Chapter 3

Wave Packets and Uncertainty Principle

Imagination is more important than knowledge.

—Albert Einstein

3.1  INTRODUCTION

What we have learnt in the previous chapter is the fact that the behaviour of microscopic system depends upon the type of experimental set up we choose to probe the system. The electrons are detected like particles in a detector, whereas the same electrons behave like propagating waves in a two-slit experiment (similar is the case with radiations). We have to describe dynamics of microscopic particles e.g. electrons. The dynamics shall be described through some equations of motion. Therefore, we have to find out equation of motion of microscopic particles which should be valid under all circumstances. When the electrons are found behaving like waves, we shall have to seek equation of motion of these (electron) waves. But when the electrons are found behaving like particles, the equation of motion should be written for the particles.

Well, the experiments which show particle properties of electrons, protons, neutrons, etc. do not really suggest or dictate any particular size or shape of these particles. Therefore, in the absence of any such compulsions, we should feel free to assume that while showing particle nature, the electrons, protons, neutrons, etc. are in the form of a tiny wave packet—a compact wave, whose amplitude is zero everywhere except in a very small region. We shall see in the next section that such a wave packet can be formed by superposing a large number of plane waves of varying wavelengths propagating simultaneously in the same direction.

But does it not seem to be an arbitrary proposition to picturise electrons as tiny wave packets when electrons are detected as particles? Not really. If this picture (i) does not violate any physical law and (ii) is not in contradiction with any experimental observation, and moreover (iii) helps us in making a consistent description of microscopic objects, we should definitely have it as a possible valid picture. In fact, by adopting such a picture, we can make statements like ‘electrons are always there in the form of waves, sometimes in the form of extended propagating plane waves (as in the two-slit experiment), and sometimes in the form of tiny wave packets (as in the experiments detecting electrons as particles)’. If that is the case, that is, if we can always treat a microscopic particle as a wave (sometimes ‘extended’ and sometimes ‘compact’), it will be a uniform description/picture of microscopic particles in the form of ‘waves’. We shall need equation of motion of waves only—of course of different types of waves.

Firstly, we proceed to study in the next section the superposition of waves which will give wave packets of various sizes and shapes.

3.2   SUPERPOSITION OF WAVES

3.2.1   Superposition of Two Plane Waves

Before considering electron waves or matter waves in general, we consider ordinary waves (e.g. mechanical waves) at macroscopic level. Let us consider two plane waves propagating simultaneously along x-axis [see Figure (3.1)]. Let the two waves have same amplitude A but slightly different angular frequencies (ω – ∆ω) and (ω + ∆ω). The two waves are represented by the wave equations

 

 

 

According to the principle of linear superposition of waves, the resultant displacement y at a position x at time t is the sum of y₁(x, t) and y₂(x, t), that is,

 

 

The resultant wave, represented by Eq. (3.2), may be thought of a wave of angular frequency ω and wave vector k that has superposed upon it a modulation of angular frequency ∆ω and of wave vector ∆k. The resultant wave thus consists of successive wave groups (which, for example, produce the familiar phenomenon of beats, if y₁ and y₂ represent sound waves).

Whereas each component wave is moving with its own phase velocity [v_p1 = (ω – ∆ω)/(k – ∆k) and v_p2 = (ω + ∆ω)/(k + ∆k)], the resultant wave, that is, the wave group, is moving with the so-called group velocity

Figure 3.1 Superposition of two plane waves of slightly different frequencies

3.2.2  Superposition of Large Number of Plane Waves

Let us now consider a large number of plane waves propagating simultaneously along x-axis. These waves are differing in their wavelengths and frequencies slightly from each other. We superpose all these waves with probability distribution g(k). The resultant amplitude y(x, t) may be obtained by integrating over the entire range of wavelengths

where Re represents the real part, Let us consider the superposition at time t = 0.

As a special case, let us take the following form of g(k)

 

We shall study its time variation a bit later. Now, we study how the resultant displacement looks like as a function of x. We plot schematically, y(x) as a function of x for two different values of Δk. The plots are shown below.

Figure 3.2(a) shows the resultant displacement for the case when all waves with their wave vectors lying in the range (k₀ – π) to (k₀ + π) are superposed, while Figure 3.2(b) shows the corresponding displacement of the resultant wave for the wave vector range (k₀ – 2π) to (k₀ + 2π). It is evident that larger is the range of wave vector (i.e. larger is the number of waves superposed) narrower is the region in which resultant wave has appreciable amplitude (i.e. narrower is the resultant wave packet).

Figure 3.2 Superposition of plane waves gives rise to wave packets: (a) a broader wave packet (obtained from lesser number of plane waves) (b) a narrow wave packet (obtained from large number of plane waves)

3.3  PHASE VELOCITY AND GROUP VELOCITY

In Section 3.2.1, we studied the superposition of two propagating plane waves y₁(x, t) = A cos[(k – ∆k) x – (ω – ∆ω)t] and y₂(x, t) = A cos [(k + ∆k)x – (ω + ∆ω)t] giving rise to the resultant wave.

 

 

where

 

 

In Figure 3.1, we had plotted y₁(x, 0), y₂(x, 0), and y(x, 0). Now we plot A′(x, 0), cos(kx), and y(x, 0) in Figure 3.3. Also plotted in Figure (3.3a) is the displacement. A′(x, t₁) at a bit later time t₁ (than t = 0). Similar plots of cos(kx – ωt₁) and y(x, t₁) at a bit later time t₁, may be thought of. It may be easily concluded that A′(x, t) = 2A cos(∆kx – ∆wt) is moving with velocity ∆ω/∆k = v_g. In Figure 3.3(c), the resultant displacement (i.e. the modulated amplitude shown by dotted curve) is propagating with velocity v_g = ∆ω/∆k. Thus, while the two individual waves are moving with their phase velocities v_p1 = (ω – ∆ω)/(k – ∆k) and v_p2 = (ω + ∆ω)/(k + ∆k), their resultant is moving with the group velocity v_g = ∆ω/∆k.

The concept of group velocity becomes more evident, when we consider (as in Section 3.2.2) superposition of a large number of plane waves propagating in same direction. As we had seen in Section 3.2.2, the resultant displacement y(x, 0) is large in a small region and becomes very small outside this region. At a bit later time t₁, the resultant displacement y(x, t₁) will show its peak at a displaced position. In fact, we shall see later in details that because of, a bit, different velocities of all individual waves, the resultant wave packet shall have a wider shape (at later time) than that at t = 0. However, the maxima of the wave packet is moving with a velocity, the group velocity,

Figure 3.3 Schematic plots of (a) 2A cos ∆kx and 2A cos (∆kx– ∆ωt) (b) cos kx and (c) y(x, 0) = 2A cos ∆kx cos kx

3.4   DE BROGLIE RELATION

It is surprising to note that even prior to the actual experiment showing electron beam diffraction (Davison-Germer experiment, 1926), French physicist L. de Broglie made a very bold unifying hypothesis that microscopic material particles might also exhibit wave-like properties. This hypothesis was based on the logic: ‘when radiations show dual nature, the material particles should also show dual nature and, thereby, the wave-particle duality should become a universal characteristic of nature’.

Thus, propagating particle may be thought of as comprising of wave or waves or wave packet. In general, we may take a wave packet representing a particle. So the group velocity of the wave packet should be equal to the velocity of the particle, which is p/m (p = linear momentum of the particle, m = mass of the particle).

 

∴

If E is (kinetic) energy of the particle, and if we assume similar relation (between energy and angular frequency) as that for radiations, we have

which gives

For a particle, the kinetic energy is

∴

From Eqs (3.12) and (3.15), we get

or

or

The wavelength λ = h/p is called the de Broglie wavelength of the particle moving with linear momentum p. If the particle of (rest) mass m is moving with non-relativistic speed v, p = mv. However, if the speed is high (i.e. relativistic), the linear momentum Then, the de Broglie wavelength is

If E is the kinetic energy of the particle, then E = p²/2m

or

Therefore, de Broglie wavelength is

3.4.1  de Broglie Wavelength Associated with Electrons

Let the electron of rest mass m and electric charge – e be accelerated from rest by a potential V to velocity v, then

or

Its wavelength

or

3.4.2   Phase Velocity and Group Velocity of Free Particles

In Sections 3.2 and 3.3, we argued that a propagating particle may be assumed to be represented by a wave-packet; the wave-packet, itself, is composed of a large number of propagating plane waves. The group velocity v_g (= dω/dk) of the packet represents the velocity of the particle, whereas each component wave propagates with its phase velocity v_p (= ω/k). In fact, the group velocity may be less, equal, or greater than the phase velocities of the component waves; it depends upon the (dispersion) relation between frequency ω and the wave vector k of the waves.

Let us consider a free particle of rest mass m moving with (a constant) velocity v. We shall consider both the cases (i) v is not too large (non-relativistic case) and (ii) v is too large (relativistic case).

1. Non-relativistic case:

   The frequency ω and the wave vector k of the (associated) matter waves are

   

   and

   

   Now group velocity

   

   and the phase velocity

   
2. Relativistic case:

   In this case the total (kinetic) energy of the particle

   ∴

   

   so

   

   and

   

   ∴

   

   and

   

   It can be easily checked that for (v/c) << 1 (the non-relativistic limit), Eq.(3.20b) gives

   

   the same result as Eq. (3.18b).

3.5   MEASUREMENT AND UNCERTAINTY PRINCIPLE

We have seen in the previous three sections of this chapter that it is possible to get a compact wave packet (moving with the group velocity v_g = dω/dk) by superposing a large number of plane propagating waves, each moving with its own phase velocity v_p (= ω/k). It may also be seen there that larger is the span of wave vector ∆k (i.e. larger is the number of plane waves involved) more localized is the resultant wave packet. From Figures 3.2(a,b), it is clear that

 

 

where ∆x is the wave packet width, the region where its amplitude is appreciable.

It simply means if we have a smaller number of plane waves (with their wave vectors lying around k_o), their superposition results in a (propagating) wave packet which is quite wide. The resulting propagating wave packet has its amplitude larger in a certain region and vanishingly smaller everywhere outside this region. On the other hand, if the number of plane waves superposed is very large (i.e. the wave vector interval ∆k is very large), the resultant propagating wave packet is very narrow.

Now following de Broglie hypothesis (that a moving particle is nothing but a propagating matter wave), broadly speaking, we may consider three situations:

1. The moving particle may be represented by a single propagating plane wave. In such a case, the wave and, therefore, the particle is spread throughout the region. In fact, if now we insist to ask the question as to ‘where is the particle’? the answer can only be ‘it can be anywhere’. That means the position of the particle is totally uncertain, that is, ∆x = ∞. But the plane wave, representing the particle, has well-defined single value of its wavelength and hence well-defined value of the wave vector of the particle (k = 2π/λ). That means the uncertainty in its wave vector ∆k = 0.
2. The moving particle may be represented by a broad wave packet. In this case, the wave packet (or the particle) is spread in a region ∆x₁, which is, in fact, inversely proportional to the wave vector interval ∆k₁ (around k₀) in which the wave vectors of all plane waves (superposed) lie. As we had seen in Section 3.2.2, ∆x₁ will be large if ∆k₁ is small, that is, if a smaller number of plane waves are superposed. Now we can not assign a single wave vector k (or wavelength λ or linear momentum p) to the wave packet or particle. Its wave vector is uncertain within the interval ∆k₁, as is its position within the length ∆x₁.
3. The moving particle may be represented by a very narrow wave packet. In this case, ∆x₁ is extremely small, whereas the wave vector spread ∆k₁ is very large. In this situation, the wave packet (and therefore the particle) at any time has very well-defined position in space but has quite uncertain value of its wave vector.

Realizing that the linear momentum of the particle is equal to ħ multiplied by the wave vector k, the above mentioned observations may be summarized as follows:

When a particle is propagating such that it is represented by a single plane wave (of a fixed wave vector k), the linear momentum of the particle p (= ħk) has a single well-defined value (i.e. ∆p = ∆k = 0), whereas the position of the particle is totally uncertain, in fact, equally probable everywhere in the region (i.e. ∆x = ∞). On the other hand, when the propagating particle may be represented by a narrow wave packet, the position of the particle is relatively well defined; it is somewhere within the small region ∆x (= 1/∆k), where the wave packet has large amplitude, but the linear momentum of the particle is very uncertain; in fact, it is uncertain within ∆p (= ħ∆k, ∆k being the range around k₀) in which the wave vectors of all plane waves (participating in forming the wave packet) lie. In terms of the uncertainty in linear momentum ∆p_x, Eq. (3.22) may be written as

 

 

We shall see in the next section that Heisenberg’s uncertainty principle, which is the result of experimental observations, just dictates similar limitations on the accuracy of position and linear momentum values of the particle.

3.6   UNCERTAINTY PRINCIPLE: THOUGHT EXPERIMENTS

In the previous sections, we have seen that, in general, a propagating material particle may be represented by a single plane wave, or by a broad wave packet or by a very narrow wave packet. In each case, the restriction ∆k∆x ≥ 1 (or ∆p∆x ≥ ħ, as p = ħk) holds good. It seems the description of the particle can not be made more precise than this. That this is a fundamental limitation on the precision of simultaneous description of the position (x) and linear momentum (p) of the particle, shall be even more clear if we go through the following two (thought) experiments.

3.6.1   Position-momentum Measurement by Heisenberg’s Microscope

In spite of convincing arguments put forward in this chapter, let us still stick to the particle picture of microscopic entities such as electrons. So, we have electron envisioned as a particle, and we want to find out its position with the help of a microscope. Let the electron be moving as shown in Figure 3.4(a). To obtain the information about the position of the electron at a certain time, we should shine light on it, which after reflection from the electron shall enter the microscope and shall give information about its position. Let us see, what happens in details. Let us use light of wavelength λ (i.e. frequency v = c/λ, energy hc/λ and linear momentum h/λ).

Figure 3.4 Heisenberg’s gamma-ray microscope

For the electron to be ‘seen’ by the observer, one photon must be scattered into the microscope. Now in scattering process of one photon by the electron, some linear momentum from the photon shall be transferred to the electron according to the rules of ‘Compton scattering’. The exact amount of the change ∆p in the x-component of linear momentum of the electron cannot be found. However, we know that the photon was scattered in a direction into the microscope; the uncertainty in the direction of scattering being of order θ. As a result, the x-component of the linear momentum of electron shall have uncertainty of the order of

The position of the electron, which is to be ascertained with the help of scattered photons, may not be known exactly. In fact, the accuracy in the position is limited by the resolving power of the microscope. The resolving power of a microscope is inversely proportional to the wavelength of the light used. A reasonable estimate of the minimum uncertainty in the position of electron ∆x, using physical optics is

 

 

Combining the two results, Eqs (3.23) and (3.24), we get

 

 

The result is consistent with Eq. (3.23).

We may get tempted to conclude from the above discussions that the electron has well-defined position and linear momentum and it is only because of uncontrollable effects of the measuring probe (photon beam in this case) that the uncertainty in the position and momentum measurements, is introduced.

This conclusion is neither correct nor consistent with any of the observations on microscopic particles mentioned earlier in this chapter. The consistent physical picture, which emerges out of all experimental observations, is the same as mentioned in Section 3.5 viz. the moving particle manifests itself in some form of wave; may it be in the form of plane wave, or broad wave packet or very narrow wave packet.

It is impossible to find a way to determine the position and linear momentum of the particle accurately and simultaneously. The uncertainty principle is the universal principle that can not be defied under any circumstances.

3.6.2   Position-momentum Measurement by Single-slit Experiment

Let us try to determine the position of a particle (electron) not by using light, which we have seen disturbs the particle, but simply by passing the particle through a narrow slit of width b. When the particle is passing through the slit of width b shown in Figure 3.5, we are sure its position along y axis is known within the accuracy b, that is, the uncertainty in position in y-direction

 

 

If we do this experiment with a mono-energetic beam of electrons passing through the slit, we shall find the electrons, after passing through the Slit S, spread all along the Screen D in a fashion shown in Figure 3.5. In fact, as discussed in Section 2.7 (Chapter 2), the mono-energetic beam of electrons shall produce single slit diffraction pattern, suggesting that the electrons are propagating in the form of plane waves. But here in this experiment, our motivation is to find precise value of the position of the electron. If we make the slit more narrow, that is, make b small, we shall be able to locate y-position of electron more precisely, that is, ∆y will be smaller if b is made small. But let us see the consequences. If we make b smaller, the angle θ through which the electron can deviate (from its initial x-direction) after passing through the slit, will be larger. And, therefore, larger will be the y-component of linear momentum attained by the electron. If the mono-energetic electron beam has linear momentum p(λ = h/p) in x-direction, the position of first minima of the diffraction pattern θ is given by

 

 

or

 

Figure 3.5 Diffraction of electrons passing through a slit

But, p sin θ is the maximum uncertainty in the y-component of the linear momentum of the electron. Before passing through the slit, the electron was propagating in x-direction. After passing through the slit, the electron may propagate in any direction within the angle θ about x-axis. So,

 

 

From Eqs (3.27) and (3.29), we get

 

3.6.3  The Energy – Time Uncertainty Relation

Let us start with position–momentum uncertainly relation

 

 

or

Now for a free particle moving in x-direction, its total energy E is

so the uncertainty in its energy is related with that in its linear momentum

If we assume a monitor be located at a fixed position, the time at which the particle or the wave packet passes the monitor, must be uncertain by an amount ∆t, where

With Eqs (3.32) and (3.33), Eq. (3.31) gives

 

 

It may be mentioned here that a rigorous proof of the so-called generalized uncertainty principle shall be given in next chapter, which states that

3.7   CONCLUSIONS

It becomes quite clear by now, whether we have a theoretical description of a microscopic particle or seek experimental information about it (e.g. with respect to its position and linear momentum), it is never possible to assign precise values to its position and linear momentum simultaneously. Theoretical description of a particle has always to be in the form of (matter) wave; may be a plane (extended) wave or a broad wave packet or a very narrow wave packet. In every case, either we have precise information about its linear momentum (and absolutely no information about its position) or we have precise information about its position (and then no information about its linear momentum). Therefore, the description of microscopic particle through waves (or wave packet) already has full flavour of uncertainty principle in it. In a way, we may say, the uncertainty principle is already built in the wave description of particles.

Exercise

Exercise 3.1

What is the de Broglie wavelength of a metallic ball of mass 1.0 gram moving at a speed of 10 m/sec?

Exercise 3.2

A proton is accelerated from rest by a potential of 100 volts. What is its de Broglie wavelength?

Exercise 3.3

A beam of mono-energetic neutrons is having de Broglie wavelength of 1 Å. Find the kinetic energy of neutrons in the units of electron volts.

Exercise 3.4

The kinetic energy of an atom in an ideal gas at absolute temperature T is of the order of kT Calculate the de Broglie wavelength of an atom of mass 6.7 × 10^–27 kg in the gas of temperature 27°C.

Exercise 3.5

An electron is moving with kinetic energy of 400 keV. Find out its phase velocity and group velocity (rest mass energy of electron mc² = 511 keV).

Exercise 3.6

An electron is confined in a one-dimensional potential box of width 0.1 nm (= 1.0 Ǻ). What is the minimum kinetic energy of electron?

Exercise 3.7

A neutron is confined in a one-dimensional potential box of width 5.0 × 10^–15 m (= 5 fm ~ the size of a nucleus). Calculate the minimum kinetic energy of the neutron. If, instead, an electron is confined within this box, what would be its minimum kinetic energy?

Exercise 3.8

An electron in an atom is excited to next higher energy state. The electron stays in that excited state for 10^–8 seconds and then goes to its previous state; in this process, the excess energy is emitted in the form of a photon. Find the inherent uncertainty in the frequency of the photon.

Exercise 3.9

An electron is known to have a speed of 200 m/s to an accuracy of 1%:

• What is the minimum uncertainty with which its position is known?
• What is the minimum uncertainty in the time the electron would take to travel 100 m?

Exercise 3.10

Find the energy in eV of

• a photon of wavelength 5000 Å
• an electron of wavelength 5000 Å
• a proton of wavelength 5000 Å

Exercise 3.11

Find the energy in eV of

• a photon of wavelength 1 Ǻ
• an electron of wavelength 1 Ǻ
• a proton of wavelength 1 Ǻ

SOLUTIONS

Solution 3.1

Solution 3.2

The kinetic energy E of the proton is

 

Solution 3.3

Solution 3.4

For an ideal gas

∴

Solution 3.5

Rest mass energy of electron E_o = mc² = 511 k eV. Total energy of relativistic electron:

 

Now

or

or

So,

 

and phase velocity from Eq. (3.20b) is

Solution 3.6

We know that

Now

 

 

∴

and

The size of hydrogen atom is around 1 Ǻ. The kinetic energy of an electron in the ground state of hydrogen atom is 13.6 e V.

Solution 3.7

∴

The momentum p of the electron is of the order of

As mass of the electron is much smaller than that of the neutron, its velocity with the above momentum will be very large. Hence we are supposed to use relativistic expression for kinetic energy of electron.

Hence if an electron is to be confined in a region of ~ 5 f m, that is, inside a nucleus of this size, its kinetic energy must be around 20 MeV. However, it has been observed that kinetic energy of electrons emitted from certain unstable nuclei is only a small fraction of the above mentioned energy. This simply suggests that the emitted electrons in such experiments were not confined in the nucleus but were created at the time of nuclear transformation (e.g. in β decay process where a neutron transforms into a proton and an electron).

Solution 3.8

or

or

Solution 3.9

•  
  
  ∆v = 2 m/s
  ∆p = m∆v = 9.109 × 10^–31 × 2
  

  so

  
  = 2.89 × 10^–5 m
• From part (a)
  

  ∴

  
  = 0.005 s

Solution 3.10

•  
  
  = 2.34 eV

   

• Using formula [Eq. (3.16d)]
  

  We may also use

  
  = 6.05 × 10^–6 eV

   

•  
  
  = 3.29 × 10^–9 eV

Solution 3.11

•  
  E_ph = 11.7 × 10³ eV = 11.7 keV
•  
  E_el = 151.25 eV
•  
  E_pr = 0.083 eV

REFERENCES

1. Beiser, A. 2003. Concepts of Modern Physics, 6th ed., McGraw-Hill.
2. Powell, J.L. and Crasemann, B. 1961. Quantum Mechanics. Reading, MA: Addison-Wesley.
3.  Bohm, D. 1951. Quantum Theory. New York: Prentice Hall.
4. Gasiorowicz, S. 1996. Quantum Physics, 2nd edn. New York: John Wiley.
5. Merzbacher, E. 1999. Quantum Mechanics, 3rd edn. New York: John Wiley.
6. Cohen-Tannoudji, C., Diu, B. and Laloe, F. 1977. Quantum Mechanics, Vol. I. New York: John Wiley.