# 5.7 Probability: the heisenberg uncertainty principle  (Page 4/11)

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Why don’t we notice Heisenberg’s uncertainty principle in everyday life? The answer is that Planck’s constant is very small. Thus the lower limit in the uncertainty of measuring the position and momentum of large objects is negligible. We can detect sunlight reflected from Jupiter and follow the planet in its orbit around the Sun. The reflected sunlight alters the momentum of Jupiter and creates an uncertainty in its momentum, but this is totally negligible compared with Jupiter’s huge momentum. The correspondence principle tells us that the predictions of quantum mechanics become indistinguishable from classical physics for large objects, which is the case here.

## Heisenberg uncertainty for energy and time

There is another form of Heisenberg’s uncertainty principle     for simultaneous measurements of energy and time . In equation form,

$\Delta E\Delta t\ge \frac{h}{4\pi },$

where $\Delta E$ is the uncertainty in energy    and $\Delta t$ is the uncertainty in time    . This means that within a time interval $\Delta t$ , it is not possible to measure energy precisely—there will be an uncertainty $\Delta E$ in the measurement. In order to measure energy more precisely (to make $\Delta E$ smaller), we must increase $\Delta t$ . This time interval may be the amount of time we take to make the measurement, or it could be the amount of time a particular state exists, as in the next [link] .

## Heisenberg uncertainty principle for energy and time for an atom

An atom in an excited state temporarily stores energy. If the lifetime of this excited state is measured to be ${\text{1.0×10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}s$ , what is the minimum uncertainty in the energy of the state in eV?

Strategy

The minimum uncertainty in energy $\Delta E$ is found by using the equals sign in $\Delta E\Delta t\ge h\text{/4}\pi$ and corresponds to a reasonable choice for the uncertainty in time. The largest the uncertainty in time can be is the full lifetime of the excited state, or $\Delta t={\text{1.0×10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}s$ .

Solution

Solving the uncertainty principle for $\Delta E$ and substituting known values gives

$\Delta E=\frac{h}{4\pi \Delta t}=\frac{6\text{.}\text{63}×{\text{10}}^{\text{–34}}\phantom{\rule{0.25em}{0ex}}\text{J}\cdot \text{s}}{4\pi \left({\text{1.0×10}}^{\text{–10}}\phantom{\rule{0.25em}{0ex}}\text{s}\right)}=\text{5}\text{.}\text{3}×{\text{10}}^{\text{–25}}\phantom{\rule{0.25em}{0ex}}\text{J.}$

Now converting to eV yields

$\Delta E=\text{(5.3}×{\text{10}}^{\text{–25}}\phantom{\rule{0.25em}{0ex}}\text{J)}\left(\frac{\text{1 eV}}{1\text{.}\text{6}×{\text{10}}^{\text{–19}}\phantom{\rule{0.25em}{0ex}}\text{J}}\right)=\text{3}\text{.}\text{3}×{\text{10}}^{\text{–6}}\phantom{\rule{0.25em}{0ex}}\text{eV}\text{.}$

Discussion

The lifetime of ${\text{10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}\text{s}$ is typical of excited states in atoms—on human time scales, they quickly emit their stored energy. An uncertainty in energy of only a few millionths of an eV results. This uncertainty is small compared with typical excitation energies in atoms, which are on the order of 1 eV. So here the uncertainty principle limits the accuracy with which we can measure the lifetime and energy of such states, but not very significantly.

The uncertainty principle for energy and time can be of great significance if the lifetime of a system is very short. Then $\Delta t$ is very small, and $\Delta E$ is consequently very large. Some nuclei and exotic particles have extremely short lifetimes (as small as ${\text{10}}^{-\text{25}}\phantom{\rule{0.25em}{0ex}}\text{s}$ ), causing uncertainties in energy as great as many GeV ( ${\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}\text{eV}$ ). Stored energy appears as increased rest mass, and so this means that there is significant uncertainty in the rest mass of short-lived particles. When measured repeatedly, a spread of masses or decay energies are obtained. The spread is $\Delta E$ . You might ask whether this uncertainty in energy could be avoided by not measuring the lifetime. The answer is no. Nature knows the lifetime, and so its brevity affects the energy of the particle. This is so well established experimentally that the uncertainty in decay energy is used to calculate the lifetime of short-lived states. Some nuclei and particles are so short-lived that it is difficult to measure their lifetime. But if their decay energy can be measured, its spread is $\Delta E$ , and this is used in the uncertainty principle ( $\Delta E\Delta t\ge h\text{/4}\pi$ ) to calculate the lifetime $\Delta t$ .

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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