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I WAVES (ENGEL & REID, 13.2, 13.3 AND 12.6) I.1 Introduction A significant part of the lecture “From Quantum to Matter” is devoted to developing the basic concepts of quantum mechanics. This is not possible without a clear understanding of classical waves. This is the purpose of these additional lecture notes.

Fig. I.1: Circular wave produced by the impact of a water drop on a pond.

A typical example of a circular wave at the surface of a pond is shown in Fig. I.1. Although this example triggers naturally an association with a wave it involves also the complex phenomena near the place where the drop felt in the water that is not describable as a wave. To introduce the concept of a wave we consider thus a much simpler example: that of a propagating disturbance along a stretched rope. It is characterized by a wavelength λ and a propagation velocity V. We use here a capital “V” instead of “v” to avoid confusions with the frequency ν. For an observer at a fixed place along the x-axis the rope is moving up and down with a frequency ν.

Fig.I.2: Sinusoidal disturbance along a stretched rope propagating at a velocity V.

V

x

2

Fig. I.3: For an observer sitting at a fixed position in space the rope moves up and down at a certain frequency ν. If the top picture corresponds to time t = 0, then the bottom picture corresponds to t =T/2 where T is the period of the oscillation.

It takes a certain time T for the fixed observer to see locally the rope to go up, pass through a maximum deflection, move down, pass through a minimum and to come back to the original position. This time T is called the period of an oscillation. During the time T the wave has moved by exactly one wavelength λ. We have therefore the fundamental relation

Vλν = ( I.1

The displacement u=u(x,t) of the rope can mathematically be described with

( ),u u x t=

xo

x

3

( ) ⎟⎠⎞

⎜⎝⎛ −==

TtxAtxuu π

λπ 22sin, ( I.2

In this form λ and T are readily identifiable as

λ → wavelength [m] ( I.3

T → period [s] ( I.4

For an observer at position x0 the displacement varies sinusoidaly with a period T

( ) 2 2, sin oo

x tu u x t AT

π πλ

⎛ ⎞= = −⎜ ⎟⎝ ⎠

( I.5

The following physical quantities are introduced for a more elegant notation

1

1

1

1 frequency [s or Hz]

2 / wavenumber [m ]2 angular frequency [s ]

Tk

ν

π λ

ω πν

−

−

−

≡ =

≡ =

≡ =

( I.6

The displacement ( )txuu ,= can then be written as

( ) ( ), sinu u x t A kx tω= = − ( I.7

Consider now a characteristic point with a given value for u. For example the red dot in Fig. I.3, which corresponds to the top of the wave, moves with the velocity V determined by the condition that

( ) constant

constanttop topu u kx t

kx t

ω

ω

= − = ⇒

− = ( I.8

and thus

V topdxdt k

ω= = ( I.9

4

For reasons that will become clear later, the velocity Vkω

= is called the phase

velocity. So far we have claimed without demonstration that the disturbance of a stretched rope could be described by a sinusoidal expression as in Eq.I.2. In the next section we show that such an expression is a solution of the so-called wave equation. I.2 The wave equation for a rope The string is pulled on both ends by a force F. The mass of the string per meter is called the specific mass ρ. A blow-up of the string is shown below

Fig. I.4: Detail of the disturbance of a stretched rope. The force F used to stretch the rope is indicated by red arrows. The displacement at position x is ( , )u x t and at

position x+ ∆x it is ( ),u u x x t∆= + . Both displacements are strongly exagerated for clarity. In reality they are much smaller than the length of the rope. Note that we do not assume that ( , )u x t is a sine function of x and t.

A string element of length ∆x has a mass ∆m = ρ ∆x . The total force (broad green arrow) along the u-direction acting on the rope element between x and x+ ∆x is

),( txu

),( txxu ∆+

∆F

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x x x

du duF F Fdx dx∆

∆+

= − ( I.10

As, by definition, in the limit ∆x → 0

2

2x x x

du dud u dx dxdx x

∆

∆+

−= ( I.11

we obtain that

2

2x x x x

du du d u xdx dx dx∆

∆+

= + ( I.12

and consequently

2

2d uF F xdx

∆ ∆= ( I.13

From Newton’s law we know, however, that Force=mass× acceleration. For the rope element this implies that

2

2d uF xdt

∆ ρ∆= ( I.14

Equating Eqs.I13 and 14 we obtain

2 2

2 2F d u d u

dx dtρ= ( I.15

INTERMEZZO: At this point we must be very careful with our notation.. When we write 2

2

dxud

we mean

in fact a derivative with respect to x at constant t and when we write 2

2

dtud

we mean in fact a derivative with

respect to t at constant x

To make this clear we use the notation 2

2

xu

∂∂

and 2

2

tu

∂∂

For a function depending on two variables, for example ),( txuu = we mean that

xtxu

xu

∂∂

=∂∂ ),(

= partial derivative with respect to x while t is kept constant

and

ttxu

tu

∂∂

=∂∂ ),(

= partial derivative with respect to t while x is kept constant

with this notation convention, Eq.I.15 can be re-written as

2 2

2 2F u u

x tρ∂ ∂

=∂ ∂

6

In fact, as explained in the intermezzo, we should have written

2 2

2 2F u u

x tρ∂ ∂

=∂ ∂

( I.16

Since [ ][ ]

2N m=kg/m s

Fρ

⎛ ⎞= ⎜ ⎟⎝ ⎠

has the dimension of a velocity squared we write

2 2

22 2V u u

x t∂ ∂

=∂ ∂ ( I.17

This equation is called the wave equation. It is a partial differential equation. All solutions of this equation are called waves. We discuss now some interesting features. I.2.1 A special solution You can for example check that ( ) ( ), sinu u x t A kx tω= = − is indeed a solution of Eq.I.19 since

cos( )u A kx tt

ω ω∂= − −

∂

2

22 sin( )u A kx t

tω ω∂

= − −∂

cos( )u Ak kx tx

ω∂= −

∂

2

22 sin( )u Ak kx t

xω∂

= − −∂

and thus

2 2 2

2 2 2

sin( ) V ( )sin( )VV

( , ) sin[ ( V )]

A kx t Ak kx tkk

u u x t A k x t

ω ω ω

ωω

− − = − −

⇒ =⇒ = ±⇒ = = ±

( I.18

The + sign corresponds to a wave moving to the left while the – sign corresponds to a wave moving to the right. The relation Vkω = can also be written as

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22 V Vπω πν λνλ

= = ⇒ = ( I.19

in words (wavelength) × (frequency) = (phase velocity) I.2.2 Another special solution Similarly to what is done in I.2.2 you could check that ( ) ( ), cosu u x t A kx tω= = − is also solution. This is left as an exercise I.2.3 Complex notation for waves We can simplify our lives by using complex functions. Instead of checking first that a sine-function is a solution of the wave equation, and, subsequently that a cosine-function is also a solution, we could do both at once by using

( )( , ) i kx tu x t Ae ω−= ( I.20

Remember that by definition

cos sinie iα α α= + ( I.21

Using Eq.I.20 in Eq.I.17 we obtain

2

2

( )

2( )

2

( )

2( )

2

( )

( )( )

( )

( )( )

i kx t

i kx t

i kx t

i kx t

k

u A i etu A i i e

t

u A ik exu A ik ik e

x

ω

ω

ω

ω

ω

ω

ω ω

−

−

−

−

−

−

∂= −

∂∂

= − −∂

∂=

∂∂

=∂

( I.22

from which follows again that 2 2 2 VV k kω ω= ⇒ = ± I.2.4 General solution Not only sine- and cosine-functions are solutions of the wave equation. In fact all functions of the form

1 2( , ) ( V ) ( V )u x t g x t g x t= − + + ( I.23

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where g1 and g2 are arbitrary functions, are solutions. Simply substitute this expression into Eq.I.17. I.2.5 Power and intensity The total force ∆F acting on the string element of length ∆x is given by (see Fig. I.4)

xxuFF ∆

∂∂

=∆ 2

2

( I.24

For sin( )u A kx tω= − this means

2

2

sin( ) ( , )

F FAk x kx tFk x u x t

∆ ∆ ω

∆

= − −

= − × ( I.25

In other words the force in proportional to the displacement u. It is a restoring force that wants to bring the string element back to its original position on the x-axis. Since

Force displacement u∝ ∝ ( I.26

2

work (force) 2

A

o

Adu∝ ∝∫ ( I.27

The energy of the string element of length ∆x is thus

xAFkE ∆=∆ 22

21 ( I.28

and the energy in one wavelength λ is

λ22

21 AFkE = ( I.29

Since it takes a time t = T for one wavelength to pass by an observer one defines the power P carried by a wave as

TAFk

TEP λ22

21

=≡ ( I.30

However, we know that VTλ λν= = . Thus

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V21 22 AFkP = ( I.31

or

2 21 V2

P Aρω= ( I.32

since Vkω = and ρF

=2V . Here it is important that you remember that power is

proportional to the square of the amplitude. The intensity I is defined as “power/m2 “. I.2.6 Beating of two waves So far we have only considered a wave with one frequency. What happens if we add two waves of similar wave numbers and frequencies? To be specific we consider

[ ] [ ]( ) ( ) ( ) ( )( , ) k k x t k k x ti iu x t e e∆ ω ∆ω ∆ ω ∆ω+ − + − − −= + ( I.33

with and k k∆ ∆ω ω . Rearranging terms we have

( )[ ]

( ) ( ) ( )

( )

( , )

2cos

i kx t i x k t i x k t

i kx t

u x t e e e

e x k t

ω ∆ ∆ω ∆ ∆ω

ω ∆ ∆ω

− − − −

−

= +

= − (I.34

Since and k k∆ ∆ω ω<< << we obtain a wave with wavenumber k and angular frequency ω slowly modulated in amplitude as shown in Fig. I.5.

Fig. I.5: The sum of two sinusoidal waves with slightly different frequencies leads to a single wave with modulated amplitude. This effect is called beating.

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I.2.7 Phase velocity and group velocity The simple example treated in the previous section is very useful to illustrate the difference between phase velocity and group velocity. For this we ask the following question: what is the velocity at which the maximum (red point in Fig. I.5) of the modulated wave is moving? To answer this question we only need to consider the cosine term in Eq.I.35

[ ]( )( , ) 2cosi kx tu x t e x k tω ∆ ∆ω−= − ( I.35

and to request that it stays constant. Then

[ ] constantx k t∆ ∆ω− = ( I.36

implies that it moves with the velocity

Vg k∆ω∆

= ( I.37

This is called the group velocity, as it originates from the interaction of a “group” (here only 2) of waves. The group velocity is fundamentally different from the phase velocity

Vkω

= ( I.38

Assume now that we add many waves with ω=ω1 , ω2 , ω3 ,....... and k = k1, k2, k3,….. then the result still remains that the amplitude modulation moves with a velocity Vg=dω/dk if all the ωi (and ki) are close to each other. The notion of group velocity will play an essential role the description of a particle in quantum mechanics. It will also play a crucial role in discussing metals, semiconductors and insulators. I.3 Fourier generator With many frequencies and wave numbers it is e.g. possible to construct a wave of any shape. This part of mathematics is called Fourier analysis. A useful applet for the synthesis of periodic functions can be found at http://www.falstad.com/fourier/ In the figures below the black dot correspond to the keys of an virtual piano. Each one corresponds to a given frequency. The intensity is given by the position of the black dot. A high position means a large intensity. The examples shown in Fig. I.6 illustrate nicely

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Fig. I.6: Fourier synthesis of periodic functions. (top panel): with two periodic functions with similar frequencies one obtains an amplitude modulated wave; (middle panel): with 11 frequencies with amplitudes distributed like a Gaussian function one obtains also a modulation. However, now the resulting periodic function has a very small amplitude over a substantial interval of the x -axis. (bottom panel): this effect is even more apparent when a wider range of frequencies is used.

frequency f

amplitude

( , )u x t

x

12

the fact that the spread in space x∆ of a wave decreases when the spread in frequencies ∆ν increases and vice versa. Very roughly one expects from these examples that

constantx∆ ∆ν = ( I.39

We shall see that this property has far reaching consequences in quantum physics. It is in fact at the basis of the so-called uncertainty principle of Heisenberg. I.4 Waves in 2D and 3D So far we have only considered waves in 1D. From your daily experience you know that waves can also propagate in more than one dimension. Think for example of the waves on a lake or of sound waves, or of the radio waves used in your mobile phones. All these waves are described by a wave equation of the same form as that derived for the stretched rope, i.e. they are of the form

2 22

2 2V u ux t∂ ∂

=∂ ∂ ( I.40

For sound in 3D we have

2 2 2 22

2 2 2 2Vx y z tρ ρ ρ ρ⎛ ⎞∂ ∂ ∂ ∂+ + =⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

( I.41

where v is the velocity of sound and ρ the density of the medium (for example air) in which sound is propagating. A typical solution of this partial differential equation is

1 2 3( , , , ) sin( )x y z t A k x k y k z tρ ω= + + − ( I.42

which is more elegantly written as

( , , , ) sin( )x y z t A tρ ω= −kr ( I.43

where ( , , )x y z=r and 1 2 3( , , )k k k=k is the wave vector. Furthermore, instead of using sine and cosine functions it is more advantageous to switch to a complex notation.

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One writes then

( )( , , , ) i tx y z t Ae ωρ • −= k r ( I.44

For electromagnetic waves in 3D we have similarly

2 2 2 22

2 2 2 2cx y z t

⎛ ⎞∂ ∂ ∂ ∂+ + =⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

E E E E ( I.45

where c is the velocity of light and E is the electric field associated with the electromagnetic wave. The same equation applies to the magnetic field B associated with the electromagnetic wave. The fact that 3 partial derivatives appear in these equations is due to the x, y, and z dependence of the medium density for sound and of the electromagnetic field for the case of radio waves. I.5 Interference effects in 2D and 3D We shall restrict ourselves to waves in 2D since visualisation of waves in 3D is difficult. A nice applet for 2D waves can be found in http://www.falstad.com/ripple/ . We present here some examples.

Fig. I.7: Circular wave produced by a pulsating point source. The intensity of the colours indicates the amplitude of the wave. This picture is somewhat similar to that shown in Fig. I.1.

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I.5.1 Interference between two sources of same frequency Consider now two sources pulsating in phase at the same frequency. A phenomenon somewhat similar to that leading to the appearance of nodes in Fig. I.5 is observed: there are lines of zero amplitude. Along these lines, which have the shape of hyperboles, the waves produced by each source interfere destructively. In between, the two separate waves interact constructively and the amplitude is maximum.

Fig. I.8: Interference pattern obtained with two point sources pulsating in phase.

I.5.2 Diffraction by one slit We consider in Fig. I.9 and Fig. I.10 what happens with a plane wave coming from the top of the figures when it passes through a slit. When the slit is much larger that the wavelength the wave passes essentially undisturbed through the slit. There are only disturbances on the edges of the wave. The situation is quite different when the width of the slit is comparable to the wavelength. then the slit acts essentially like a point source.

Fig. I.9: Plane wave passing through a wide slit

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Fig. I.10: Diffraction of a plane wave by a narrow slit.

I.5.3 Diffraction by two narrow slits From the pattern shown in Fig. I.10 one expects that two narrow slits will act as two point sources. Since these two point sources are excited by the same plane wave they are in phase and their effect should comparable to that shown in Fig. I.8. This is indeed what is observed.

Fig. I.11: Diffraction pattern of a plane wave when it passes two narrow slits.

I.6 Summary In these additional lecture notes I gave all the ingredients necessary to describe the propagation of classical waves. The basic characteristics of a wave such as wavelength, frequency, wave number, wave vector, angular frequency, intensity and power have been defined. The wave equation has been derived for the case of a stretched rope and the wave equations for sound and electromagnetic waves in 3D have been indicated. Furthermore you have “seen” that localized waves can be generated by adding several waves: the localisation is strongest when the spread in frequency is largest. Finally, you can explore with two applets the propagation of 2D waves through obstacles such as slits. You are now ready for the lecture From Quantum to Matter!