The Uncertainty Principle

The corner stone of nineteenth century physics was the idea of the uniformity of nature that identical causes always produce identical effects. This idea was expressed by Clerk-Maxwell in the following words  

All matter was supposed to consists of particles which moved according to definite laws so that if the positions and motions of all the particles concerned were known at any time, then the positions and motions at any future time were exactly fixed and could, theoretically at least, be predicted. 

We are not now in a position to support any such dogmatic statement as that of Clerk-Maxwell. The idea of the uniformity of nature was based on superficial observations of large scale phenomena, and it now appears that with given initial conditions many future events are possible. At best all we can hope to do is to calculate the relative probabilities or chances of th.e different possible events. Also Clerk-Maxwell s statement involves the assumption that it is possible to fix the initial configuration and motion of the bodies concerned exactly, and as we shall see this is not the case. 

Consider, for example, the decomposition of radium into helium and radon. It is found that about one radium atom in two thousand decomposes in a year. Suppose we are asked to predict the state of a particular radium atom at some future time, say after ten years. All we can say is that it will either be unchanged or will have decomposed, and that the chance of 

Let us consider a very simple experiment which will serve 

The discovery of the wave properties of matter raised some new and interesting questions. Consider, for example, a ball rolling down a ramp. Using the equations of classical physics, we can calculate, with great accuracy, the ball s position, direction of motion, and speed at any instant. Can we do the same for an electron, which exhibits wave properties A wave extends in space and its location is not precisely defined. We might therefore anticipate that it is impossible to determine exactly where an electron is located at a specific instant. 

The German physicist Werner Heisenberg ( FIGURE 6.14) proposed that the dual nature of matter places a fundamental limitation on how precisely we can know both the location and 

During his postdoctoral assistantship with Niels Bohr, Heisenberg fonnulated his famous uncertainty principle. At 32 he was one of the youngest scientists to receive a Nobel Prize. 

Heisenberg mathematically related the uncertainty in position. Ax, and the uncertainty in momentum, A(mv), to a quantity involving Planck s constant  

A brief calculation illustrates the dramatic implications of the uncertainty principle. The electron has a mass of 9.11 X 10 kg and moves at an average speed of about 5X10 m/s in a hydrogen atom. Let s assume that we know the speed to an uncertainty of 1% [that is, an uncertainty of (0.01 )(5 X 10 m/s) = 5 X 10 m/s] and that this is the only important source of uncertainty in the momentum, so that A(mv) = m Av. We can use Equation 6.9 to calculate the uncertainty in the position of the electron  

Because the diameter of a hydrogen atom is about 1 X 10 m, the uncertainty in the position of the electron in the atom is an order of magnitude greater than the size of the atom. Thus, we have essentially no idea where the electron is located in the atom. On the other hand, if we were to repeat the calculation with an object of ordinary mass, such as a tennis ball, the uncertainty would be so small that it would be inconsequential. In that case, m is large and Ax is out of the realm of measurement and therefore of no practical consequence. 

De Broglie s hypothesis and Heisenberg s uncertainty principle set the stage for a new and more broadly applicable theory of atomic structure. In this approach, any attempt to define precisely the instantaneous location and momentum of the electron is abandoned. The wave nature of the electron is recognized, and its behavior is described in terms appropriate to waves. The result is a model that precisely describes the energy of the electron while describing its location not precisely but rather in terms of probabilities. 

The wavelength (A) of an electron of mass m moving at velocity v is given by the de Broglie relation  

Calculate the wavelength of an electron traveling with a speed of 2.65 X 10 m/s. 

SORT You are given the speed of an electron and asked to calculate its wavelength. 

STRATEGIZE The conceptual plan shows how the de Broglie relation relates the wavelength of an electron to its mass and velocity. 

SOLVE Substitute the velocity, Planck s constant, and the mass of an electron to calculate the electron s wavelength. To correctly cancel the units, break down the J in Planck s constant into its SI base units (1 J = 1 kg m /s ). 

Since the radical is greater than unity for all values of n, we have the result, 

The inequality (21.24) is the statement of the Heisenberg uncertainty principle for the particle in the box. 

The situation for the free particle compared with the particle in the box may be summarized as follows. 

The free particle has an exactly defined momentum, but the position is completely indefinite. 

When we try to gain information about the position of the particle by confining it within the length L, an indefiniteness or uncertainty is introduced in the momentum. The product of these uncertainties is given by the inequality (21.24) Ap Ax h/4n. 

The uncertainty principle, an important relation that is a consequence of quantum mechanics, was discovered by the German physicist Werner Heisenberg (born 1901) in 1927. Heisenberg showed that as a result of the wave-particle duality of matter it is impossible to carry out simultaneously a precise determination of the position of a particle and of its velocity. He also showed that it is impossible to determine exactly the energy of a system at an instant of time. 

The uncertainty principle in quantum mechanics is closely related to an uncertainty equation between frequency and time for any sort of waves. Let us consider, for example, a train of ocean waves passing a buoy that is anchored at a fixed point. An observer on the buoy could measure the frequency (number of waves passing the buoy per unit time) at time / by counting the number of crests and troughs passing the buoy between the times t ) — At and / -I- At, dividing by 2 to obtain the number of waves in time 2Ar, and by 2At to obtain the frequency (i ), which is defined as the number of waves in unit time  

A diagram illustrating the uncertainty in determining the frequency by counting the number of waves passing a point during a period of time. 

We may rewrite this equation as X A/ = i. A more detailed discussion based on an error-function definition of Ai and A/ leads to the uncertainty equation for frequency and time in its customary form  

By use of quantum theory this equation can be at once converted into the uncertainty equation for energy and time for photons. The energy of a photon with frequency v is hv. The uncertainty in frequency Ai when multiplied by li is the uncertainty in energy AE  

Classically, if you know the position and momentum of a mass at any one time (that is, if you know those quantities simultaneously), you know everything about 

Unless otherwise noted, all art on this page is Cengage Learning 2014. 

Heisenberg realized this and in 1927 announced his uncertainty principle. (The principle can be derived mathematically, so it is not a postulate of quantum mechanics. We will not cover the derivation here.) The uncertainty principle deals only with certain observables that might be measured simultaneously. Two of these observables are position x (in the x direction), and momentum (also in the X direction). If the uncertainty in the position is given the symbol Ax and the uncertainty in the momentum is termed Ap, then Heisenberg s uncertainty principle is 

Because the classical definition of momentum p is mv, equation 10.5 is sometimes written as 

You may want to verify not just the numbers but how the units work out. This minimum uncertainty is undetectable even using modern measuring devices, so this uncertainty in position would never be noticed. 

Let us consider what effect the wave-particle duality has on attempts to measure simultaneously the X coordinate and the JC component of linear momentum of a microscopic particle. We start with a beam of particles with momentum p, traveling in the y direction, and we let the beam fall on a narrow slit. Behind this slit is a photographic plate. See Fig. 1.1. 

Particles that pass through the slit of width w have an uncertainty w in their Jc coordinate at the time of going through the slit. Calling this spread in x values Ajc, we have Ax = w. 

Solution (a) When r = 0,y/ x 1.0 (because e = 1) and the probability of finding the electron at the nucleus is proportional to 1.0 x SV. (b) At a distance r= Uq in an arbitrary direction, y/ °c e, so the probability of being found there is proportional to e x 5U= 0.14 x 5V. Therefore, the ratio of probabilities is 1.0/0.14= 7.1. It is more probable (by a factor of 7.1) that the electron will be found at the nucleus than in the same tiny volume located at a distance from the nucleus. 

I Self-test 9.2 ) The wavefunction for the lowest energy state in the ion He+ is proportional to Calculate the ratio of probabilities as in Example 9.2, by 

Answer The ratio of probabilities is 55 a more compact wavefunction on account of the higher nuclear charge. 

Given that electrons behave like waves, we need to be able to reconcile the predictions of quantum mechanics with the existence of objects, such as biological cells and the organelles within them. 

We have seen that, according to the de Broglie relation, a wave of constant wavelength, the wavefunction sin(2nx/X), corresponds to a particle with a definite linear momentum p = h/L However, a wave does not have a definite location at a single point in space, so we cannot speak of the precise position of the particle if it has a definite momentum. Indeed, because a sine wave spreads throughout the whole of space, we cannot say anything about the location of the particle because the wave spreads everywhere, the particle maybe found anywhere in the whole of space. This statement is one half of the uncertainty principle, proposed by Werner Heisenberg in 1927, in one of the most celebrated results of quantum mechanics  

In 1927, Davisson and Germer experimentally confirmed de Broglie s hypothesis by reflecting electron from metals and observing diffraction effects. In 1932, Stem observed the same effects with helium atoms and hydrogen molecules, thus verifying that the wave effects are not peculiar to electrons, but result from some general law of motion for microscopic particles. 

Thus electrons behave in some respects like particles and in other respects like waves. We are faced with the apparently contradictory wave-particle duality of matter (and of light). How can an electron be both a particle, which is a localized entity, and a wave, which is nonlocalized The qpswer is that an electron is neither a wave nor a particle, but something else. An accurate pictorial description of an electron s behavior is impossible using the wave or particle concept of classical physics. Hie concepts of classical physics have been developed from experience in the macroscopic world and do not properly describe the microscopic world. Evolution has shaped the human brain to allow it to understand and deal effectively with macroscopic phenomena. The human nervous system was not developed to deal with phenomena at the atomic and molecular level, so it is not surprising if we cannot fully understand such phenomena. 

Although both photons and electrons show an apparent duality, they are not the same kinds of entities. Photons always travel at speed c and have zero rest mass electrons always have v c and a nonzero rest mass. Photons must always be treated relativistically, but electrons whose speed is much less than c can be treated nonrela-tivistically. 

In 1927, Davisson and Germer demonstrated that electrons are diffracted by crystals in a manner similar to the diffraction of X rays. These electron-diffraction experiments substantiated de Broglie s suggestion that an electron has wave properties such as wavelength, frequency, phase, and interference. In seemingly direct contradiction, however, certain other experiments, particularly those of J. J. Thomson, showed that an electron is a particle with mass, energy, and momentum. 

As an attempt at an explanation of the above situation, Bohr put forward the principle of complementarity, in which he postulated that 

A consequence of the apparently dual nature of an electron is the uncertainty principle, developed by Werner Heisenberg. The essential idea of the uncertainty principle is that it is impossible to specify at any given moment both the position and the momentum of an electron. The lower limit of this uncertainty is Planck s constant divided by 4tt. In equational form, 

Here Apx is the uncertainty in the momentum and Ax is the uncertainty in the position. Thus, at any instant, the more accurately it is possible to measure the momentum of an electron, the more uncertain the exact position becomes. The uncertainty principle means that we cannot think of an electron as traveling around from point to point, with a certain momentum at each point. Rather we are forced to consider the electron as having only a certain probability of being found at each fixed point in space. We must also realize that it is not possible to measure simultaneously, and to any desired accuracy, the physical quantities that would allow us to decide whether the electron is a particle or a wave. We thus carry forth the idea that the electron is both a particle and a wave. 

To describe the problem of trying to locate a subatomic particle that behaves like a wave, Werner Heisenberg formulated what is now known as the Heisenberg uncertainty principle It is impossible to know simultaneously both the momentum p (defined as mass times velocity, m X ) and the position x of a particle with certainty. Stated mathematically. 

If the Heisenbetg uncertainty principle is applied to the hydrogen atom, we find that the electron cannot orbit the nucleus in a well-defined path, as Bohr thought. If it did, we could determine precisely both the position of the electron (from the radius of the orbit) and its speed (fixjm its kinetic energy) at the same time. This would violate the uncertainty principle. 

Sample Problem 6.6 shows how to use the Heisenberg uncertainty principle. 

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. 

Strategy The uncertainty in the velocity, 1 percent of 5 X 10 m/s, is Azr. Using Equation 6.11, calculate Ax and compare it with the diameter of the hydrogen atom. 

If an electron has wave-like properties, there is an important and difficult consequence it becomes impossible to know exactly both the momentum and position of the electron at the same instant in time. This is a statement of Heisenberg s uncertainty principle. In order to get around this problem, rather than trying to define its exact position and momentum, we use the probability of finding the electron in a given volume of space. The probability of finding an electron at a given point in space is determined from the function ij where is a mathematical function which describes the behaviour of an electron-wave ip is the wavefunction. 

The probability of finding an electron at a given point in space is determined from the function ip where ip is the wavefunction. 

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One feature of this inequality warrants special attention. In the previous paragraph it was shown that the precise measurement of A made possible when v is an eigenfiinction of A necessarily results in some uncertainty in a simultaneous measurement of B when the operators /land fido not conmuite. However, the mathematical statement of the uncertainty principle tells us that measurement of B is in fact completely uncertain one can say nothing at all about B apart from the fact that any and all values of B are equally probable A specific example is provided by associating A and B with the position and momentum of a particle moving along the v-axis. It is rather easy to demonstrate that [p, x]=- ih, so that If... 

For each degree of freedom, classical states within a small volume A/ij Aq- h merge into a single quantum state which cannot be fiirther distinguished on account of the uncertainty principle. For a system with /... 

Main J, Mandelshtam V A and Taylor H S 1997 High resolution quantum recurrence spectra beyond the uncertainty principle Phys. Rev. Lett. 78 4351... 

The force F which has to be applied to a molecular lever requires accurate knowledge of its position x if reversible work is to be perfonned. Specifying the positional accuracy as Ax, the uncertainty principle gives the energy requirement as... 

Another feature of the spectrum shown in Figure 10.19 is the narrow width of the absorption lines, which is a consequence of the fixed difference in energy between the ground and excited states. Natural line widths for atomic absorption, which are governed by the uncertainty principle, are approximately 10 nm. Other contributions to broadening increase this line width to approximately 10 nm. 

If the radiofrequency spectmm is due to emission of radiation between pairs of states - for example nuclear spin states in NMR spectroscopy - the width of a line is a consequence of the lifetime, t, of the upper, emitting state. The lifetime and the energy spread, AE, of the upper state are related through the uncertainty principle (see Equation 1.16) by... 

An important consequence of shortening a laser pulse is that the line width is increased as a result of the uncertainty principle as stated in Equation (1.16). When the width of the pulse is very small there is difficulty in measuring the energy precisely because of the rather small number of wavelengths in the pulse. For example, for a pulse width of 40 ps there is a frequency spread of the laser, given approximately by (2 iAt), of about 4.0 GFIz (0.13 cm ). 

The uncertainty principle, according to which either the position of a confined microscopic particle or its momentum, but not both, can be precisely measured, requires an increase in the carrier energy. In quantum wells having abmpt barriers (square wells) the carrier energy increases in inverse proportion to its effective mass (the mass of a carrier in a semiconductor is not the same as that of the free carrier) and the square of the well width. The confined carriers are allowed only a few discrete energy levels (confined states), each described by a quantum number, as is illustrated in Eigure 5. Stimulated emission is allowed to occur only as transitions between the confined electron and hole states described by the same quantum number. 

The uncertainty principle necessitates that any extremal trajectory should be spread , and the next step in our calculation is to find the prefactor by incorporating the small fluctuations around... 

Equation (4.24) indicates that the quantum number of the transverse x-vibration is an adiabatic invariant of the trajectory. At T=0 becomes the instantaneous zero-point spread of the transverse vibration (2co,) in agreement with the uncertainty principle. 

Moderately slow exchange. The state lifetime is 2t we ask how the absorption band is affected as this becomes smaller. The uncertainty principle argument given earlier is applicable here lifetime broadening will occur as the state lifetime decreases. Thus, we expect resonance absorption at (or near) frequencies Va nnd Vb but the bands will be broader than in the very slow exchange limit. Equation (4-68) is applicable in this regime. 

Whether this concept can stand up under a rigorous psychological analysis has never been discussed, at least in the literature of theoretical physics. It may even be inconsistent with quantum mechanics in that the creation of a finite mass is equivalent to the creation of energy that, by the uncertainty principle, requires a finite time A2 A h. Thus the creation of an electron would require a time of the order 10 20 second. Higher order operations would take more time, and the divergences found in quantum field theory due to infinite series of creation operations would spread over an infinite time, and so be quite unphysical. 

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... 

According to the uncertainty principle the non-exponential short-time behaviour of Kt determines the deviation of the high-frequency spectral wings from Lorentzian shape. The actual spectrum obtained by substitution of Eq. (2.53) into Eq. (2.13) is bi-Lorentzian ... 

The uncertainty principle has negligible practical consequences for macroscopic objects, but it is of profound importance for subatomic particles such as the electrons in atoms and for a scientific understanding of the nature of the world. 

FIGURE 1.22 A representation of the uncertainty principle, (a) The location ot the particle is ill defined and so the momentum of the particle (represented by the arrow) can be specified reasonably precisely, (b) The location of the particle is well defined, and so the momentum cannot be specified very precisely. 

In an early model of the hydrogen atom proposed by Niels Bohr, the electron traveled in a circular orbit of radius uncertainty principle rules out this model. 

The uncertainty principle is negligible for macroscopic objects. Electronic devices, however, are being manufactured on a smaller and smaller scale, and the properties of nanoparticles, particles with sizes that range from a few to several hundred nanometers, may be different from those of larger particles as a result of quantum mechanical phenomena, (a) Calculate the minimum uncertainty in the speed of an electron confined in a nanoparticle of diameter 200. nm and compare that uncertainty with the uncertainty in speed of an electron confined to a wire of length 1.00 mm. (b) Calculate the minimum uncertainty in the speed of a I.i+ ion confined in a nanoparticle that has a diameter of 200. nm and is composed of a lithium compound through which the lithium ions can move at elevated temperatures (ionic conductor), (c) Which could be measured more accurately in a nanoparticle, the speed of an electron or the speed of a Li+ ion ... 

Rodebush has also implied that the accuracy with which very low temperatures can be measured is restricted by the uncertainty principle and by the nature of the substance under investigation. However, the accuracy of a temperature measurement is not limited in a serious way by the uncertainty principle for energy, inasmuch as the relation between the uncertainty in temperature and the length of time involved in the measurement depends on the size of the thermometer, and the uncertainty in temperature can be made arbitrarily small by sufficiently increasing the size of the thermometer we assume as the temperature of the substance the temperature of the surrounding thermostat with which it is in either stable or metastable equilibrium, provided that thermal equilibrium effective for the time of the investigation is reached. 

In Science, every concept, question, conclusion, experimental result, method, theory or relationship is always open to reexamination. Molecules do exist Nevertheless, there are serious questions about precise definition. Some of these questions lie at the foundations of modem physics, and some involve states of aggregation or extreme conditions such as intense radiation fields or the region of the continuum. There are some molecular properties that are definable only within limits, for example, the geometrical stmcture of non-rigid molecules, properties consistent with the uncertainty principle, or those limited by the negleet of quantum-field, relativistic or other effects. And there are properties which depend specifically on a state of aggregation, such as superconductivity, ferroelectric (and anti), ferromagnetic (and anti), superfluidity, excitons. polarons, etc. Thus, any molecular definition may need to be extended in a more complex situation. 

Apply the uncertainty principle to the operators Lx and Ly to obtain an expression for ALj ALy. Evaluate the expression for a system in state Im). 

Heisenberg, the atom-bomb man. The uncertainty principle. The more accurately you measure the position of something at a particular moment, the less accurately you can measure where it s going the velocity—the trajectory. At least, that s roughly it. My father could tell you more. ... 

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. 

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