<< Chapter < Page | Chapter >> Page > |
Another kind of uncertainty principle concerns uncertainties in simultaneous measurements of the energy of a quantum state and its lifetime,
where $\text{\Delta}E$ is the uncertainty in the energy measurement and $\text{\Delta}t$ is the uncertainty in the lifetime measurement. The energy-time uncertainty principle does not result from a relation of the type expressed by [link] for technical reasons beyond this discussion. Nevertheless, the general meaning of the energy-time principle is that a quantum state that exists for only a short time cannot have a definite energy. The reason is that the frequency of a state is inversely proportional to time and the frequency connects with the energy of the state, so to measure the energy with good precision, the state must be observed for many cycles.
To illustrate, consider the excited states of an atom. The finite lifetimes of these states can be deduced from the shapes of spectral lines observed in atomic emission spectra. Each time an excited state decays, the emitted energy is slightly different and, therefore, the emission line is characterized by a distribution of spectral frequencies (or wavelengths) of the emitted photons. As a result, all spectral lines are characterized by spectral widths. The average energy of the emitted photon corresponds to the theoretical energy of the excited state and gives the spectral location of the peak of the emission line. Short-lived states have broad spectral widths and long-lived states have narrow spectral widths.
Check Your Understanding A sodium atom makes a transition from the first excited state to the ground state, emitting a 589.0-nm photon with energy 2.105 eV. If the lifetime of this excited state is $1.6\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-8}}\text{s}$ , what is the uncertainty in energy of this excited state? What is the width of the corresponding spectral line?
$4.1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-8}}\text{eV}$ ; $1.1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-5}}\text{nm}$
If the formalism of quantum mechanics is ‘more exact’ than that of classical mechanics, why don’t we use quantum mechanics to describe the motion of a leaping frog? Explain.
Can the de Broglie wavelength of a particle be known precisely? Can the position of a particle be known precisely?
Yes, if its position is completely unknown. Yes, if its momentum is completely unknown.
Can we measure the energy of a free localized particle with complete precision?
Can we measure both the position and momentum of a particle with complete precision?
No. According to the uncertainty principle, if the uncertainty on the particle’s position is small, the uncertainty on its momentum is large. Similarly, if the uncertainty on the particle’s position is large, the uncertainty on its momentum is small.
A velocity measurement of an $\alpha $ -particle has been performed with a precision of 0.02 mm/s. What is the minimum uncertainty in its position?
A gas of helium atoms at 273 K is in a cubical container with 25.0 cm on a side. (a) What is the minimum uncertainty in momentum components of helium atoms? (b) What is the minimum uncertainty in velocity components? (c) Find the ratio of the uncertainties in (b) to the mean speed of an atom in each direction.
a. $\text{\Delta}p\ge 2.11\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-34}}\text{N}\xb7\text{s}$ ; b. $\text{\Delta}v\ge 6.31\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-8}}\text{m}$ ; c. $\text{\Delta}v\text{/}\sqrt{{k}_{\text{B}}T\text{/}{m}_{\alpha}}=5.94\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-11}}$
If the uncertainty in the $y$ -component of a proton’s position is 2.0 pm, find the minimum uncertainty in the simultaneous measurement of the proton’s $y$ -component of velocity. What is the minimum uncertainty in the simultaneous measurement of the proton’s $x$ -component of velocity?
Some unstable elementary particle has a rest energy of 80.41 GeV and an uncertainty in rest energy of 2.06 GeV. Estimate the lifetime of this particle.
$\text{\Delta}\tau \ge 1.6\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-25}}\text{s}$
An atom in a metastable state has a lifetime of 5.2 ms. Find the minimum uncertainty in the measurement of energy of the excited state.
Measurements indicate that an atom remains in an excited state for an average time of 50.0 ns before making a transition to the ground state with the simultaneous emission of a 2.1-eV photon. (a) Estimate the uncertainty in the frequency of the photon. (b) What fraction of the photon’s average frequency is this?
a. $\text{\Delta}f\ge 1.59\phantom{\rule{0.2em}{0ex}}\text{MHz}$ ; b. $\text{\Delta}\omega \text{/}{\omega}_{0}=3.135\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-9}}$
Suppose an electron is confined to a region of length 0.1 nm (of the order of the size of a hydrogen atom) and its kinetic energy is equal to the ground state energy of the hydrogen atom in Bohr’s model (13.6 eV). (a) What is the minimum uncertainty of its momentum? What fraction of its momentum is it? (b) What would the uncertainty in kinetic energy of this electron be if its momentum were equal to your answer in part (a)? What fraction of its kinetic energy is it?
Notification Switch
Would you like to follow the 'University physics volume 3' conversation and receive update notifications?