\psi
\rangle ,}
where Î is the identity operator.
Suppose, for the sake of proof by contradiction,
that
|
ψ
⟩
{\displaystyle |\psi \rangle }
is also a right eigenstate of momentum, with
constant eigenvalue p0. If this were true,
then one could write
(
x
^
−
x
0
I
^
)
p
^
|
ψ
⟩
=
(
x
^
−
x
0
I
^
)
p
0
|
ψ
⟩
=
(
x
0
I
^
−
x
0
I
^
)
p
0
|
ψ
⟩
=
0.
{\displaystyle ({\hat {x}}-x_{0}{\hat {I}}){\hat
{p}}\,|\psi \rangle =({\hat {x}}-x_{0}{\hat
{I}})p_{0}\,|\psi \rangle =(x_{0}{\hat {I}}-x_{0}{\hat
{I}})p_{0}\,|\psi \rangle =0.}
On the other hand, the above canonical commutation
relation requires that
[
x
^
,
p
^
]
|
ψ
⟩
=
i
ℏ
|
ψ
⟩
≠
0.
{\displaystyle [{\hat {x}},{\hat {p}}]|\psi
\rangle =i\hbar |\psi \rangle \neq 0.}
This implies that no quantum state can simultaneously
be both a position and a momentum eigenstate.
When a state is measured, it is projected
onto an eigenstate in the basis of the relevant
observable. For example, if a particle's position
is measured, then the state amounts to a position
eigenstate. This means that the state is not
a momentum eigenstate, however, but rather
it can be represented as a sum of multiple
momentum basis eigenstates. In other words,
the momentum must be less precise. This precision
may be quantified by the standard deviations,
σ
x
=
⟨
x
^
2
⟩
−
⟨
x
^
⟩
2
{\displaystyle \sigma _{x}={\sqrt {\langle
{\hat {x}}^{2}\rangle -\langle {\hat {x}}\rangle
^{2}}}}
σ
p
=
⟨
p
^
2
⟩
−
⟨
p
^
⟩
2
.
{\displaystyle \sigma _{p}={\sqrt {\langle
{\hat {p}}^{2}\rangle -\langle {\hat {p}}\rangle
^{2}}}.}
As in the wave mechanics interpretation above,
one sees a tradeoff between the respective
precisions of the two, quantified by the uncertainty
principle.
== Robertson–Schrödinger uncertainty relations
==
The most common general form of the uncertainty
principle is the Robertson uncertainty relation.For
an arbitrary Hermitian operator
O
^
{\displaystyle {\hat {\mathcal {O}}}}
we can associate a standard deviation
σ
O
=
⟨
O
^
2
⟩
−
⟨
O
^
⟩
2
,
{\displaystyle \sigma _{\mathcal {O}}={\sqrt
{\langle {\hat {\mathcal {O}}}^{2}\rangle
-\langle {\hat {\mathcal {O}}}\rangle ^{2}}},}
where the brackets
⟨
O
⟩
{\displaystyle \langle {\mathcal {O}}\rangle
}
indicate an expectation value. For a pair
of operators
A
^
{\displaystyle {\hat {A}}}
and
B
^
{\displaystyle {\hat {B}}}
, we may define their commutator as
[
A
^
,
B
^
]
=
A
^
B
^
−
B
^
A
^
,
{\displaystyle [{\hat {A}},{\hat {B}}]={\hat
{A}}{\hat {B}}-{\hat {B}}{\hat {A}},}
In this notation, the Robertson uncertainty
relation is given by
σ
A
σ
B
≥
|
1
2
i
⟨
[
A
^
,
B
^
]
⟩
|
=
1
2
|
⟨
[
A
^
,
B
^
]
⟩
|
,
{\displaystyle \sigma _{A}\sigma _{B}\geq
\left|{\frac {1}{2i}}\langle [{\hat {A}},{\hat
{B}}]\rangle \right|={\frac {1}{2}}\left|\langle
[{\hat {A}},{\hat {B}}]\rangle \right|,}
The Robertson uncertainty relation immediately
follows from a slightly stronger inequality,
the Schrödinger uncertainty relation,
where we have introduced the anticommutator,
{
A
^
,
B
^
}
=
A
^
B
^
+
B
^
A
^
.
{\displaystyle \{{\hat {A}},{\hat {B}}\}={\hat
{A}}{\hat {B}}+{\hat {B}}{\hat {A}}.}
=== Examples ===
Since the Robertson and Schrödinger relations
are for general operators, the relations can
be applied to any two observables to obtain
specific uncertainty relations. A few of the
most common relations found in the literature
are given below.
For position and linear momentum, the canonical
commutation relation
[
x
^
,
p
^
]
=
i
ℏ
{\displaystyle [{\hat {x}},{\hat {p}}]=i\hbar
}
implies the Kennard inequality from above:
σ
x
σ
p
≥
ℏ
2
.
{\displaystyle \sigma _{x}\sigma _{p}\geq
{\frac {\hbar }{2}}.}
For two orthogonal components of the total
angular momentum operator of an object:
σ
J
i
σ
J
j
≥
ℏ
2
|
⟨
J
k
⟩
|
,
{\displaystyle \sigma _{J_{i}}\sigma _{J_{j}}\geq
{\frac {\hbar }{2}}{\big |}\langle J_{k}\rangle
{\big |},}
where i, j, k are distinct, and Ji denotes
angular momentum along the xi axis. This relation
implies that unless all three components vanish
together, only a single component of a system's
angular momentum can be defined with arbitrary
precision, normally the component parallel
to an external (magnetic or electric) field.
Moreover, for
[
J
x
,
J
y
]
=
i
ℏ
ε
x
y
z
J
z
{\displaystyle [J_{x},J_{y}]=i\hbar \varepsilon
_{xyz}J_{z}}
, a choice
A
^
=
J
x
{\displaystyle {\hat {A}}=J_{x}}
,
B
^
=
J
y
{\displaystyle {\hat {B}}=J_{y}}
, in angular momentum multiplets, ψ = |j,
m〉, bounds the Casimir invariant (angular
momentum squared,
⟨
J
x
2
+
J
y
2
+
J
z
2
⟩
{\displaystyle \langle J_{x}^{2}+J_{y}^{2}+J_{z}^{2}\rangle
}
) from below and thus yields useful constraints
such as j(j + 1) ≥ m(m + 1), and hence j
≥ m, among others. In non-relativistic mechanics,
time is privileged as an independent variable.
Nevertheless, in 1945, L. I. Mandelshtam and
I. E. Tamm derived a non-relativistic time–energy
uncertainty relation, as follows. For a quantum
system in 
a non-stationary state ψ and an observable
B represented by a self-adjoint operator
B
^
{\displaystyle {\hat {B}}}
, the following formula holds:
σ
E
σ
B
|
d
⟨
B
^
⟩
d
t
|
≥
ℏ
2
,
{\displaystyle \sigma _{E}{\frac {\sigma _{B}}{\left|{\frac
{\mathrm {d} \langle {\hat {B}}\rangle }{\mathrm
{d} t}}\right|}}\geq {\frac {\hbar }{2}},}
where σE is the standard deviation of the
energy operator (Hamiltonian) in the state
ψ, σB stands for the standard deviation
of B. Although the second factor in the left-hand
side has dimension of time, it is different
from the time parameter that enters the Schrödinger
equation. It is a lifetime of the state ψ
with respect to the observable B: In other
words, this is the time interval (Δt) after
which the expectation value
⟨
B
^
⟩
{\displaystyle \langle {\hat {B}}\rangle }
changes appreciably.
An informal, heuristic meaning of the principle
is the following: A state that only exists
for a short time cannot have a definite energy.
To have a definite energy, the frequency of
the state must be defined accurately, and
this requires the state to hang around for
many cycles, the reciprocal of the required
accuracy. For example, in spectroscopy, excited
states have a finite lifetime. By the time–energy
uncertainty principle, they do not have a
definite energy, and, each time they decay,
the energy they release is slightly different.
The average energy of the outgoing photon
has a peak at the theoretical energy of the
state, but the distribution has a finite width
called the natural linewidth. Fast-decaying
states have a broad linewidth, while slow-decaying
states have a narrow linewidth.
The same linewidth effect also makes it difficult
to specify the rest mass of unstable, fast-decaying
particles in particle physics. The faster
the particle decays (the shorter its lifetime),
the less certain is its mass (the larger the
particle's width).For the number of electrons
in a superconductor and the phase of its Ginzburg–Landau
order parameter
Δ
N
Δ
φ
≥
1.
{\displaystyle \Delta N\,\Delta \varphi \geq
1.}
=== A counterexample ===
Suppose we consider a quantum particle on
a ring, where the wave function depends on
an angular variable
θ
{\displaystyle \theta }
, which we may take to lie in the interval
[
0
,
2
π
]
{\displaystyle [0,2\pi ]}
. Define "position" and "momentum" operators
A
^
{\displaystyle {\hat {A}}}
and
B
^
{\displaystyle {\hat {B}}}
by
A
^
ψ
(
θ
)
=
θ
ψ
(
θ
)
,
θ
∈
[
0
,
2
π
]
,
{\displaystyle {\hat {A}}\psi (\theta )=\theta
\psi (\theta ),\quad \theta \in [0,2\pi ],}
and
B
^
ψ
=
−
i
ℏ
d
ψ
d
θ
,
{\displaystyle {\hat {B}}\psi =-i\hbar {\frac
{d\psi }{d\theta }},}
where we impose periodic boundary conditions
on
B
^
{\displaystyle {\hat {B}}}
. Note that the definition of
A
^
{\displaystyle {\hat {A}}}
depends on our choice to have
θ
{\displaystyle \theta }
range from 0 to
2
π
{\displaystyle 2\pi }
. These operators satisfy the usual commutation
relations for position and momentum operators,
[
A
^
,
B
^
]
=
i
ℏ
{\displaystyle [{\hat {A}},{\hat {B}}]=i\hbar
}
.Now let
ψ
{\displaystyle \psi }
be any of the eigenstates of
B
^
{\displaystyle {\hat {B}}}
, which are given by
ψ
(
θ
)
=
e
2
π
i
n
θ
{\displaystyle \psi (\theta )=e^{2\pi in\theta
}}
. Note that these states are normalizable,
unlike the eigenstates of the momentum operator
on the line. Note also that the operator
A
^
{\displaystyle {\hat {A}}}
is bounded, since
θ
{\displaystyle \theta }
ranges over a bounded interval. Thus, in the
state
ψ
{\displaystyle \psi }
, the uncertainty of
B
{\displaystyle B}
is zero and the uncertainty of
A
{\displaystyle A}
is finite, so that
σ
A
σ
B
=
0.
{\displaystyle \sigma _{A}\sigma _{B}=0.}
Although this result appears to violate the
Robertson uncertainty principle, the paradox
is resolved when we note that
ψ
{\displaystyle \psi }
is not in the domain of the operator
B
^
A
^
{\displaystyle {\hat {B}}{\hat {A}}}
, since multiplication by
θ
{\displaystyle \theta }
disrupts the periodic boundary conditions
imposed on
B
^
{\displaystyle {\hat {B}}}
. Thus, the derivation of the Robertson relation,
which requires
A
^
B
^
ψ
{\displaystyle {\hat {A}}{\hat {B}}\psi }
and
B
^
A
^
ψ
{\displaystyle {\hat {B}}{\hat {A}}\psi }
to be defined, does not apply. (These also
furnish an example of operators satisfying
the canonical commutation relations but not
the Weyl relations.)
For the usual position and momentum operators
X
^
{\displaystyle {\hat {X}}}
and
P
^
{\displaystyle {\hat {P}}}
on the real line, no such counterexamples
can occur. As long as
σ
x
{\displaystyle \sigma _{x}}
and
σ
p
{\displaystyle \sigma _{p}}
are defined in the state
ψ
{\displaystyle \psi }
, the Heisenberg uncertainty principle holds,
even if
ψ
{\displaystyle \psi }
fails to be in the domain of
X
^
P
^
{\displaystyle {\hat {X}}{\hat {P}}}
or of
P
^
X
^
{\displaystyle {\hat {P}}{\hat {X}}}
.
== Examples ==
(Refs )
=== Quantum harmonic oscillator stationary
states ===
Consider a one-dimensional quantum harmonic
oscillator (QHO). It is possible to express
the position and momentum operators in terms
of the creation and annihilation operators:
x
^
=
ℏ
2
m
ω
(
a
+
a
†
)
{\displaystyle {\hat {x}}={\sqrt {\frac {\hbar
}{2m\omega }}}(a+a^{\dagger })}
p
^
=
i
m
ω
ℏ
2
(
a
†
−
a
)
.
{\displaystyle {\hat {p}}=i{\sqrt {\frac {m\omega
\hbar }{2}}}(a^{\dagger }-a).}
Using the standard rules for creation and
annihilation operators on the eigenstates
of the QHO,
a
†
|
n
⟩
=
n
+
1
|
n
+
1
⟩
{\displaystyle a^{\dagger }|n\rangle ={\sqrt
{n+1}}|n+1\rangle }
a
|
n
⟩
=
n
|
n
−
1
⟩
,
{\displaystyle a|n\rangle ={\sqrt {n}}|n-1\rangle
,}
the variances may be computed directly,
σ
x
2
=
ℏ
m
ω
(
n
+
1
2
)
{\displaystyle \sigma _{x}^{2}={\frac {\hbar
}{m\omega }}\left(n+{\frac {1}{2}}\right)}
σ
p
2
=
ℏ
m
ω
(
n
+
1
2
)
.
{\displaystyle \sigma _{p}^{2}=\hbar m\omega
\left(n+{\frac {1}{2}}\right)\,.}
The product of these standard deviations is
then
σ
x
σ
p
=
ℏ
(
n
+
1
2
)
≥
ℏ
2
.
{\displaystyle \sigma _{x}\sigma _{p}=\hbar
\left(n+{\frac {1}{2}}\right)\geq {\frac {\hbar
}{2}}.~}
In particular, the above Kennard bound is
saturated for the ground state n=0, for which
the probability density is just the normal
distribution.
=== Quantum harmonic oscillator with Gaussian
initial condition ===
In a quantum harmonic oscillator of characteristic
angular frequency ω, place a state that is
offset from the bottom of the potential by
some displacement x0 as
ψ
(
x
)
=
(
m
Ω
π
ℏ
)
1
/
4
exp
⁡
(
−
m
Ω
(
x
−
x
0
)
2
2
ℏ
)
,
{\displaystyle \psi (x)=\left({\frac {m\Omega
}{\pi \hbar }}\right)^{1/4}\exp {\left(-{\frac
{m\Omega (x-x_{0})^{2}}{2\hbar }}\right)},}
where Ω describes the width of the initial
state but need not be the same as ω. Through
integration over the propagator, we can solve
for the full time-dependent solution. After
many cancelations, the probability densities
reduce to
|
Ψ
(
x
,
t
)
|
2
∼
N
(
x
0
cos
⁡
(
ω
t
)
,
ℏ
2
m
Ω
(
cos
2
⁡
(
ω
t
)
+
Ω
2
ω
2
sin
2
⁡
(
ω
t
)
)
)
{\displaystyle |\Psi (x,t)|^{2}\sim {\mathcal
{N}}\left(x_{0}\cos {(\omega t)},{\frac {\hbar
}{2m\Omega }}\left(\cos ^{2}(\omega t)+{\frac
{\Omega ^{2}}{\omega ^{2}}}\sin ^{2}{(\omega
t)}\right)\right)}
|
Φ
(
p
,
t
)
|
2
∼
N
(
−
m
x
0
ω
sin
⁡
(
ω
t
)
,
ℏ
m
Ω
2
(
cos
2
⁡
(
ω
t
)
+
ω
2
Ω
2
sin
2
⁡
(
ω
t
)
)
)
,
{\displaystyle |\Phi (p,t)|^{2}\sim {\mathcal
{N}}\left(-mx_{0}\omega \sin(\omega t),{\frac
{\hbar m\Omega }{2}}\left(\cos ^{2}{(\omega
t)}+{\frac {\omega ^{2}}{\Omega ^{2}}}\sin
^{2}{(\omega t)}\right)\right),}
where we have used the notation
N
(
μ
,
σ
2
)
{\displaystyle {\mathcal {N}}(\mu ,\sigma
^{2})}
to denote a normal distribution of mean μ
and variance σ2. Copying the variances above
and applying trigonometric identities, we
can write the product of the standard deviations
as
σ
x
σ
p
=
ℏ
2
(
cos
2
⁡
(
ω
t
)
+
Ω
2
ω
2
sin
2
⁡
(
ω
t
)
)
(
cos
2
⁡
(
ω
t
)
+
ω
2
Ω
2
sin
2
⁡
(
ω
t
)
)
=
ℏ
4
3
+
1
2
(
Ω
2
ω
2
+
ω
2
Ω
2
)
−
(
1
2
(
Ω
2
ω
2
+
ω
2
Ω
2
)
−
1
)
cos
⁡
(
4
ω
t
)
{\displaystyle {\begin{aligned}\sigma _{x}\sigma
_{p}&={\frac {\hbar }{2}}{\sqrt {\left(\cos
^{2}{(\omega t)}+{\frac {\Omega ^{2}}{\omega
^{2}}}\sin ^{2}{(\omega t)}\right)\left(\cos
^{2}{(\omega t)}+{\frac {\omega ^{2}}{\Omega
^{2}}}\sin ^{2}{(\omega t)}\right)}}\\&={\frac
{\hbar }{4}}{\sqrt {3+{\frac {1}{2}}\left({\frac
{\Omega ^{2}}{\omega ^{2}}}+{\frac {\omega
^{2}}{\Omega ^{2}}}\right)-\left({\frac {1}{2}}\left({\frac
{\Omega ^{2}}{\omega ^{2}}}+{\frac {\omega
^{2}}{\Omega ^{2}}}\right)-1\right)\cos {(4\omega
t)}}}\end{aligned}}}
From the relations
Ω
2
ω
2
+
ω
2
Ω
2
≥
2
,
|
cos
⁡
(
4
ω
t
)
|
≤
1
,
{\displaystyle {\frac {\Omega ^{2}}{\omega
^{2}}}+{\frac {\omega ^{2}}{\Omega ^{2}}}\geq
2,\quad |\cos(4\omega t)|\leq 1,}
we can conclude the following: (the right
most equality holds only when Ω = ω) .
σ
x
σ
p
≥
ℏ
4
3
+
1
2
(
Ω
2
ω
2
+
ω
2
Ω
2
)
−
(
1
2
(
Ω
2
ω
2
+
ω
2
Ω
2
)
−
1
)
=
ℏ
2
.
{\displaystyle \sigma _{x}\sigma _{p}\geq
{\frac {\hbar }{4}}{\sqrt {3+{\frac {1}{2}}\left({\frac
{\Omega ^{2}}{\omega ^{2}}}+{\frac {\omega
^{2}}{\Omega ^{2}}}\right)-\left({\frac {1}{2}}\left({\frac
{\Omega ^{2}}{\omega ^{2}}}+{\frac {\omega
^{2}}{\Omega ^{2}}}\right)-1\right)}}={\frac
{\hbar }{2}}.}
=== Coherent states ===
A coherent state is a right eigenstate of
the annihilation operator,
a
^
|
α
⟩
=
α
|
α
⟩
{\displaystyle {\hat {a}}|\alpha \rangle =\alpha
|\alpha \rangle }
,which may be represented in terms of Fock
states as
|
α
⟩
=
e
−
|
α
|
2
2
∑
n
=
0
∞
α
n
n
!
|
n
⟩
{\displaystyle |\alpha \rangle =e^{-{|\alpha
|^{2} \over 2}}\sum _{n=0}^{\infty }{\alpha
^{n} \over {\sqrt {n!}}}|n\rangle }
In the picture where the coherent state is
a massive particle in a QHO, the position
and momentum operators may be expressed in
terms of the annihilation operators in the
same formulas above and used to calculate
the variances,
σ
x
2
=
ℏ
2
m
ω
,
{\displaystyle \sigma _{x}^{2}={\frac {\hbar
}{2m\omega }},}
σ
p
2
=
ℏ
m
ω
2
.
{\displaystyle \sigma _{p}^{2}={\frac {\hbar
m\omega }{2}}.}
Therefore, every coherent state saturates
the Kennard bound
σ
x
σ
p
=
ℏ
2
m
ω
ℏ
m
ω
2
=
ℏ
2
.
{\displaystyle \sigma _{x}\sigma _{p}={\sqrt
{\frac {\hbar }{2m\omega }}}\,{\sqrt {\frac
{\hbar m\omega }{2}}}={\frac {\hbar }{2}}.}
with position and momentum each contributing
an amount
ℏ
/
2
{\displaystyle {\sqrt {\hbar /2}}}
in a "balanced" way. Moreover, every squeezed
coherent state also saturates the Kennard
bound although the individual contributions
of position and momentum need not be balanced
in general.
=== Particle in a box ===
Consider a particle in a one-dimensional box
of length
L
{\displaystyle L}
. The eigenfunctions in position and momentum
space are
ψ
n
(
x
,
t
)
=
{
A
sin
⁡
(
k
n
x
)
e
−
i
ω
n
t
,
0
<
x
<
L
,
0
,
otherwise,
{\displaystyle \psi _{n}(x,t)={\begin{cases}A\sin(k_{n}x)\mathrm
{e} ^{-\mathrm {i} \omega _{n}t},&0<x<L,\\0,&{\text{otherwise,}}\end{cases}}}
and
φ
n
(
p
,
t
)
=
π
L
ℏ
n
(
1
−
(
−
1
)
n
e
−
i
k
L
)
e
−
i
ω
n
t
π
2
n
2
−
k
2
L
2
,
{\displaystyle \varphi _{n}(p,t)={\sqrt {\frac
{\pi L}{\hbar }}}\,\,{\frac {n\left(1-(-1)^{n}e^{-ikL}\right)e^{-i\omega
_{n}t}}{\pi ^{2}n^{2}-k^{2}L^{2}}},}
where
ω
n
=
π
2
ℏ
n
2
8
L
2
m
{\displaystyle \omega _{n}={\frac {\pi ^{2}\hbar
n^{2}}{8L^{2}m}}}
and we have used the de Broglie relation
p
=
ℏ
k
{\displaystyle p=\hbar k}
. The variances of
x
{\displaystyle x}
and
p
{\displaystyle p}
can be calculated explicitly:
σ
x
2
=
L
2
12
(
1
−
6
n
2
π
2
)
{\displaystyle \sigma _{x}^{2}={\frac {L^{2}}{12}}\left(1-{\frac
{6}{n^{2}\pi ^{2}}}\right)}
σ
p
2
=
(
ℏ
n
π
L
)
2
.
{\displaystyle \sigma _{p}^{2}=\left({\frac
{\hbar n\pi }{L}}\right)^{2}.}
The product of the standard deviations is
therefore
σ
x
σ
p
=
ℏ
2
n
2
π
2
3
−
2
.
{\displaystyle \sigma _{x}\sigma _{p}={\frac
{\hbar }{2}}{\sqrt {{\frac {n^{2}\pi ^{2}}{3}}-2}}.}
For all
n
=
1
,
2
,
3
,
…
{\displaystyle n=1,\,2,\,3,\,\ldots }
, the quantity
n
2
π
2
3
−
2
{\displaystyle {\sqrt {{\frac {n^{2}\pi ^{2}}{3}}-2}}}
is greater than 1, so the uncertainty principle
is never violated. For numerical concreteness,
the smallest value occurs when
n
=
1
{\displaystyle n=1}
, in which case
σ
x
σ
p
=
ℏ
2
π
2
3
−
2
≈
0.568
ℏ
>
ℏ
2
.
{\displaystyle \sigma _{x}\sigma _{p}={\frac
{\hbar }{2}}{\sqrt {{\frac {\pi ^{2}}{3}}-2}}\approx
0.568\hbar >{\frac {\hbar }{2}}.}
=== Constant momentum ===
Assume a particle initially has a momentum
space wave function described by a normal
distribution around some constant momentum
p0 according to
φ
(
p
)
=
(
x
0
ℏ
π
)
1
/
2
⋅
exp
⁡
(
−
x
0
2
(
p
−
p
0
)
2
2
ℏ
2
)
,
{\displaystyle \varphi (p)=\left({\frac {x_{0}}{\hbar
{\sqrt {\pi }}}}\right)^{1/2}\cdot \exp {\left({\frac
{-x_{0}^{2}(p-p_{0})^{2}}{2\hbar ^{2}}}\right)},}
where we have introduced a reference scale
x
0
=
ℏ
/
m
ω
0
{\displaystyle x_{0}={\sqrt {\hbar /m\omega
_{0}}}}
, with
ω
0
>
0
{\displaystyle \omega _{0}>0}
describing the width of the distribution−−cf.
nondimensionalization. If the state is allowed
to evolve in free space, then the time-dependent
momentum and position space wave functions
are
Φ
(
p
,
t
)
=
(
x
0
ℏ
π
)
1
/
2
⋅
exp
⁡
(
−
x
0
2
(
p
−
p
0
)
2
2
ℏ
2
−
i
p
2
t
2
m
ℏ
)
,
{\displaystyle \Phi (p,t)=\left({\frac {x_{0}}{\hbar
{\sqrt {\pi }}}}\right)^{1/2}\cdot \exp {\left({\frac
{-x_{0}^{2}(p-p_{0})^{2}}{2\hbar ^{2}}}-{\frac
{ip^{2}t}{2m\hbar }}\right)},}
Ψ
(
x
,
t
)
=
(
1
x
0
π
)
1
/
2
⋅
e
−
x
0
2
p
0
2
/
2
ℏ
2
1
+
i
ω
0
t
⋅
exp
⁡
(
−
(
x
−
i
x
0
2
p
0
/
ℏ
)
2
2
x
0
2
(
1
+
i
ω
0
t
)
)
.
{\displaystyle \Psi (x,t)=\left({\frac {1}{x_{0}{\sqrt
{\pi }}}}\right)^{1/2}\cdot {\frac {e^{-x_{0}^{2}p_{0}^{2}/2\hbar
^{2}}}{\sqrt {1+i\omega _{0}t}}}\cdot \exp
{\left(-{\frac {(x-ix_{0}^{2}p_{0}/\hbar )^{2}}{2x_{0}^{2}(1+i\omega
_{0}t)}}\right)}.}
Since
⟨
p
(
t
)
⟩
=
p
0
{\displaystyle \langle p(t)\rangle =p_{0}}
and
σ
p
(
t
)
=
ℏ
/
x
0
2
,
{\displaystyle \sigma _{p}(t)=\hbar /x_{0}{\sqrt
{2}},}
this can be interpreted as a particle moving
along with constant momentum at arbitrarily
high precision. On the other hand, the standard
deviation of the position is
σ
x
=
x
0
2
1
+
ω
0
2
t
2
{\displaystyle \sigma _{x}={\frac {x_{0}}{\sqrt
{2}}}{\sqrt {1+\omega _{0}^{2}t^{2}}}}
such that the uncertainty product can only
increase with time as
σ
x
(
t
)
σ
p
(
t
)
=
ℏ
2
1
+
ω
0
2
t
2
{\displaystyle \sigma _{x}(t)\sigma _{p}(t)={\frac
{\hbar }{2}}{\sqrt {1+\omega _{0}^{2}t^{2}}}}
== Additional uncertainty relations ==
=== Mixed states ===
The Robertson–Schrödinger uncertainty relation
may be generalized in a straightforward way
to describe mixed states.
σ
A
2
σ
B
2
≥
(
1
2
tr
⁡
(
ρ
{
A
,
B
}
)
−
tr
⁡
(
ρ
A
)
tr
⁡
(
ρ
B
)
)
2
+
(
1
2
i
tr
⁡
(
ρ
[
A
,
B
]
)
)
2
{\displaystyle \sigma _{A}^{2}\sigma _{B}^{2}\geq
\left({\frac {1}{2}}\operatorname {tr} (\rho
\{A,B\})-\operatorname {tr} (\rho A)\operatorname
{tr} (\rho B)\right)^{2}+\left({\frac {1}{2i}}\operatorname
{tr} (\rho [A,B])\right)^{2}}
=== The Maccone–Pati uncertainty relations
===
The Robertson–Schrödinger uncertainty relation
can be trivial if the state of the system
is chosen to be eigenstate of one of the observable.
The stronger uncertainty relations proved
by Maccone and Pati give non-trivial bounds
on the sum of the variances for two incompatible
observables. For two non-commuting observables
A
{\displaystyle A}
and
B
{\displaystyle B}
the first stronger uncertainty relation is
given by
σ
A
2
+
σ
B
2
≥
±
i
⟨
Ψ
∣
[
A
,
B
]
|
Ψ
⟩
+
∣
⟨
Ψ
∣
(
A
±
i
B
)
∣
Ψ
¯
⟩
|
2
,
{\displaystyle \sigma _{A}^{2}+\sigma _{B}^{2}\geq
\pm i\langle \Psi \mid [A,B]|\Psi \rangle
+\mid \langle \Psi \mid (A\pm iB)\mid {\bar
{\Psi }}\rangle |^{2},}
where
σ
A
2
=
⟨
Ψ
|
A
2
|
Ψ
⟩
−
⟨
Ψ
∣
A
∣
Ψ
⟩
2
{\displaystyle \sigma _{A}^{2}=\langle \Psi
|A^{2}|\Psi \rangle -\langle \Psi \mid A\mid
\Psi \rangle ^{2}}
,
σ
B
2
=
⟨
Ψ
|
B
2
|
Ψ
⟩
−
⟨
Ψ
∣
B
∣
Ψ
⟩
2
{\displaystyle \sigma _{B}^{2}=\langle \Psi
|B^{2}|\Psi \rangle -\langle \Psi \mid B\mid
\Psi \rangle ^{2}}
,
|
Ψ
¯
⟩
{\displaystyle |{\bar {\Psi }}\rangle }
is a normalized vector that is orthogonal
to the state of the system
|
Ψ
⟩
{\displaystyle |\Psi \rangle }
and one should choose the sign of
±
i
⟨
Ψ
∣
[
A
,
B
]
∣
Ψ
⟩
{\displaystyle \pm i\langle \Psi \mid [A,B]\mid
\Psi \rangle }
to make this real quantity a positive number.
The second stronger uncertainty relation is
given by
σ
A
2
+
σ
B
2
≥
1
2
|
⟨
Ψ
¯
A
+
B
∣
(
A
+
B
)
∣
Ψ
⟩
|
2
{\displaystyle \sigma _{A}^{2}+\sigma _{B}^{2}\geq
{\frac {1}{2}}|\langle {\bar {\Psi }}_{A+B}\mid
(A+B)\mid \Psi \rangle |^{2}}
where
|
Ψ
¯
A
+
B
⟩
{\displaystyle |{\bar {\Psi }}_{A+B}\rangle
}
is a state orthogonal to
|
Ψ
⟩
{\displaystyle |\Psi \rangle }
.
The form of
|
Ψ
¯
A
+
B
⟩
{\displaystyle |{\bar {\Psi }}_{A+B}\rangle
}
implies that the right-hand side of the new
uncertainty relation
is nonzero unless
|
Ψ
⟩
{\displaystyle |\Psi \rangle }
is an eigenstate of
(
A
+
B
)
{\displaystyle (A+B)}
. One may note that
|
Ψ
⟩
{\displaystyle |\Psi \rangle }
can be an eigenstate of
(
A
+
B
)
{\displaystyle (A+B)}
without being an eigenstate of either
A
{\displaystyle A}
or
B
{\displaystyle B}
. However, when
|
Ψ
⟩
{\displaystyle |\Psi \rangle }
is an eigenstate of one of the two observables
the Heisenberg–Schrödinger uncertainty
relation becomes trivial. But the lower bound
in the new relation is nonzero
unless
|
Ψ
⟩
{\displaystyle |\Psi \rangle }
is an eigenstate of both.
=== Phase space ===
In the phase space formulation of quantum
mechanics, the Robertson–Schrödinger relation
follows from a positivity condition on a real
star-square function. Given a Wigner function
W
(
x
,
p
)
{\displaystyle W(x,p)}
with star product ★ and a function f, the
following is generally true:
⟨
f
∗
⋆
f
⟩
=
∫
(
f
∗
⋆
f
)
W
(
x
,
p
)
d
x
d
p
≥
0.
{\displaystyle \langle f^{*}\star f\rangle
=\int (f^{*}\star f)\,W(x,p)\,dx\,dp\geq 0.}
Choosing
f
=
a
+
b
x
+
c
p
{\displaystyle f=a+bx+cp}
, we arrive at
⟨
f
∗
⋆
f
⟩
=
[
a
∗
b
∗
c
∗
]
[
1
⟨
x
⟩
⟨
p
⟩
⟨
x
⟩
⟨
x
⋆
x
⟩
⟨
x
⋆
p
⟩
⟨
p
⟩
⟨
p
⋆
x
⟩
⟨
p
⋆
p
⟩
]
[
a
b
c
]
≥
0.
{\displaystyle \langle f^{*}\star f\rangle
={\begin{bmatrix}a^{*}&b^{*}&c^{*}\end{bmatrix}}{\begin{bmatrix}1&\langle
x\rangle &\langle p\rangle \\\langle x\rangle
&\langle x\star x\rangle &\langle x\star p\rangle
\\\langle p\rangle &\langle p\star x\rangle
&\langle p\star p\rangle \end{bmatrix}}{\begin{bmatrix}a\\b\\c\end{bmatrix}}\geq
0.}
Since this positivity condition is true for
all a, b, and c, it follows that all the eigenvalues
of the matrix are positive. The positive eigenvalues
then imply a corresponding positivity condition
on the determinant:
det
[
1
⟨
x
⟩
⟨
p
⟩
⟨
x
⟩
⟨
x
⋆
x
⟩
⟨
x
⋆
p
⟩
⟨
p
⟩
⟨
p
⋆
x
⟩
⟨
p
⋆
p
⟩
]
=
det
[
1
⟨
x
⟩
⟨
p
⟩
⟨
x
⟩
⟨
x
2
⟩
⟨
x
p
+
i
ℏ
2
⟩
⟨
p
⟩
⟨
x
p
−
i
ℏ
2
⟩
⟨
p
2
⟩
]
≥
0
,
{\displaystyle \det {\begin{bmatrix}1&\langle
x\rangle &\langle p\rangle \\\langle x\rangle
&\langle x\star x\rangle &\langle x\star p\rangle
\\\langle p\rangle &\langle p\star x\rangle
&\langle p\star p\rangle \end{bmatrix}}=\det
{\begin{bmatrix}1&\langle x\rangle &\langle
p\rangle \\\langle x\rangle &\langle x^{2}\rangle
&\left\langle xp+{\frac {i\hbar }{2}}\right\rangle
\\\langle p\rangle &\left\langle xp-{\frac
{i\hbar }{2}}\right\rangle &\langle p^{2}\rangle
\end{bmatrix}}\geq 0,}
or, explicitly, after algebraic manipulation,
σ
x
2
σ
p
2
=
(
⟨
x
2
⟩
−
⟨
x
⟩
2
)
(
⟨
p
2
⟩
−
⟨
p
⟩
2
)
≥
(
⟨
x
p
⟩
−
⟨
x
⟩
⟨
p
⟩
)
2
+
ℏ
2
4
.
{\displaystyle \sigma _{x}^{2}\sigma _{p}^{2}=\left(\langle
x^{2}\rangle -\langle x\rangle ^{2}\right)\left(\langle
p^{2}\rangle -\langle p\rangle ^{2}\right)\geq
\left(\langle xp\rangle -\langle x\rangle
\langle p\rangle \right)^{2}+{\frac {\hbar
^{2}}{4}}~.}
=== Systematic and statistical errors ===
The inequalities above focus on the statistical
imprecision of observables as quantified by
the standard deviation
σ
{\displaystyle \sigma }
. Heisenberg's original version, however,
was dealing with the systematic error, a disturbance
of the quantum system produced by the measuring
apparatus, i.e., an observer effect.
If we let
ε
A
{\displaystyle \varepsilon _{A}}
represent the error (i.e., inaccuracy) of
a measurement of an observable A and
η
B
{\displaystyle \eta _{B}}
the disturbance produced on a subsequent measurement
of the conjugate variable B by the former
measurement of A, then the inequality proposed
by Ozawa — encompassing both systematic
and statistical errors — holds:
Heisenberg's uncertainty principle, as originally
described in the 1927 formulation, mentions
only the first term of Ozawa inequality, regarding
the systematic error. Using the notation above
to describe the error/disturbance effect of
sequential measurements (first A, then B),
it could be written as
The formal derivation of the Heisenberg relation
is possible but far from intuitive. It was
not proposed by Heisenberg, but formulated
in a mathematically consistent way only in
recent years.
Also, it must be stressed that the Heisenberg
formulation is not taking into account the
intrinsic statistical errors
σ
A
{\displaystyle \sigma _{A}}
and
σ
B
{\displaystyle \sigma _{B}}
. There is increasing experimental evidence
that the total quantum uncertainty cannot
be described by the Heisenberg term alone,
but requires the presence of all the three
terms of the Ozawa inequality.
Using the same formalism, it is also possible
to introduce the other kind of physical situation,
often confused with the previous one, namely
the case of simultaneous measurements (A and
B at the same time):
The two simultaneous measurements on A and
B are necessarily unsharp or weak.
It is also possible to derive an uncertainty
relation that, as the Ozawa's one, combines
both the statistical and systematic error
components, but keeps a form very close to
the Heisenberg original inequality. By adding
Robertson
and Ozawa relations we obtain
ε
A
η
B
+
ε
A
σ
B
+
σ
A
η
B
+
σ
A
σ
B
≥
|
⟨
[
A
^
,
B
^
]
⟩
|
.
{\displaystyle \varepsilon _{A}\eta _{B}+\varepsilon
_{A}\,\sigma _{B}+\sigma _{A}\,\eta _{B}+\sigma
_{A}\sigma _{B}\geq \left|\langle [{\hat {A}},{\hat
{B}}]\rangle \right|.}
The four terms can be written as:
(
ε
A
+
σ
A
)
(
η
B
+
σ
B
)
≥
|
⟨
[
A
^
,
B
^
]
⟩
|
.
{\displaystyle (\varepsilon _{A}+\sigma _{A})\,(\eta
_{B}+\sigma _{B})\,\geq \,\left|\langle [{\hat
{A}},{\hat {B}}]\rangle \right|.}
Defining:
ε
¯
A
≡
(
ε
A
+
σ
A
)
{\displaystyle {\bar {\varepsilon }}_{A}\,\equiv
\,(\varepsilon _{A}+\sigma _{A})}
as the inaccuracy in the measured values of
the variable A and
η
¯
B
≡
(
η
B
+
σ
B
)
{\displaystyle {\bar {\eta }}_{B}\,\equiv
\,(\eta _{B}+\sigma _{B})}
as the resulting fluctuation in the conjugate
variable B,
Fujikawa established
an uncertainty relation similar to the Heisenberg
original one, but valid both for systematic
and statistical errors:
=== Quantum entropic uncertainty principle
===
For many distributions, the standard deviation
is not a particularly natural way of quantifying
the structure. For example, uncertainty relations
in which one of the observables is an angle
has little physical meaning for fluctuations
larger than one period. Other examples include
highly bimodal distributions, or unimodal
distributions with divergent variance.
A solution that overcomes these issues is
an uncertainty based on entropic uncertainty
instead of the product of variances. While
formulating the many-worlds interpretation
of quantum mechanics in 1957, Hugh Everett
III conjectured a stronger extension of the
uncertainty principle based on entropic certainty.
This conjecture, also studied by Hirschman
and proven in 1975 by Beckner and by Iwo Bialynicki-Birula
and Jerzy Mycielski is that, for two normalized,
dimensionless Fourier transform pairs f(a)
and g(b) where
f
(
a
)
=
∫
−
∞
∞
g
(
b
)
e
2
π
i
a
b
d
b
{\displaystyle f(a)=\int _{-\infty }^{\infty
}g(b)\ e^{2\pi iab}\,db}
and
g
(
b
)
=
∫
−
∞
∞
f
(
a
)
e
−
2
π
i
a
b
d
a
{\displaystyle \,\,\,g(b)=\int _{-\infty }^{\infty
}f(a)\ e^{-2\pi iab}\,da}
the Shannon information entropies
H
a
=
∫
−
∞
∞
f
(
a
)
log
⁡
(
f
(
a
)
)
d
a
,
{\displaystyle H_{a}=\int _{-\infty }^{\infty
}f(a)\log(f(a))\,da,}
and
H
b
=
∫
−
∞
∞
g
(
b
)
log
⁡
(
g
(
b
)
)
d
b
{\displaystyle H_{b}=\int _{-\infty }^{\infty
}g(b)\log(g(b))\,db}
are subject to the following constraint,
where the logarithms may be in any base.
The probability distribution functions associated
with the position wave function ψ(x) and
the momentum wave function φ(x) have dimensions
of inverse length and momentum respectively,
but the entropies may be rendered dimensionless
by
H
x
=
−
∫
|
ψ
(
x
)
|
2
ln
⁡
(
x
0
|
ψ
(
x
)
|
2
)
d
x
=
−
⟨
ln
⁡
(
x
0
∣
ψ
(
x
)
|
2
)
⟩
{\displaystyle H_{x}=-\int |\psi (x)|^{2}\ln(x_{0}\,|\psi
(x)|^{2})\,dx=-\left\langle \ln(x_{0}\mid
\psi (x)|^{2})\right\rangle }
H
p
=
−
∫
|
φ
(
p
)
|
2
ln
⁡
(
p
0
|
φ
(
p
)
|
2
)
d
p
=
−
⟨
ln
⁡
(
p
0
|
φ
(
p
)
|
2
)
⟩
{\displaystyle H_{p}=-\int |\varphi (p)|^{2}\ln(p_{0}\,|\varphi
(p)|^{2})\,dp=-\left\langle \ln(p_{0}\left|\varphi
(p)\right|^{2})\right\rangle }
where x0 and p0 are some arbitrarily chosen
length and momentum respectively, which render
the arguments of the logarithms dimensionless.
Note that the entropies will be functions
of these chosen parameters. Due to the Fourier
transform relation between the position wave
function ψ(x) and the momentum wavefunction
φ(p), the above constraint can be written
for the corresponding entropies as
where h is Planck's constant.
Depending on one's choice of the x0 p0 product,
the expression may be written in many ways.
If x0 p0 is chosen to be h, then
H
x
+
H
p
≥
log
⁡
(
e
2
)
{\displaystyle H_{x}+H_{p}\geq \log \left({\frac
{e}{2}}\right)}
If, instead, x0 p0 is chosen to be ħ, then
H
x
+
H
p
≥
log
⁡
(
e
π
)
{\displaystyle H_{x}+H_{p}\geq \log(e\,\pi
)}
If x0 and p0 are chosen to be unity in whatever
system of units are being used, then
H
x
+
H
p
≥
log
⁡
(
e
h
2
)
{\displaystyle H_{x}+H_{p}\geq \log \left({\frac
{e\,h}{2}}\right)}
where h is interpreted as a dimensionless
number equal to the value of Planck's constant
in the chosen system of units.
The quantum entropic uncertainty principle
is more restrictive than the Heisenberg uncertainty
principle. From the inverse logarithmic Sobolev
inequalities
H
x
≤
1
2
log
⁡
(
2
e
π
σ
x
2
/
x
0
2
)
,
{\displaystyle H_{x}\leq {\frac {1}{2}}\log(2e\pi
\sigma _{x}^{2}/x_{0}^{2})~,}
H
p
≤
1
2
log
⁡
(
2
e
π
σ
p
2
/
p
0
2
)
,
{\displaystyle H_{p}\leq {\frac {1}{2}}\log(2e\pi
\sigma _{p}^{2}/p_{0}^{2})~,}
(equivalently, from the fact that normal distributions
maximize the entropy of all such with a given
variance), it readily follows that this entropic
uncertainty principle is stronger than the
one based on standard deviations, because
σ
x
σ
p
≥
ℏ
2
exp
⁡
(
H
x
+
H
p
−
log
⁡
(
e
h
2
x
0
p
0
)
)
≥
ℏ
2
.
{\displaystyle \sigma _{x}\sigma _{p}\geq
{\frac {\hbar }{2}}\exp \left(H_{x}+H_{p}-\log
\left({\frac {e\,h}{2\,x_{0}\,p_{0}}}\right)\right)\geq
{\frac {\hbar }{2}}~.}
In other words, the Heisenberg uncertainty
principle, is a consequence of the quantum
entropic uncertainty principle, but not vice
versa. A few remarks on these inequalities.
First, the choice of base e is a matter of
popular convention in physics. The logarithm
can alternatively be in any base, provided
that it be consistent on both sides of the
inequality. Second, recall the Shannon entropy
has been used, not the quantum von Neumann
entropy. Finally, the normal distribution
saturates the inequality, and it is the only
distribution with this property, because it
is the maximum entropy probability distribution
among those with fixed variance (cf. here
for proof).
A measurement apparatus will have a finite
resolution set by the discretization of its
possible outputs into bins, with the probability
of lying within one of the bins given by the
Born rule. We will consider the most common
experimental situation, in which the bins
are of uniform size. Let δx be a measure
of the spatial resolution. We take the zeroth
bin to be centered near the origin, with possibly
some small constant offset c. The probability
of lying within the jth interval of width
δx is
P
⁡
[
x
j
]
=
∫
(
j
−
1
/
2
)
δ
x
−
c
(
j
+
1
/
2
)
δ
x
−
c
|
ψ
(
x
)
|
2
d
x
{\displaystyle \operatorname {P} [x_{j}]=\int
_{(j-1/2)\delta x-c}^{(j+1/2)\delta x-c}|\psi
(x)|^{2}\,dx}
To account for this discretization, we can
define the Shannon entropy of the wave function
for a given measurement apparatus as
H
x
=
−
∑
j
=
−
∞
∞
P
⁡
[
x
j
]
ln
⁡
P
⁡
[
x
j
]
.
{\displaystyle H_{x}=-\sum _{j=-\infty }^{\infty
}\operatorname {P} [x_{j}]\ln \operatorname
{P} [x_{j}].}
Under the above definition, the entropic uncertainty
relation is
H
x
+
H
p
>
ln
⁡
(
e
2
)
−
ln
⁡
(
δ
x
δ
p
h
)
.
{\displaystyle H_{x}+H_{p}>\ln \left({\frac
{e}{2}}\right)-\ln \left({\frac {\delta x\delta
p}{h}}\right).}
Here we note that δx δp/h is a typical infinitesimal
phase space volume used in the calculation
of a partition function. The inequality is
also strict and not saturated. Efforts to
improve this bound are an active area of research.
== Harmonic analysis ==
In the context of harmonic analysis, a branch
of mathematics, the uncertainty principle
implies that one cannot at the same time localize
the value of a function and its Fourier transform.
To wit, the following inequality holds,
(
∫
−
∞
∞
x
2
|
f
(
x
)
|
2
d
x
)
(
∫
−
∞
∞
ξ
2
|
f
^
(
ξ
)
|
2
d
ξ
)
≥
‖
f
‖
2
4
16
π
2
.
{\displaystyle \left(\int _{-\infty }^{\infty
}x^{2}|f(x)|^{2}\,dx\right)\left(\int _{-\infty
}^{\infty }\xi ^{2}|{\hat {f}}(\xi )|^{2}\,d\xi
\right)\geq {\frac {\|f\|_{2}^{4}}{16\pi ^{2}}}.}
Further mathematical uncertainty inequalities,
including the above entropic uncertainty,
hold between a function f and its Fourier
transform ƒ̂:
H
x
+
H
ξ
≥
log
⁡
(
e
/
2
)
{\displaystyle H_{x}+H_{\xi }\geq \log(e/2)}
=== Signal processing ===
In the context of signal processing, and in
particular time–frequency analysis, uncertainty
principles are referred to as the Gabor limit,
after Dennis Gabor, or sometimes the Heisenberg–Gabor
limit. The basic result, which follows from
"Benedicks's theorem", below, is that a function
cannot be both time limited and band limited
(a function and its Fourier transform cannot
both have bounded domain)—see bandlimited
versus timelimited. Thus
σ
t
⋅
σ
f
≥
1
4
π
≈
0.08
cycles
{\displaystyle \sigma _{t}\cdot \sigma _{f}\geq
{\frac {1}{4\pi }}\approx 0.08{\text{ cycles}}}
where
σ
t
{\displaystyle \sigma _{t}}
and
σ
f
{\displaystyle \sigma _{f}}
are the standard deviations of the time and
frequency estimates respectively .
Stated alternatively, "One cannot simultaneously
sharply localize a signal (function f ) in
both the time domain and frequency domain
( ƒ̂, its Fourier transform)".
When applied to filters, the result implies
that one cannot achieve high temporal resolution
and frequency resolution at the same time;
a concrete example are the resolution issues
of the short-time Fourier transform—if one
uses a wide window, one achieves good frequency
resolution at the cost of temporal resolution,
while a narrow window has the opposite trade-off.
Alternate theorems give more precise quantitative
results, and, in time–frequency analysis,
rather than interpreting the (1-dimensional)
time and frequency domains separately, one
instead interprets the limit as a lower limit
on the support of a function in the (2-dimensional)
time–frequency plane. In practice, the Gabor
limit limits the simultaneous time–frequency
resolution one can achieve without interference;
it is possible to achieve higher resolution,
but at the cost of different components of
the signal interfering with each other.
=== DFT-Uncertainty principle ===
There is an uncertainty principle that uses
signal sparsity (or the number of non-zero
coefficients).Let
{
x
n
}
:=
x
0
,
x
1
,
…
,
x
N
−
1
{\displaystyle \left\{\mathbf {x_{n}} \right\}:=x_{0},x_{1},\ldots
,x_{N-1}}
be a sequence of N complex numbers and
{
X
k
}
:=
X
0
,
X
1
,
…
,
X
N
−
1
,
{\displaystyle \left\{\mathbf {X_{k}} \right\}:=X_{0},X_{1},\ldots
,X_{N-1},}
its discrete Fourier transform.
Denote by
‖
x
‖
0
{\displaystyle \|x\|_{0}}
the number of non-zero elements in the time
sequence
x
0
,
x
1
,
…
,
x
N
−
1
{\displaystyle x_{0},x_{1},\ldots ,x_{N-1}}
and by
‖
X
‖
0
{\displaystyle \|X\|_{0}}
the number of non-zero elements in the frequency
sequence
X
0
,
X
1
,
…
,
X
N
−
1
{\displaystyle X_{0},X_{1},\ldots ,X_{N-1}}
. Then,
N
≤
‖
x
‖
0
⋅
‖
X
‖
0
.
{\displaystyle N\leq \|x\|_{0}\cdot \|X\|_{0}.}
=== Benedicks's theorem ===
Amrein–Berthier and Benedicks's theorem
intuitively says that the set of points where
f is non-zero and the set of points where
ƒ̂ is non-zero cannot both be small.
Specifically, it is impossible for a function
f in L2(R) and its Fourier transform ƒ̂
to both be supported on sets of finite Lebesgue
measure. A more quantitative version is
‖
f
‖
L
2
(
R
d
)
≤
C
e
C
|
S
|
|
Σ
|
(
‖
f
‖
L
2
(
S
c
)
+
‖
f
^
‖
L
2
(
Σ
c
)
)
.
{\displaystyle \|f\|_{L^{2}(\mathbf {R} ^{d})}\leq
Ce^{C|S||\Sigma |}{\bigl (}\|f\|_{L^{2}(S^{c})}+\|{\hat
{f}}\|_{L^{2}(\Sigma ^{c})}{\bigr )}~.}
One expects that the factor CeC|S||Σ| may
be replaced by CeC(|S||Σ|)1/d,
which is only known if either S or Σ is convex.
=== Hardy's uncertainty principle ===
The mathematician G. H. Hardy formulated the
following uncertainty principle: it is not
possible for f and ƒ̂ to both be "very rapidly
decreasing". Specifically, if f in
L
2
(
R
)
{\displaystyle L^{2}(\mathbb {R} )}
is such that
|
f
(
x
)
|
≤
C
(
1
+
|
x
|
)
N
e
−
a
π
x
2
{\displaystyle |f(x)|\leq C(1+|x|)^{N}e^{-a\pi
x^{2}}}
and
|
f
^
(
ξ
)
|
≤
C
(
1
+
|
ξ
|
)
N
e
−
b
π
ξ
2
{\displaystyle |{\hat {f}}(\xi )|\leq C(1+|\xi
|)^{N}e^{-b\pi \xi ^{2}}}
(
C
>
0
,
N
{\displaystyle C>0,N}
an integer),then, if ab > 1, f = 0, while
if ab = 1, then there is a polynomial P of
degree ≤ N such that
f
(
x
)
=
P
(
x
)
e
−
a
π
x
2
.
{\displaystyle f(x)=P(x)e^{-a\pi x^{2}}.}
This was later improved as follows: if
f
∈
L
2
(
R
d
)
{\displaystyle f\in L^{2}(\mathbb {R} ^{d})}
is such that
∫
R
d
∫
R
d
|
f
(
x
)
|
|
f
^
(
ξ
)
|
e
π
|
⟨
x
,
ξ
⟩
|
(
1
+
|
x
|
+
|
ξ
|
)
N
d
x
d
ξ
<
+
∞
,
{\displaystyle \int _{\mathbb {R} ^{d}}\int
_{\mathbb {R} ^{d}}|f(x)||{\hat {f}}(\xi )|{\frac
{e^{\pi |\langle x,\xi \rangle |}}{(1+|x|+|\xi
|)^{N}}}\,dx\,d\xi <+\infty ~,}
then
f
(
x
)
=
P
(
x
)
e
−
π
⟨
A
x
,
x
⟩
,
{\displaystyle f(x)=P(x)e^{-\pi \langle Ax,x\rangle
}~,}
where P is a polynomial of degree (N − d)/2
and A is a real d×d positive definite matrix.
This result was stated in Beurling's complete
works without proof and proved in Hörmander
(the case
d
=
1
,
N
=
0
{\displaystyle d=1,N=0}
) and Bonami, Demange, and Jaming for the
general case. Note that Hörmander–Beurling's
version implies the case ab > 1 in Hardy's
Theorem while the version by Bonami–Demange–Jaming
covers the full strength of Hardy's Theorem.
A different proof of Beurling's theorem based
on Liouville's theorem appeared in
ref.A full description of the case ab < 1
as well as the following extension to Schwartz
class distributions appears in ref.Theorem.
If a tempered distribution
f
∈
S
′
(
R
d
)
{\displaystyle f\in {\mathcal {S}}'(\mathbb
{R} ^{d})}
is such that
e
π
|
x
|
2
f
∈
S
′
(
R
d
)
{\displaystyle e^{\pi |x|^{2}}f\in {\mathcal
{S}}'(\mathbb {R} ^{d})}
and
e
π
|
ξ
|
2
f
^
∈
S
′
(
R
d
)
,
{\displaystyle e^{\pi |\xi |^{2}}{\hat {f}}\in
{\mathcal {S}}'(\mathbb {R} ^{d})~,}
then
f
(
x
)
=
P
(
x
)
e
−
π
⟨
A
x
,
x
⟩
,
{\displaystyle f(x)=P(x)e^{-\pi \langle Ax,x\rangle
}~,}
for some convenient polynomial P and real
positive definite matrix A of type d × d.
== History ==
Werner Heisenberg formulated the uncertainty
principle at Niels Bohr's institute in Copenhagen,
while working on the mathematical foundations
of quantum mechanics.
In 1925, following pioneering work with Hendrik
Kramers, Heisenberg developed matrix mechanics,
which replaced the ad hoc old quantum theory
with modern quantum mechanics. The central
premise was that the classical concept of
motion does not fit at the quantum level,
as electrons in an atom do not travel on sharply
defined orbits. Rather, their motion is smeared
out in a strange way: the Fourier transform
of its time dependence only involves those
frequencies that could be observed in the
quantum jumps of their radiation.
Heisenberg's paper did not admit any unobservable
quantities like the exact position of the
electron in an orbit at any time; he only
allowed the theorist to talk about the Fourier
components of the motion. Since the Fourier
components were not defined at the classical
frequencies, they could not be used to construct
an exact trajectory, so that the formalism
could not answer certain overly precise questions
about where the electron was or how fast it
was going.
In March 1926, working in Bohr's institute,
Heisenberg realized that the non-commutativity
implies the uncertainty principle. This implication
provided a clear physical interpretation for
the non-commutativity, and it laid the foundation
for what became known as the Copenhagen interpretation
of quantum mechanics. Heisenberg showed that
the commutation relation implies an uncertainty,
or in Bohr's language a complementarity. Any
two variables that do not commute cannot be
measured simultaneously—the more precisely
one is known, the less precisely the other
can be known. Heisenberg wrote:It can be expressed
in its simplest form as follows: One can never
know with perfect accuracy both of those two
important factors which determine the movement
of one of the smallest particles—its position
and its velocity. It is impossible to determine
accurately both the position and the direction
and speed of a particle at the same instant.
In his celebrated 1927 paper, "Über den anschaulichen
Inhalt der quantentheoretischen Kinematik
und Mechanik" ("On the Perceptual Content
of Quantum Theoretical Kinematics and Mechanics"),
Heisenberg established this expression as
the minimum amount of unavoidable momentum
disturbance caused by any position measurement,
but he did not give a precise definition for
the uncertainties Δx and Δp. Instead, he
gave some plausible estimates in each case
separately. In his Chicago lecture he refined
his principle:
Kennard in 1927 first proved the modern inequality:
where ħ = h/2π, and σx, σp are the standard
deviations of position and momentum. Heisenberg
only proved relation (2) for the special case
of Gaussian states.
=== Terminology and translation ===
Throughout the main body of his original 1927
paper, written in German, Heisenberg used
the word, "Ungenauigkeit" ("indeterminacy"),
to describe the basic theoretical principle.
Only in the endnote did he switch to the word,
"Unsicherheit" ("uncertainty"). When the English-language
version of Heisenberg's textbook, The Physical
Principles of the Quantum Theory, was published
in 1930, however, the translation "uncertainty"
was used, and it became the more commonly
used term in the English language thereafter.
=== Heisenberg's microscope ===
The principle is quite counter-intuitive,
so the early students of quantum theory had
to be reassured that naive measurements to
violate it were bound always to be unworkable.
One way in which Heisenberg originally illustrated
the intrinsic impossibility of violating the
uncertainty principle is by utilizing the
observer effect of an imaginary microscope
as a measuring device.He imagines an experimenter
trying to measure the position and momentum
of an electron by shooting a photon at it.
Problem 1 – If the photon has a short wavelength,
and therefore, a large momentum, the position
can be measured accurately. But the photon
scatters in a random direction, transferring
a large and uncertain amount of momentum to
the electron. If the photon has a long wavelength
and low momentum, the collision does not disturb
the electron's momentum very much, but the
scattering will reveal its position only vaguely.Problem
2 – If a large aperture is used for the
microscope, the electron's location can be
well resolved (see Rayleigh criterion); but
by the principle of conservation of momentum,
the transverse momentum of the incoming photon
affects the electron's beamline momentum and
hence, the new momentum of the electron resolves
poorly. If a small aperture is used, the accuracy
of both resolutions is the other way around.The
combination of these trade-offs implies that
no matter what photon wavelength and aperture
size are used, the product of the uncertainty
in measured position and measured momentum
is greater than or equal to a lower limit,
which is (up to a small numerical factor)
equal to Planck's constant. Heisenberg did
not care to formulate the uncertainty principle
as an exact limit (which is elaborated below),
and preferred to use it instead, as a heuristic
quantitative statement, correct up to small
numerical factors, which makes the radically
new noncommutativity of quantum mechanics
inevitable.
== Critical reactions ==
The Copenhagen interpretation of quantum mechanics
and Heisenberg's Uncertainty Principle were,
in fact, seen as twin targets by detractors
who believed in an underlying determinism
and realism. According to the Copenhagen interpretation
of quantum mechanics, there is no fundamental
reality that the quantum state describes,
just a prescription for calculating experimental
results. There is no way to say what the state
of a system fundamentally is, only what the
result of observations might be.
Albert Einstein believed that randomness is
a reflection of our ignorance of some fundamental
property of reality, while Niels Bohr believed
that the probability distributions are fundamental
and irreducible, and depend on which measurements
we choose to perform. Einstein and Bohr debated
the uncertainty principle for many years.
=== The ideal of the detached observer ===
Wolfgang Pauli called Einstein's fundamental
objection to the uncertainty principle "the
ideal of the detached observer" (phrase translated
from the German):
"Like the moon has a definite position" Einstein
said to me last winter, "whether or not we
look at the moon, the same must also hold
for the atomic objects, as there is no sharp
distinction possible between these and macroscopic
objects. Observation cannot create an element
of reality like a position, there must be
something contained in the complete description
of physical reality which corresponds to the
possibility of observing a position, already
before the observation has been actually made."
I hope, that I quoted Einstein correctly;
it is always difficult to quote somebody out
of memory with whom one does not agree. It
is precisely this kind of postulate which
I call the ideal of the detached observer.
Letter from Pauli to Niels Bohr, February
15, 1955
=== Einstein's slit ===
The first of Einstein's thought experiments
challenging the uncertainty principle went
as follows:
Consider a particle passing through a slit
of width d. The slit introduces an uncertainty
in momentum of approximately h/d because the
particle passes through the wall. But let
us determine the momentum of the particle
by measuring the recoil of the wall. In doing
so, we find the momentum of the particle to
arbitrary accuracy by conservation of momentum.Bohr's
response was that the wall is quantum mechanical
as well, and that to measure the recoil to
accuracy Δp, the momentum of the wall must
be known to this accuracy before the particle
passes through. This introduces an uncertainty
in the position of the wall and therefore
the position of the slit equal to h/Δp, and
if the wall's momentum is known precisely
enough to measure the recoil, the slit's position
is uncertain enough to disallow a position
measurement.
A similar analysis with particles diffracting
through multiple slits is given by Richard
Feynman.
=== Einstein's box ===
Bohr was present when Einstein proposed the
thought experiment which has become known
as Einstein's box. Einstein argued that "Heisenberg's
uncertainty equation implied that the uncertainty
in time was related to the uncertainty in
energy, the product of the two being related
to Planck's constant." Consider, he said,
an ideal box, lined with mirrors so that it
can contain light indefinitely. The box could
be weighed before a clockwork mechanism opened
an ideal shutter at a chosen instant to allow
one single photon to escape. "We now know,
explained Einstein, precisely the time at
which the photon left the box." "Now, weigh
the box again. The change of mass tells the
energy of the emitted light. In this manner,
said Einstein, one could measure the energy
emitted and the time it was released with
any desired precision, in contradiction to
the uncertainty principle."Bohr spent a sleepless
night considering this argument, and eventually
realized that it was flawed. He pointed out
that if the box were to be weighed, say by
a spring and a pointer on a scale, "since
the box must move vertically with a change
in its weight, there will be uncertainty in
its vertical velocity and therefore an uncertainty
in its height above the table. ... Furthermore,
the uncertainty about the elevation above
the earth's surface will result in an uncertainty
in the rate of the clock," because of Einstein's
own theory of gravity's effect on time.
"Through this chain of uncertainties, Bohr
showed that Einstein's light box experiment
could not simultaneously measure exactly both
the energy of the photon and the time of its
escape."
=== EPR paradox for entangled particles ===
Bohr was compelled to modify his understanding
of the uncertainty principle after another
thought experiment by Einstein. In 1935, Einstein,
Podolsky and Rosen (see EPR paradox) published
an analysis of widely separated entangled
particles. Measuring one particle, Einstein
realized, would alter the probability distribution
of the other, yet here the other particle
could not possibly be disturbed. This example
led Bohr to revise his understanding of the
principle, concluding that the uncertainty
was not caused by a direct interaction.But
Einstein came to much more far-reaching conclusions
from the same thought experiment. He believed
the "natural basic assumption" that a complete
description of reality would have to predict
the results of experiments from "locally changing
deterministic quantities" and therefore would
have to include more information than the
maximum possible allowed by the uncertainty
principle.
In 1964, John Bell showed that this assumption
can be falsified, since it would imply a certain
inequality between the probabilities of different
experiments. Experimental results confirm
the predictions of quantum mechanics, ruling
out Einstein's basic assumption that led him
to the suggestion of his hidden variables.
These hidden variables may be "hidden" because
of an illusion that occurs during observations
of objects that are too large or too small.
This illusion can be likened to rotating fan
blades that seem to pop in and out of existence
at different locations and sometimes seem
to be in the same place at the same time when
observed. This same illusion manifests itself
in the observation of subatomic particles.
Both the fan blades and the subatomic particles
are moving so fast that the illusion is seen
by the observer. Therefore, it is possible
that there would be predictability of the
subatomic particles behavior and characteristics
to a recording device capable of very high
speed tracking....Ironically this fact is
one of the best pieces of evidence supporting
Karl Popper's philosophy of invalidation of
a theory by falsification-experiments. That
is to say, here Einstein's "basic assumption"
became falsified by experiments based on Bell's
inequalities. For the objections of Karl Popper
to the Heisenberg inequality itself, see below.
While it is possible to assume that quantum
mechanical predictions are due to nonlocal,
hidden variables, and in fact David Bohm invented
such a formulation, this resolution is not
satisfactory to the vast majority of physicists.
The question of whether a random outcome is
predetermined by a nonlocal theory can be
philosophical, and it can be potentially intractable.
If the hidden variables are not constrained,
they could just be a list of random digits
that are used to produce the measurement outcomes.
To make it sensible, the assumption of nonlocal
hidden variables is sometimes augmented by
a second assumption—that the size of the
observable universe puts a limit on the computations
that these variables can do. A nonlocal theory
of this sort predicts that a quantum computer
would encounter fundamental obstacles when
attempting to factor numbers of approximately
10,000 digits or more; a potentially achievable
task in quantum mechanics.
=== Popper's criticism ===
Karl Popper approached the problem of indeterminacy
as a logician and metaphysical realist. He
disagreed with the application of the uncertainty
relations to individual particles rather than
to ensembles of identically prepared particles,
referring to them as "statistical scatter
relations". In this statistical interpretation,
a particular measurement may be made to arbitrary
precision without invalidating the quantum
theory. This directly contrasts with the Copenhagen
interpretation of quantum mechanics, which
is non-deterministic but lacks local hidden
variables.
In 1934, Popper published Zur Kritik der Ungenauigkeitsrelationen
(Critique of the Uncertainty Relations) in
Naturwissenschaften, and in the same year
Logik der Forschung (translated and updated
by the author as The Logic of Scientific Discovery
in 1959), outlining his arguments for the
statistical interpretation. In 1982, he further
developed his theory in Quantum theory and
the schism in Physics, writing:
[Heisenberg's] formulae are, beyond all doubt,
derivable statistical formulae of the quantum
theory. But they have been habitually misinterpreted
by those quantum theorists who said that these
formulae can be interpreted as determining
some upper limit to the precision of our measurements.
[original emphasis]
Popper proposed an experiment to falsify the
uncertainty relations, although he later withdrew
his initial version after discussions with
Weizsäcker, Heisenberg, and Einstein; this
experiment may have influenced the formulation
of the EPR experiment.
=== Many-worlds uncertainty ===
The many-worlds interpretation originally
outlined by Hugh Everett III in 1957 is partly
meant to reconcile the differences between
Einstein's and Bohr's views by replacing Bohr's
wave function collapse with an ensemble of
deterministic and independent universes whose
distribution is governed by wave functions
and the Schrödinger equation. Thus, uncertainty
in the many-worlds interpretation follows
from each observer within any universe having
no knowledge of what goes on in the other
universes.
=== Free will ===
Some scientists including Arthur Compton and
Martin Heisenberg have suggested that the
uncertainty principle, or at least the general
probabilistic nature of quantum mechanics,
could be evidence for the two-stage model
of free will. One critique, however, is that
apart from the basic role of quantum mechanics
as a foundation for chemistry, nontrivial
biological mechanisms requiring quantum mechanics
are unlikely, due to the rapid decoherence
time of quantum systems at room temperature.
The standard view, however, is that this decoherence
is overcome by both screening and decoherence-free
subspaces found in biological cells.
== The second law of thermodynamics ==
There is reason to believe that violating
the uncertainty principle also strongly implies
the violation of the second law of thermodynamics.
== See also ==
== Notes ==
== References ==
== External links ==
Hazewinkel, Michiel, ed. (2001) [1994], "Uncertainty
principle", Encyclopedia of Mathematics, Springer
Science+Business Media B.V. / Kluwer Academic
Publishers, ISBN 978-1-55608-010-4
Matter as a Wave – a chapter from an online
textbook
Quantum mechanics: Myths and facts
Stanford Encyclopedia of Philosophy entry
Fourier Transforms and Uncertainty at MathPages
aip.org: Quantum mechanics 1925–1927 – The
uncertainty principle
Eric Weisstein's World of Physics – Uncertainty
principle
John Baez on the time–energy uncertainty
relation
The certainty principle
Common Interpretation of Heisenberg's Uncertainty
Principle Is Proved False
