Elements of Quantum Theory
77
Similarly, f (k – k 0 ), which is essentially the wave function i the momentum
space, has nonzero values only for |(k – k 0 )x| b
<
, so that the spread in the
x-component of momentum of the particle is
∆p x ≈ b
(3.62)
From Eqs. (3.61) and (3.62), one gets
(∆x) (∆p x ) ≈
(3.63)
Similar results are valid for the measurements of y or z components. Thus,
there is an inherent uncertainty in the determination of the position and
momentum of a particle. The position and momentum of a particle cannot be
simultaneously determined with infinite accuracy. The product of the uncertainties
or allowed errors in their measurements must satisfy the Heisenberg uncertainty
principle whose special case is stated in Eq. (3.63). According to the Heisenberg
uncertainty principle, the product of the uncertainties in the values of two
canonically conjugate variables whose operators are hermitian, cannot
be less than h
–
in the order of magnitude. Examples of canonically conjugate
variables are (x, p x ), (y, p y ) and (z, p z ). A similar relation for time and energy
results from the analysis of the response of a state to a time dependent
interaction,
(∆t) (∆E) ≈
(3.64)
The interpretation of this relation is that if it takes time ∆t to measure the
energy of a system, there is an inherent uncertainty in the measured value of the
energy, given by Eq. (3.64). In particular, this relation implies that unstable particles
with lifetime τ have an associated uncertainty in their energy, of order / τ
.
The Heisenberg uncertainty principle can be made into a quantitative statement
if ∆x, ∆p x , etc. are defined as the standard deviations, i.e.
2 1/2
(
)
x
x x
∆ = 〈 −
〉 ,
2 1/2
(
)
x
x
x
p
p
p
∆ =〈
−
〉 , etc. It can then be shown rigorously that
(∆x) (∆p x ) ≥ / 2
(3.65)
a result which is valid for pairs of canonically conjugate, hermitian operators
(see e.g. Ref. 18).
The Heisenberg uncertainty principle can be easily demonstrated by the
thought experiment of Sec. 3.1, which may be regarded as an experiment for
determining the position and the momentum of the particle. The position of the
particle in the experiment has an uncertainty of
∆x ≈ a
(3.66)
since it is not known whether the particle passed through slit S 1 or S 2 . Similarly,
the momentum of the particle also is undetermined to the extent
77
Similarly, f (k – k 0 ), which is essentially the wave function i the momentum
space, has nonzero values only for |(k – k 0 )x| b
<
, so that the spread in the
x-component of momentum of the particle is
∆p x ≈ b
(3.62)
From Eqs. (3.61) and (3.62), one gets
(∆x) (∆p x ) ≈
(3.63)
Similar results are valid for the measurements of y or z components. Thus,
there is an inherent uncertainty in the determination of the position and
momentum of a particle. The position and momentum of a particle cannot be
simultaneously determined with infinite accuracy. The product of the uncertainties
or allowed errors in their measurements must satisfy the Heisenberg uncertainty
principle whose special case is stated in Eq. (3.63). According to the Heisenberg
uncertainty principle, the product of the uncertainties in the values of two
canonically conjugate variables whose operators are hermitian, cannot
be less than h
–
in the order of magnitude. Examples of canonically conjugate
variables are (x, p x ), (y, p y ) and (z, p z ). A similar relation for time and energy
results from the analysis of the response of a state to a time dependent
interaction,
(∆t) (∆E) ≈
(3.64)
The interpretation of this relation is that if it takes time ∆t to measure the
energy of a system, there is an inherent uncertainty in the measured value of the
energy, given by Eq. (3.64). In particular, this relation implies that unstable particles
with lifetime τ have an associated uncertainty in their energy, of order / τ
.
The Heisenberg uncertainty principle can be made into a quantitative statement
if ∆x, ∆p x , etc. are defined as the standard deviations, i.e.
2 1/2
(
)
x
x x
∆ = 〈 −
〉 ,
2 1/2
(
)
x
x
x
p
p
p
∆ =〈
−
〉 , etc. It can then be shown rigorously that
(∆x) (∆p x ) ≥ / 2
(3.65)
a result which is valid for pairs of canonically conjugate, hermitian operators
(see e.g. Ref. 18).
The Heisenberg uncertainty principle can be easily demonstrated by the
thought experiment of Sec. 3.1, which may be regarded as an experiment for
determining the position and the momentum of the particle. The position of the
particle in the experiment has an uncertainty of
∆x ≈ a
(3.66)
since it is not known whether the particle passed through slit S 1 or S 2 . Similarly,
the momentum of the particle also is undetermined to the extent