5 Heisenberg’s Uncertainty Principle

Prof. V.S. Bhasin

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TABLE OF CONTENTS

 

1. Learning Outcomes

2. Introduction

3. Heisenberg’s Uncertainty Principle

3.1 Measurement Experiments

3.1.1 Determination of Position when momentum is precisely known

3.1.2 Determination of Momentum when position is precisely given

4. A Few Examples Illustrating Uncertainty Principle

5. Complementarity Principle

6.  Summary

 

1. Learning Outcomes

 

  • After studying this module, you shall be able to
  • Know the statement of Heisenberg’s Uncertainty Principle
  • Learn through a measurement experiment ( Heisenberg’s γ-ray microscope),how the process of trying to determine position using a precise value of momentum, not only gives an inaccurate position determination but at the same time introduces the uncertainty into the momentum.
  • Learn through another measurement experiment ( Doppler shift)that even if the position is known precisely the process of measuring momentum not only gives uncertainty in the momentum measurement but at the same time introduces uncertainty in its position
  • Know through a few typical examples the implications /consequences of using the uncertainty principle
  • Learn the complimentary principle given by Bohr

 

2.  Introduction

 

In this module we introduce the concepts such as Heisenberg’s uncertainty principle and the complementary principle enunciated by Bohr. While constructing the wave packets in space and time, we noticed that there are fundamental limitations placed by the wave packet representing the particle on the accuracy with which both the position and momentum of a particle can be specified. We found that if we construct a wave packet in space which has a spread Δx , then the range of width Δk (k being the wave number of harmonic waves superposed) is related with ∆   ≥ (1⁄∆ ). In fact, this is a result quite generally known from the theory of Fourier transform. In other words, the spread in momenta (Δp) of the de-Broglie waves contributing to wave packet satisfies an inequality (Δp) (Δx) ≥ Similar inequality between energy and time also holds when we consider a wave packet in time constructed from the superposition of a spectrum of harmonic waves in frequencies (or angular frequencies). This is the essential content of the uncertainty principle, which we shall be studying in the present module. The  underlying idea that the physical variables describing the basic properties of matter, such as momentum and position, do not exist in a precisely defined form in quantum theory constitute a far-reaching change that led Bohr to enunciate it in terms of what he called the complementary principle. This will be illustrated through the example of a double slit experiment in the present module.

 

3    Heisenbrg’s Uncertainty Principle:

 

One of the basic contents of the theory of quantum mechanics is the uncertainty principle, developed by W. Heisenberg in 1927. Stated in simple terms, it says that if the x-coordinate of the position of a quantum mechanical particle is known to an accuracy, ∆x, then the x-component of the corresponding momentum variable cannot be determined to an accuracy better than ∆ ≈ ℎ /∆ . In other words, one cannot determine precisely and simultaneously the values of both the members of particular pairs of physical variables, known as canonically conjugate to each other, which define a given atomic system. The x- (or y- or z-) component of the position coordinate of a particle and the corresponding component of momentum, ( or ) , a component of angular momentum and its angular coordinate ϕ ( in the xy- plane), the energy E of a particle and the time t at which the energy is measured – are some of the examples of canonically conjugate pairs. Thus, expressed in quantitative terms, the uncertainty principle states that,

 

Similar inequalities between corresponding components of the   y- and z- components of the position vector and the the momentum vector, viz.,  can also be written. Notice, however, that there is no relation between the uncertainty on the component, ∆x, of the position vector and the uncertainty in a different Cartesian component of the momentum, for example p y . The fact that Planck’s constant ‘h’ has a value so small that makes the uncertainty principle relevant primarily to the systems of atomic size.

 

Let us illustrate this principle first by discussing some measurement experiments and then present a few typical examples.

 

3.1 Measurement Experiments

 

3.1.1 Determination of Position when momentum is precisely known:The γ-ray microscope

 

Let us illustrate the uncertainty principle through a measurement experiment, which was first discussed by Heisenberg to measure the position of a particle as accurately as possible. The basic idea is to show how quantum effects intervene to prevent measurements of unlimited precision. Suppose we wish to determine the position of a small object with precision using a microscope. The limit to the accuracy, ∆x, with which a position determination can be made is given by its resolving power

where   is the wavelength of the radiation that enters the objective lens and α is half the angle subtended at the position of the particle P, by the objective lens, as is shown in Fig.

 

 

Fig. (4.1Experimental arrangement for the localization of a particle P illuminated by means of one of the scattered quanta Q. The image is focused on the screen by the lens L

 

 

the object P itself recoils, the x-component of the recoil momentum is also uncertain to the same extent, viz.,

wherein we have used Eq. (5.2). Thus, Eqn. (5.3) shows that in the very process of trying to determine position precisely, the momentum is made uncertain and the extent of this uncertainty is determined by the relation ∆  ∆  ~ℎ.

 

One way of trying to reduce the uncertainty in momentum transfer is to narrow the aperture of the lens so as to reduce the range of scattered angles. This would decrease the resolving power of the lens due to increased diffraction effects. As a result there would be a proportional decrease in the accuracy of the position measurements. Thus no matter how we perform the experiment a limit on the accuracy of the measurements corresponding to the uncertainty principle would always be introduced at some point in the experiment.

 

3.1.2 Momentum Determination Experiment

 

In the above experiment it is assumed that the momentum of the particle is precisely known before the measurement is performed. We found that the measurement not only gives an inaccurate position determination but at the same time introduces the uncertainty into the momentum.

 

Let us now consider a different experiment in which the position is accurately known initially and the momentum is measured. Consider the method of measuring momentum by determining the velocity by means of Doppler shift of the light radiated by the particle. ( We know the velocity of a star, for example, is generally obtained by measuring the red shift.) Here the particle is, in fact, an atom when it is in an excited state. Suppose the frequency of the photon radiated by the atom, when it is at rest, is ν . Because of the Doppler effect, motion of the atom towards the observer with speed v means that the observed frequency is

 

The uncertainty in the velocity depends on the accuracy with which we measure ν′ and it is known that the uncertainty in ν′ is

 

where is the measure of uncertainty in time or the instant at which the photon is emitted; at this instant the momentum of the atom decreases by hν’/c and its velocity decreases by hν’/(mc). This makes the position of the atom uncertain by the amount

 

Thus the position uncertainty arises due to the finiteness of .; it is because of the fact that is finite that we do not know when the velocity changed and therefore where the atom at later times is.

 

The uncertainty in momentum is obtained by the Eqs.(5.4) and (5.5):

Since we here consider a approximation, Eq.(5.4) gives uncertainty Δx ≈ ℎ. non-relativistic case, v/c<<1 and therefore, in this ν′≅ . Thus, combining Eqs. (5.4) and (5.7) leads to the  Δx Δ?? ≈ ℎ

 

4    A Few Examples Illustrating Uncertainty Principle

 

Let us consider a few examples to study the implications of the uncertainty principle:

 

Example 1

 

Let us apply the first of the inequalities given above to study its implication in the case of hydrogen atom. Suppose the spread Δx in the position of the electron in hydrogen atom is of the order of its radius. Now experimentally, the size of hydrogen atom is determined to be of the order of 1 A . T hus, uncertainty in the position of the electron must be of the order , Δx∼ 10−10 . Using the first inequality in (5.1), would then imply the uncertainty in momentum Δp≥ 1010ℏ. Since the mass of the electron is approximately 10−30 , the uncertainty in the value of velocity of the electron 106 m / s , which turns out to be much higher than the known value. It is thus evident that the electron in hydrogen atom can not be described even approximately in classical terms and it makes no sense to talk of a well defined trajectory.

 

Example 2

 

In a Stern-Gerlach experiment, a beam of silver atoms emerges from an oven which contain silver vapour at a temperature 1200 °K. The beam is collimated by passing through a small aperture. Let us see how far the spot on the screen can be localized by narrowing the aperture in accordance with the uncertainty principle.

Fig. Arrangement of Stern-Gerlach Experiment

 

Suppose the atoms have a component of momentum along the beam. Then by equipartition of energy,

The y-coordinate of an atom at the moment when it passes through the aperture is determined to a precision Δy=a, where a is the diameter of the aperture. There is therefore, an uncertainty p y in the corresponding component of the momentum such that Δp y~ h / a . This leads to a spread of the beam with angle α, where

 

Now, the spot of diameter D on the screen, which is at a distance l from the oven, is given by

The minimum value of D of the spot is obtained by differentiating this expression with respect to a, which gives

Substituting this value of a, we get

Since the right hand side of this expression has all fixed constants, it is clear one can not minimize the spot on the screen beyond a certain value.

 

Example 3:

 

It is believed that the origin of fundamental forces of Nature taking place between different constituents is due to the exchanged quanta. One can make use of the uncertainty principle, E t h , to estimate the rest mass of the exchanged quanta for the given range of interaction. The range of the interaction is related with the energy E of the field quantum exchanged. The exchanged quantum, being virtual, exists only for a short time t ≤ h/E. Assuming that the field quanta travel with velocity of light c, the range

If the field quantum has rest mass m, it must have at least , energy, E=m 2. Using this value of E in the above relation gives

 

This relation clearly shows that if the range of interaction is infinite, the mass of the exchanged quantum is zero as is known in the case of electromagnetic or gravitational forces. For a finite range, as in the case of nuclear force, if we take the value of 0 ≅ 1.4   , the mass of the exchanged quantum , using the above relation, gives the value to be about 140 MeV/ 2. This particle is identified as pion, originally proposed by Yukawa as a mediator responsible for nuclear interaction occurring between nucleons.

 

Example 4:

 

Consider, in a Frank-Hertz experiment, a mono-energetic beam of electrons incident on hydrogen atoms. As a result these atoms are raised to their first excited state by colliding with a beam of electrons. Suppose mean life of the excited hydrogen atoms is τ. This means that these atoms on an average take a time τ to radiate their energy and return to the ground state. The excited state has therefore a life time with a precision Δt≈ and therefore its energy in accordance with the uncertainty principle will have an uncertainty ΔE≥ ℏ . Due to short life time, the energy fluctuations in the energy of the excited atoms are thus appreciable. In fact it is the energy fluctuations ΔE due to the short lives of excited states which give rise to the natural widths of spectral lines.

Fig.    Spectraline of an excited state having a width due to energy fluctuation

 

5 Complementarity Principle : The double slit experiment

 

Let us revisit Young’s double slit experiment showing the interference of light ( or , equivalently of a beam of electrons instead of light) in the context of uncertainty principle, and probe the question: can we say that the photon (or electron) which reaches some point of the interference pattern has followed a definite trajectory passing through a particular one of the two slits? The probability interpretation, which was earlier introduced, makes it abundantly clear that we cannot.

 

Let us assume that the particle has a trajectory. Because of the wave nature of the particle, a sharp trajectory is not possible. Let us suppose that the trajectory is a moving wave packet with lateral extension less than half the separation ‘a’ between the two slits. This assumption is made to ensure that it can go through only one of the two slits. Can there be an interference pattern then? Let us try to answer this question from the point of view of uncertainty principle. As we have assumed that the extension of the wave packet is less than a/2, the uncertainty in a component of the position vector perpendicular to the direction( assumed to be along x-axis) of motion of the particle is ∆  <   /2. Therefore, according to the uncertainty principle, the corresponding transverse component of momentum has uncertainty, ∆  , i.e., at least ( h / ∆y) = 2h / a. Therefore the angle, θ, giving the direction of motion of the particle is uncertain by an amount

where   is the de- Broglie wave length of the particle. But we know the angular separation between successive maxima of the interference pattern in Young’s double slitis /a. Clearly, as the above uncertainty exceeds this value, the interference pattern will be washed out. Note that the above analysis holds for a material particle as well as for a photon.

 

The situation analyzed above clearly shows that when the wave packet is made compact enough to make it behave like a particle, it then loses the ability to display wave properties. In other words, the particle and the wave aspect of a physical entity are complementary and cannot be demonstrated at the same time, both aspects are needed for a complete description. This is the complementary principle of Bohr.

 

Bohr, thus, introduced this principle to understand in more physical terms the implications of Heisenberg’s uncertainty principle. According to this, atomic phenomena cannot be described with the precision demanded by classical dynamics: the complementary elements which are necessary for a complete description are mutually exclusive. Looking from an experimentalist’s point of view, this uncertainty or inaccuracy in measurement is not because of any deficiency in his skill or because of the limitation in the apparatus available to him but rather may be regarded as a law of nature that measurements more precise than those demanded by the uncertainty principle can not be made.

 

6. Summary

  • Having studied this module, you should be able to
  • Know the statement of Heisenberg’s Uncertainty Principle
  • Learn through a measurement experiment how the process of trying to determine position precisely using a precise value of momentum, not only gives an inaccurate position determination but at the same time introduces the uncertainty into the momentum.
  • Learn through another measurement experiment that even if the position is known precisely the process of measuring momentum not only gives uncertainty in the momentum measurement but at the same time introduces uncertainty in its position
  • Know through a few typical examples the implications /consequences of using the uncertainty principle
  • Learn the complementary principle given by Bohr

 

you can view video on The Heisenberg Uncertainty Principle

 

Suggested Reading

  • Visit the following web sites
  • https://www.aip.org/history/heisenberg/voice1.htm
  • Heisenberg Recalls His Early Thoughts on the Uncertainty Principle
  • https://www.aip.org/history/heisenberg/p08c.htm
  • Implications of Uncertainty Principle;
  • Practical Value of Uncertainty Principle:
  • “Even the Uncertainty Principle isn’t “merely” philosophy: it predicts real properties of electrons. Electrons jump at random from one energy state to another state which they could never reach except that their energy is momentarily uncertain. This “tunneling” makes possible the nuclear reactions that power the sun and many other processes. Physicists have put some of these processes to practical use in microelectronics. For example, delicate superconducting instruments that use electron tunneling to detect tiny magnetic fields are enormously helpful for safely scanning the human brain.”

 

References

 

For More Details ( on this topic and other topics discussed in Text Module) See

1.Quantum Mechanics by Bransden and Joachain, Pearson Education Ltd(2000).

2.Quantum Mechanics ,Vol. I, by C. Cohen- Tanoudji , John Wiley & Sons

3.A Text Book of Quantum Mechanics by P M Mathews and K Venkatesan; Tata McG raw-Hill Publishing Co.

Ltd.

4. L.D. Landau & E.M.Lifshitz (1977). Quantum Mechanics: Non-Relativistic Theory. Vol. 3 (3rd ed.). Pergman

Press

5. Messiah, Quantum Mechanics, Vol. 1, Wiley, NY, 1961