Using the uncertainty principle

To gain some appreciation of the biological importance—or lack of it—of the uncertainty principle, estimate the minimum uncertainty in the position of 

Strategy We can estimate Ap from mAv, where Av is the uncertainty in the speed v then we use eqn 9.5 to estimate the minimum uncertainty in position. Ax, where x is the direction in which the projectile is traveling. 

For the electron, the uncertainty in position is far larger than the diameter of the atom, which is about 100 pm. Therefore, the concept of a trajectory—the simultaneous possession of a precise position and momentum—is untenable. However, the degree of uncertainty is completely negligible for all practical purposes in the case of the bacterium. Indeed, the position of the cell can be known to within 0.05 per cent of the diameter of a hydrogen atom. It follows that the uncertainty principle plays no direct role in cell biology. However, it plays a major role in the description of the motion of electrons around nuclei in atoms and molecules and, is we shall see soon, the transfer of electrons between molecules and proteins during metabolism. 

Self-test 9.3 ) Estimate the minimum imcertainty in the speed of an electron that can move along the carbon skeleton of a conjugated polyene (such as P-carotene) of length 2.0 run. 

The imcertcunty principle epitomizes the difference between classiccil cuid qucuitiun mechcuiics. Clcissical mechanics supposed, falsely as we now know, that the position cuid momentum of a pcutide can be specified simultcuieously with cubitTcuy precision. However, quantum mechanics shows that position cuid momentiun are complementary, that is, not simultaneously specifiable. Qucuitum mechcUiics requires us to make a choice we can specify position at the expense of momentiun or momentum at the expense of position. 

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This potential is the same as a flat-bottomed container with infinitely high walls separating inside from outside. Here we will use the Uncertainty Principle to estimate the minimum energy later (in Chapter 6) we will find all of the possible energies and states for this system, using a differential equation known as... 

The neutron persists outside of a nucleus for approximately 12 minutes before decaying. Use the uncertainty principle to estimate the fundamental limitation to measurements of its mass. 

The shortest laser pulse created to date has a duration (full width at half maximum) of 3.5 femtoseconds, and a center wavelength of approximately 800 nm (v 375 THz). However, because of the uncertainty principle, such a pulse has a very large range of frequencies Av. Use the uncertainty principle to determine... 

Use the Uncertainty Principle to calculate minimum values for these uncertainties... 

The intrinsic linewidth of a peak measures the lifetime of a spin in a given configuration. Lifetimes may depend on chemical processes such as the proton exchange between water and the ethanol OH shown in Figure 1. In such cases, an order of magnitude estimate of the kinetics of the exchange process can be made using the uncertainty principle... 

A 1-liter cube contains Ng at 300 K and 10" atm. Using the uncertainty principle compute the minimum uncertainty in the specification of x component of velocity u. Accepting this quantity as the minimum possible value of Au, compute the fraction of molecules with x component of velocity between u and w + zlw for (a) = = 0, (b) for u = rms = (kTlmY, How many molecules are in the two velocity domains ... 

Section 3.2 uses the uncertainty principle to stipulate that the orbital angular momentum vector L cannot be exactly parallel to the z axis. Start with the Bohr model of a point electron in a circular orbit, and assume that the uncertainty principle requires this orbit to be tilted out of the xy plane by some minimum angle jS. The vector L must be exactly perpendicular to the plane of the orbit, and so it will be at an angle p from the z axis. As the electron orbits the nucleus in this tilted plane, its z value then varies between z = —rsin/3 and z = rsin/3, and the projection of the linear momentum p onto the z axis varies from... 

An electron in a hydrogen atom is known to have a velocity of 5 x 10 m/s 1 percent. Using the uncertainty principle, calculate the minimum uncertainty in the position of the electron and, given that the diameter of the hydrogen atom is less than 1 angstrom (A), comment on the magnitude of this uncertainty compared to the size of the atom. 

Before the discovery of the neutron by James Chadwick in 1932, it was thought that the nucleus contained electrons and neutrons. Use the uncertainty principle to show that an electron cannot exist inside the nucleus. (Compare the attractive Coulomb potential to the energy imcertainty resulting from the confinement within the nucleus. Take the radius of the nucleus to be 10 m.)... 

Using the uncertainty principle of energy and time, define the natural width F of a spectral line in the excited state and at its transition from the excited state to the ground state. Define also the corresponding A A. Assume x equal to 10 sec and wavelength A = 600 nm. 

This result holds equally well, of course, when R happens to be the operator representing the entropy of an ensemble. Both Tr Wx In Wx and Tr WN In WN are invariant under unitary transformations, and so have no time dependence arising from the Schrodinger equation. This implies a paradox with the second law of thermodynamics in that apparently no increase in entropy can occur in an equilibrium isolated system. This paradox has been resolved by observing that no real laboratory system can in fact be conceived in which the hamiltonian is truly independent of time the uncertainty principle allows virtual fluctuations of the hamiltonian with time at all boundaries that are used to define the configuration and isolate the system, and it is easy to prove that such fluctuations necessarily increase the entropy.30... 

If we increase the accuracy with which the position of the electron is determined by decreasing the wavelength of the light that is used to observe the electron, then the photon has a greater momentum, since p = hiA. The photon can then transfer a larger amount of momentum to the electron, and so the uncertainty in the momentum of the electron increases. Thus any reduction in the uncertainty in the position of the electron is accompanied by an increase in the uncertainty in the momentum of the electron, in accordance with the uncertainty principle relationship. We may summarize by saying that there is no way of accurately measuring simultaneously both the position and velocity of an electron the more closely we attempt to measure its position, the more we disturb its motion and the less accurately therefore we are able to define its velocity. 

The major shortcoming of the spectral method is the rate of convergence. Its ability to resolve eigenvalues is restricted by the width of the filter, which in turn is inversely proportional to the length of the Fourier series (the uncertainty principle). Thus, to accurately characterize an eigenpair in a dense spectrum, one might have to use a very long Chebyshev recursion. 

For FTS data, artifact removal is a consideration that is as important as resolution improvement for most researchers in this field. Interferogram continuation methods are not as yet widely known in this area. Methods currently in widespread use that are effective in artifact removal involve the multiplication of the interferogram by various window functions, an operation called apodization. A carefully chosen window function can be very effective in suppressing the artifacts. However, the peaks are almost always broadened in the process. This can be understood from the uncertainty principle. A window that reduces the function most strongly closest to the end points will yield a transform for the modified function that must be broader than it was originally. Alternatively we may employ the convolution... 

Use of the Uncertainty Principle to Measure the Rates of Chemical Transformations... 

USE OF THE UNCERTAINTY PRINCIPLE TO MEASURE THE RATES OF CHEMICAL TRANSFORMATIONS... 

Exercise 27-10 The purpose of this exercise is to investigate the importance of the uncertainty principle for some kinds of spectroscopy other than nmr, as discussed in Section 27-1. (You may wish to use the wavelength-energy conversion factors given in Sections 9-3 and 9-4.)... 

The symbol h, which is read h bar, means h/ln, a useful combination that occurs widely in quantum mechanics. From inside the back cover, we see that ti = 1.054 X 10-34 J-s. Equation 6 tells us that if the uncertainty in position is very small (Ax very small), then the uncertainty in linear momentum must be large, and vice versa (Fig. 1.11). The uncertainty principle has negligible practical consequences for macroscopic objects, but it is of profound importance for electrons in atoms and for a scientific understanding of the nature of the world. 

I the Uncertainty Principle Using Diffraction of Light Waves," /. Chem. Educ., Vol. 77,2000, 1025-1027. 

The uncertainty principle shows that the classical trajectory of a particle, with a precisely determined position and momentum, is really an illusion. It is a very good approximation, however, for macroscopic bodies. Consider a particle with mass I Xg, and position known to an accuracy of 1 pm. Equation 2.41 shows that the uncertainty in momentum is at least 5 x 10 29 kg m s-1, corresponding to a velocity of 5 x 10 JO m s l. This is totally negligible for any practical purpose, and it illustrates that in the macroscopic world, even with very light objects, the uncertainty principle is irrelevant. If we wanted to, we could describe these objects by wave packets and use the quantum theory, but classical mechanics gives essentially the same answer, and is much easier. At the atomic and molecular level, however, especially with electrons, which are very light, we must abandon the idea of a classical trajectory. The statistical predictions provided by Bom s interpretation of the wavefunction are the best that can be obtained. 

A main feature of ultrafast processes under consideration takes place in the time scale shorter than picoseconds. Thus, it is necessary to employ the laser with pulse-duration 10 fsec to study these ultrafast processes. From the uncertainty principle AE At h/2 it can be seen that using this pulse-duration, numerous vibronic states can be coherently pumped (or excited) and thus the probing signal in a pump-probe experiment will contain the information of the dynamics of both population and coherence (or phase). In other words, in order to obtain the information of ultrafast dynamics it is... 

While h is quite small in the macroscopic world, it is not at all insignificant when the particle under consideration is of subatomic scale. Let us use an actual example to illustrate this point. Suppose the Ax of an electron is 10-14 m, or 0.01 pm. Then, with eq. (1.2.1), we get Apx = 5.27 x 10-21 kg m s-1. This uncertainty in momentum would be quite small in the macroscopic world. However, for subatomic particles such as an electron, with mass of 9.11 x 10-31 kg, such an uncertainty would not be negligible at all. Hence, on the basis of the Uncertainty Principle, we can no longer say that an electron is precisely located at this point with an exactly known velocity. It should be stressed that the uncertainties we are discussing here have nothing to do with the imperfection of the measuring instruments. Rather, they are inherent indeterminacies. If we recall the Bohr theory of the hydrogen atom, we find that both the radius of the orbit and the velocity of the electron can be precisely calculated. Hence the Bohr results violate the Uncertainty Principle. 

In the fast modulation limit [Equation (8)], the loss of information is fundamental. The lifetime of the frequency perturbations is short compared to their magnitude, and the uncertainty principle precludes a full characterization of w(t) by any experimental technique. However, in the slow modulation limit and in the intermediate regimes, the loss of information in the FID is not fundamental. The next section shows that the Raman echo contains additional information about the rate of the frequency fluctuations that is not present in the FID. By using a combination of Raman echo and... 

I ve been using marbles and atom-size insects as an analogy for electrons, but I don t want to leave you with the misconception that electrons can only be thought of as solid objects. In the introduction to this book and in the first chemistry book, I discussed how we can think of electrons (and all particles, for that matter) as collections of waves. It is this wave nature of electrons that is the basis for quantum mechanics, which is the math we use to come up with the uncertainty principle. So, while it is often convenient to consider electrons to be tiny, solid objects, you should always be aware of the model of electrons as waves. 

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