momentum, position and the uncertainty principle
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8/10/2019 Momentum, Position and the Uncertainty Principle
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Momentum and the momentum operator
For momentumwe write an operator
We postulate this can be written as
with
where xo, y
o, and z
oare unit vectors
in the x, y, and z directions
ˆ p
ˆ p i
o o o x y z
x y z
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Momentum and the momentum operator
With this postulated form we find that
and we have a correspondence between the
classical notion of the energy E
and the corresponding Hamiltonian operatoof the Schrödinger equation
ˆ p i 2 2
2ˆ2 2 p
m m
2
2 p
E V m
2 22 ˆˆ
2 2 p H V V
m m
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Momentum and the momentum operator
Note that
This means the plane waves arethe eigenfunctions of the operator
with eigenvaluesWe can therefore say for these eigenstates that
the momentum isNote that p is a vector, with threecomponents with scalar values
not an operator
ˆ exp exp exp p i i i i k r k r k k r
exp i k rˆ p
k
p k
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Position and the position operator
For the position operatorthe postulated operator is almost trivial
when we are working with functions ofpositionIt is simply the position vector, r , itself
At least when we are working in arepresentation that is in terms of position
we therefore typically do not writethough rigorously we shouldThe operator for the z-component of position
would, for example, also simply be z itself
r̂
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The uncertainty principle
Here we illustrate the position-momentumuncertainty principle by example
We have looked at a Gaussian wavepacket beforWe could write this as a sum over waves of
different k-values, with Gaussian weightsor we could take the limit of that process by
using an integration
2
, exp exp2
Gk
k k z t i k t k
k
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The uncertainty principle
We could rewrite
at time t = 0 as
where
2
, exp exp2Gk
k k z t i k t kk
,0 expk k
z k ikz dk
2
exp2k
k k k
k
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The uncertainty principle
In
is the representation of the wavefunction in k
is the probability P k
strictly, the probability densitythat if we measured the momentum of the particle
actually the z component of momentumit would be found to have value
k k
2
k k
k
2
exp2
k
k k k
k
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The uncertainty principle
With
then this probability (density) of finding a valfor the momentum would be
This Gaussian corresponds to the statistical
Gaussian probability distributionwith standard deviation
2
exp2k
k k k
k
22
2exp2
k k
k k P k
k
k
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The uncertainty principle
Note also that
is the Fourier transform of and, as is well knownthe Fourier transform of a Gaussian is
a Gaussianspecifically here
,0 expk k
z k ikz dk
k k
2 2,0 exp z k z
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The uncertainty principle
If we want to rewrite
in the standard form
where the parameterwould now be the standard deviation
in the probability distribution for z
then
2 2 2,0 exp 2 z k z
22
2,0 exp 2
z z z
z
1/ 2k z
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The uncertainty principle
Fromif we now multiply by to get the standard deviati
we would measure in momentumwe have
which is the relation between the standarddeviations we would see in
measurements of position andmeasurements of momentum
1/ 2k z
2 p z
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The uncertainty principle
This relation
is as good as we can get for a Gaussian
For examplea Gaussian pulse will broaden in space as
it propagateseven though the range of k valuesremains the same
2 p z
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It also turns out that the Gaussian shapeis the one with the minimum possible
product of andSo quite generally
which is the uncertainty principlefor position and momentum inone direction
The uncertainty principle
2 p z
p z
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The uncertainty principle in Fourier analysis
Uncertainty principles are well known in Fourier analyOne cannot simultaneously have both
a well defined frequency anda well defined time
If a signal is a short pulseit is necessarily made up out of a range of
frequencies
The shorter the pulse isthe larger the range of frequencies
12
t
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