29.7 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 $\Delta t$ .

what is the meaning of physics
an object that has a small mass and an object has a large mase have the same momentum which has high kinetic energy
The with smaller mass
how
Faith
Since you said they have the same momentum.. So meaning that there is more like an inverse proportionality in the quantities used to find the momentum. We are told that the the is a larger mass and a smaller mass., so we can conclude that the smaller mass had higher velocity as compared to other one
Mathamaticaly correct
Mathmaticaly correct :)
I have proven it by using my own values
Larger mass=4g Smaller mass=2g Momentum of both=8 Meaning V for L =2 and V for S=4 Now find there kinetic energies using the data presented
grateful soul...thanks alot
Faith
Welcome
2 stones are thrown vertically upward from the ground, one with 3 times the initial speed of the other. If the faster stone takes 10 s to return to the ground, how long will it take the slower stone to return? If the slower stone reaches a maximum height of H, how high will the faster stone go
30s
is speed the same as velocity
no
Nebil
in a question i ought to find the momentum but was given just mass and speed
Faith
just multiply mass and speed then you have the magnitude of momentem
Nebil
Yes
Consider speed to be velocity
it worked our . . thanks
Faith
Distinguish between semi conductor and extrinsic conductors
Suppose that a grandfather clock is running slowly; that is, the time it takes to complete each cycle is longer than it should be. Should you (@) shorten or (b) lengthen the pendulam to make the clock keep attain the preferred time?
I think you shorten am not sure
Uche
shorten it, since that is practice able using the simple pendulum as experiment
Silvia
it'll always give the results needed no need to adjust the length, it is always measured by the starting time and ending time by the clock
Paul
it's not in relation to other clocks
Paul
wat is d formular for newton's third principle
Silvia
okay
Silvia
shorten the pendulum string because the difference in length affects the time of oscillation.if short , the time taken will be adjusted.but if long ,the time taken will be twice the previous cycle.
discuss under damped
resistance of thermometer in relation to temperature
how
Bernard
that resistance is not measured yet, it may be probably in the next generation of scientists
Paul
Is fundamental quantities under physical quantities?
please I didn't not understand the concept of the physical therapy
physiotherapy - it's a practice of exercising for healthy living.
Paul
what chapter is this?
Anderson
this is not in this book, it's from other experiences.
Paul
am new in the group
Daniel
Sure
What is Boyce law
Boyles law states that the volume of a fixed amount of gas is inversely proportional to pressure acting on that given gas if the temperature remains constant which is: V<k/p or V=k(1/p)
how to convert meter per second to kilometers per hour
Divide with 3.6
Mateo
multiply by (km/1000m) x (3600 s/h) -> 3.6
2 how heat loss is prevented in a vacuum flask
what is science
Helen
logical reasoning for a particular phenomenon.
Ajay
I don't know anything about it 😔. I'm sorry, please forgive 😔
due to non in contact mean no conduction and no convection bec of non conducting base and walls and also their is a grape between the layer like to take the example of thermo flask
Abdul
dimensions v²=u²+2at