# 29.7 Probability: the heisenberg uncertainty principle  (Page 5/11)

 Page 5 / 11

There is another consequence of the uncertainty principle for energy and time. If energy is uncertain by $\Delta E$ , then conservation of energy can be violated by $\Delta E$ for a time $\Delta t$ . Neither the physicist nor nature can tell that conservation of energy has been violated, if the violation is temporary and smaller than the uncertainty in energy. While this sounds innocuous enough, we shall see in later chapters that it allows the temporary creation of matter from nothing and has implications for how nature transmits forces over very small distances.

Finally, note that in the discussion of particles and waves, we have stated that individual measurements produce precise or particle-like results. A definite position is determined each time we observe an electron, for example. But repeated measurements produce a spread in values consistent with wave characteristics. The great theoretical physicist Richard Feynman (1918–1988) commented, “What there are, are particles.” When you observe enough of them, they distribute themselves as you would expect for a wave phenomenon. However, what there are as they travel we cannot tell because, when we do try to measure, we affect the traveling.

## Section summary

• Matter is found to have the same interference characteristics as any other wave.
• There is now a probability distribution for the location of a particle rather than a definite position.
• Another consequence of the wave character of all particles is the Heisenberg uncertainty principle, which limits the precision with which certain physical quantities can be known simultaneously. For position and momentum, the uncertainty principle is $\Delta x\Delta p\ge \frac{h}{4\pi }$ , where $\Delta x$ is the uncertainty in position and $\Delta p$ is the uncertainty in momentum.
• For energy and time, the uncertainty principle is $\Delta E\Delta t\ge \frac{h}{4\pi }$ where $\Delta E$ is the uncertainty in energy and $\Delta t$ is the uncertainty in time.
• These small limits are fundamentally important on the quantum-mechanical scale.

## Conceptual questions

What is the Heisenberg uncertainty principle? Does it place limits on what can be known?

## Problems&Exercises

(a) If the position of an electron in a membrane is measured to an accuracy of $1\text{.}\text{00 μm}$ , what is the electron’s minimum uncertainty in velocity? (b) If the electron has this velocity, what is its kinetic energy in eV? (c) What are the implications of this energy, comparing it to typical molecular binding energies?

(a) 57.9 m/s

(b) $9\text{.}\text{55}×{\text{10}}^{-9}\phantom{\rule{0.25em}{0ex}}\text{eV}$

(c) From [link] , we see that typical molecular binding energies range from about 1eV to 10 eV, therefore the result in part (b) is approximately 9 orders of magnitude smaller than typical molecular binding energies.

(a) If the position of a chlorine ion in a membrane is measured to an accuracy of $1\text{.}\text{00 μm}$ , what is its minimum uncertainty in velocity, given its mass is $5\text{.}\text{86}×{\text{10}}^{-\text{26}}\phantom{\rule{0.25em}{0ex}}\text{kg}$ ? (b) If the ion has this velocity, what is its kinetic energy in eV, and how does this compare with typical molecular binding energies?

Suppose the velocity of an electron in an atom is known to an accuracy of $2\text{.}0×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{m/s}$ (reasonably accurate compared with orbital velocities). What is the electron’s minimum uncertainty in position, and how does this compare with the approximate 0.1-nm size of the atom?

29 nm,

290 times greater

The velocity of a proton in an accelerator is known to an accuracy of 0.250% of the speed of light. (This could be small compared with its velocity.) What is the smallest possible uncertainty in its position?

A relatively long-lived excited state of an atom has a lifetime of 3.00 ms. What is the minimum uncertainty in its energy?

$1\text{.}\text{10}×{\text{10}}^{-\text{13}}\phantom{\rule{0.25em}{0ex}}\text{eV}$

(a) The lifetime of a highly unstable nucleus is ${\text{10}}^{-\text{20}}\phantom{\rule{0.25em}{0ex}}\text{s}$ . What is the smallest uncertainty in its decay energy? (b) Compare this with the rest energy of an electron.

The decay energy of a short-lived particle has an uncertainty of 1.0 MeV due to its short lifetime. What is the smallest lifetime it can have?

$3\text{.}3×{\text{10}}^{-\text{22}}\phantom{\rule{0.25em}{0ex}}\text{s}$

The decay energy of a short-lived nuclear excited state has an uncertainty of 2.0 eV due to its short lifetime. What is the smallest lifetime it can have?

What is the approximate uncertainty in the mass of a muon, as determined from its decay lifetime?

$2.66×{\text{10}}^{-\text{46}}\phantom{\rule{0.25em}{0ex}}\text{kg}$

Derive the approximate form of Heisenberg’s uncertainty principle for energy and time, $\Delta E\Delta t\approx h$ , using the following arguments: Since the position of a particle is uncertain by $\Delta x\approx \lambda$ , where $\lambda$ is the wavelength of the photon used to examine it, there is an uncertainty in the time the photon takes to traverse $\Delta x$ . Furthermore, the photon has an energy related to its wavelength, and it can transfer some or all of this energy to the object being examined. Thus the uncertainty in the energy of the object is also related to $\lambda$ . Find $\Delta t$ and $\Delta E$ ; then multiply them to give the approximate uncertainty principle.

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