# Are quantum mechanical waves three-dimensional

## Quantum mechanics

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### Waves and particle beams

- introduction
- Waves - basic concepts
- Particle beams

#### introduction

##### Wave-particle dualism

Figure 1: Wave packet and particle cloud

At the beginning of quantum mechanics is the observation that in some experiments light behaves as if it were a wave, in other experiments it behaves as if it consisted of elementary particles (photons).

It was soon recognized that the movement of all elementary particles takes place in such a way that, under certain conditions, wave properties (wavelength, frequency) become visible, but under other conditions, the particle properties (momentum, energy) come to the fore.

Depending on the type of particle, it is more difficult to detect the particle properties or the wave properties while they are moving. This is why classical physics first discovered the wave properties of photon beams, but first of all the particle properties of electron beams.

In fact, the following behavior can be observed for both photons and electrons:

- They always appear individually as particles. They are point-like, ie. if they hit individually somewhere, then always at a certain point and not somehow distributed like a wave. You can measure their speed and momentum. On impact, they transfer energy to the target hit.
- However, their movement follows the laws of propagation of a wave, because the local distribution of the particles shows interference phenomena.

Today both electrons and photons are referred to as particles, the mechanical movement of which is described by the properties of a wave.

Notes for teachers

##### Chapter overview

In the following we repeat some basic terms on the subject of waves, and then show how to describe particle beams in classical physics.

The most important experiment that can prove the wave character of a particle beam is the double slit experiment. It is used to determine interference, i.e. a typical wave property. We'll cover it in pretty much detail in the next chapter.

#### Waves - basic concepts

##### Waves - The most important things in a nutshell

Figure 2: Snapshot of a sine wave

A wave is an excitation or vibration that propagates through space. We call the physical quantity that describes this excitation the wave size. It is described mathematically by a wave function.

Examples:

- In the case of water waves, the wave size is the height of the water surface above the surface of the water at rest.
- In the case of sound waves, the wave size is the air pressure.
- With electromagnetic waves (radio waves, microwaves, light) the wave size is formed from the electric and magnetic fields.

In the simplest case, a wave is a process in which the wave size changes periodically in space and time. The wave size fluctuates around a mean value, wave crests and wave troughs alternate. The area around a maximum is called the wave crest, the area around a minimum of the wave function as the wave trough.

The illustration shows a snapshot of a sine wave. It is given by the wave function

u (x, t) = A sin (kx - ωt + d).

Here A is the amplitude of the wave, k is the wave number and ω is the angular frequency. Wave number and angular frequency are related to the wavelength λ and the period of oscillation T:

k = 2π / λ

ω = 2π / T

The argument of the sine function, a = kx - ωt + d, is often called the phase angle, the number d is then called the phase shift.

The speed at which a maximum (or a minimum or a zero) of the wave propagates is called the wave speed.

###### Exercises

Exercise:

Where exactly are the maxima of the sine wave defined here at time t = 0?

Solution: Where the argument of the sine function has one of the values a_{k} = π / 2 + 2 k π, where k is any integer. So at x_{k} = (π / 2 + 2 k π - d) / k, with k = 0, ± 1, ± 2, ± 3, ...

Exercise:

Which formula is used to describe a sine wave that is not around the mean value 0, but around a mean value u_{0} oscillates?

Solution:

u (x, t) = u_{0}+ A sin (kx - ωt + d)

Exercise:

A sine wave is described by the formula u (x, t) = A sin (kx - ωt + d). What is a formula for a sine wave that is half the wavelength in comparison?

Solution:

u (x, t) = A sin (2kx - ωt + d)

Exercise:

... has twice the period of oscillation?

Solution:

u (x, t) = A sin (kx - (ω / 2) t + d)

Exercise:

... although it has the same wavelength, but a wave speed that is twice as high?

Solution: During an oscillation, a wave crest moves forward by one wavelength. If you halve the period of oscillation T, the wavelength λ is covered in half the time, so the wave speed doubles. Half the period of oscillation means double the angular frequency. So we have to replace ω by 2ω. The solution is therefore

u (x, t) = A sin (kx - 2ωt + d)

Exercise:

... has the same period of oscillation, but twice the wave speed?

Solution: If you double the wavelength, a wave crest has to cover twice the distance during the oscillation period T. The speed of the wave is thus doubled. Double the wavelength is obtained if the wave number k is halved. So the answer is

u (x, t) = A sin (kx / 2 - ωt + d)

Exercise:

... for which u (0,0) = 0 applies?

Solution:

u (x, t) = A sin (kx - ωt)

Exercise:

... for which u (0,0) = 1 applies?

Solution:

u (x, t) = sin (kx - ωt + π / 2)

Notes for teachers

##### Visualization of waves

In one-dimensional representations, we visualize a wave using a function graph. Depending on the application, the function value describes the height of a water wave, or the air pressure of a sound wave, etc. The time dependence of the wave is illustrated by the animation.

We visualize a wave in two dimensions using a density graphic. The value of the wave-like physical quantity u (the height of the wave) is represented by a gray value. Here the wave crests are light, the wave troughs are dark.

The wave shown is given by the formula

u (x, t) = u_{0} + A sin (kx - ωt)

described. At a fixed point x the quantity u oscillates around the mean value u_{0}, for example, observe the black point at the point x = 0.

The period of oscillation is the duration of an up and down movement of the black point. The wavelength is the distance between neighboring wave crests (maxima). During the period of oscillation, the wave moves forward by one wavelength.

##### Circular waves

A wave that spreads evenly in all directions is called a circular wave (density graph)In two dimensions, waves can propagate in all directions. The animation shows a wave that spreads in a circle (circular wave). Wave crests and wave troughs form ever larger concentric circles that emanate from an excitation center. The amplitude decreases towards the outside, as the wave is distributed over an ever larger area.

Such a wave occurs, for example, when a point on the surface of the water is periodically excited.

Similarly, spherical waves can form in three-dimensional space (spherical waves).

In two dimensions, waves can propagate in all directions. The animation shows a wave with circular wave fronts. The amplitude decreases towards the outside as the wave energy is distributed over an ever larger area.

Such a wave occurs, for example, when a point on the surface of the water is periodically excited.

Similarly, spherical waves can form in three-dimensional space.

##### Interference from waves

Interference when two waves with different wavelengths overlapThe film shows two well-focused wave trains that move side by side with slightly different wavelengths and propagation speeds. The wave crests are light, the wave troughs dark.

If the wave trains are superimposed, a new phenomenon appears in the overlap area: a periodic increase and decrease in the amplitude. The newly created wave pattern also seems to move at a different speed than the two partial waves.

The effects that can occur when two waves are superimposed (= superposition) are summarized under the term interference.

The superposition of two or more waves to form a new wave is called superposition. With superposition, the waves influence each other and new waveforms are created. The change in the wave movement during superposition is called interference.

In many cases, interference can be described mathematically by simply adding the wave sizes at each point ("linear superposition"):

- When two (light) wave crests come together, a particularly high wave crest is created

(in the picture: light gray + light gray = white). - Adding wave troughs (negative deflection) results in a deep wave trough

(in the picture: dark gray + dark gray = black). - However, if a wave crest and a wave valley come together at a certain place and at a certain time, the wave sizes cancel each other out at this point

(in the picture: dark gray + light gray = medium gray)

An important experiment to demonstrate interference effects is the double slit experiment.

##### Interference between two circular waves

Interference when two circular waves are superimposed (density graph) In a wave trough, a swab causes a liquid surface to vibrate at two points at the same time. Two circular waves emanate from these disturbances, the interference of which can be observed in the wave trough (3D view)The film shows two circular waves emanating from two neighboring points. Think, for example, of a water surface that is stimulated simultaneously at two points. The two circular waves propagate in the same medium and overlap. This creates an interesting pattern. Along certain lines, the two waves seem to almost cancel each other out. These lines are almost straight and seem to start from a point between the two wave centers.

This image, which arises when two waves are superimposed, is called an interference pattern. In order for a beautiful and clearly visible interference pattern to arise, the two waves must be excited with the same frequency and with the same phase (ie "in common mode").

The following graphic provides an explanation for the occurrence of the interference pattern:

Figure 3: Explanation of the interference pattern. The circle segments mark the minima of the upper circular wave (dark) and the maxima of the lower circular wave (light). The two waves cancel each other out where maxima and minima meet, provided that their amplitudes are approximately the same there.

We assume that when two waves are superimposed, the wave sizes are simply added (this is called linear superposition). Then the following general principle applies:

- When two wave crests come together, a particularly high wave crest is created
- Adding wave troughs results in a particularly deep wave trough
- However, if a wave crest and a wave valley come together at a certain place and at a certain time, the wave sizes cancel each other out at this point

The extinction is only perfect if the maximum of one wave at the observed location is just as high above the mean value as the minimum of the other wave below it (i.e. if the amplitudes of the two waves at this location are the same). This is only approximately the case for the circular waves under consideration, because the amplitude decreases with the distance from the respective center. The extinction works particularly well when the crest of the wave and the valley of the wave meet at a location that is approximately the same distance from both wave centers.

##### Stationary vibrations

A swinging ropeLet us consider a stretched elastic rope attached at one end and periodically moved up and down at the other end. If this movement occurs at the correct frequency, the rope can be excited to vibrate of the type shown here.

Every point of the rope oscillates up and down with the same frequency, but the amplitude of this oscillation differs from place to place. In some places there is no vertical movement at all - these places are called nodes of vibration. The attachment point is inevitably such a vibration node.

The rope oscillation shown is stationary. In contrast to a continuous wave, no excitation seems to be propagated from place to place. This type of movement is therefore called a stationary oscillation.

###### Exercises

Exercise:

Find a function u (x, t) that would be suitable to mathematically describe the rope movement shown in the animation.

Solution:

Since the animation does not contain exact labels, there are many different solutions depending on the assumptions made. One possible consideration would be the following:

We assume that the black point is at location x = 0 and the attachment point is at x = L. The deflection at the fastening point is of course always 0:

u (x = L, t) = 0 for all t.

For the moving end at x = 0 we assume a harmonic oscillation.For example

u (x = 0, t) = A sin (ωt) for all t.

The whole rope vibrates in this temporal rhythm, but with an amplitude that depends on the location. So we do the approach

u (x, t) = A (x) sin (ωt) for all t.

In the simplest case, we again assume a harmonic function for the dependence of the amplitude on the location, for example

A (x) = cos (kx) for all 0 ≤ x ≤ L.

Due to the cosine function, the amplitude is automatically maximum at x = 0. The constant k must now be selected in such a way that there is a vibration node (a zero of the cosine function) at the attachment point x = L. It is k = 2π / λ, where λ is the period of the cosine function (the "wavelength"). If we count the nodes of the oscillation, we see that the fifth zero of the cosine must be at x = L. From this we get the condition

L = 9 λ / 4 = 9 (2π / k) / 4 and thus

k = 9 π / 2 L.

A function that mathematically models the stationary oscillation shown here would therefore be

u (x, t) = A cos (k x) sin (ω t) with k = 9 π / 2 L.

Notes for teachers

##### Standing waves

Standing wave as superposition of opposing waves. The red wave at the bottom is the sum of the two green waves at the top.This film shows the superposition of two opposing waves with the same frequency and the same amplitude

u (x, t) = A sin (kx - ωt) + A sin (kx + ωt) = 2 A sin (kx) cos (ωt)

The superposition is created by simply adding the wave sizes of the individual waves (the deflections) at each point.

This special superposition of two continuous waves is not itself a continuous wave, but a so-called standing wave. The wave size performs a stationary movement, which is why one speaks of a stationary oscillation.

The interference of two waves running in opposite directions with the same frequency and the same amplitude creates a stationary oscillation.

There are places where the function u (x, t) always vanishes:

u (x_{n}, t) = 0 for all t if x_{n} = πn ∕ k, with n = 0, ± 1, ± 2, ± 3 ....

These places are called nodes of vibration. Everywhere else the wave size changes periodically over time. Standing waves are temporally periodic processes (oscillations) with spatially fixed ("stationary") oscillation nodes.

There are times when the function u (x, t) vanishes everywhere:

u (x, t_{n}) = 0 for all x if t_{n} = π (2n + 1) ∕ 2ω, with n = 0, ± 1, ± 2, ± 3 ....

##### Standing waves - the vibrating string

Clamped vibrating string: vibration modes with fixed vibration nodes.An elastic rope attached at two ends ("vibrating string") can form standing waves, but only with certain wavelengths.

Let us assume that the attachment points of the rope are at a distance L from each other. The attachment points are inevitably nodes of vibration (= places where the string is always at rest). A standing wave has a certain number n of "antinodes" between the attachment points. The animation shows standing waves with 1, 2, 3 and 4 antinodes (or 0, 1, 2 and 3 vibration nodes between the attachment points).

The oscillation, which (apart from the attachment points) has no further oscillation nodes and only one antinode, is called the fundamental oscillation of the string.

The length of an antinode is exactly half the wavelength of the standing wave. So L is an integral multiple of half the wavelength. The only possible wavelengths of the vibrating string are therefore

λ_{n} = 2L / n

(where n = 1,2,3, ... is the number of antinodes; n-1 is the number of nodes).

The frequency of the standing wave is directly proportional to n and also depends on the mass and elastic properties of the vibrating string. The frequency of the fundamental oscillation is called the fundamental frequency.

###### Exercises

Exercise:

A tensioned elastic rope has a characteristic wave speed v (this depends on the tension and the mass density of the rope). Find a formula that gives the fundamental frequency f_{0} of a vibrating string is related to the wave speed v.

Solution:

Speed, wavelength and frequency are linked by the following formula:

v = λ / T = λ f

Here, λ is the wavelength, T the period of oscillation, and f = 1 / T the frequency. For the fundamental oscillation, λ = 2L, so v = 2 L f_{0} or f_{0} = v / (2L)

Notes for teachers

#### Particle beams

##### What are particle beams?

The film shows the classical physical concept of an idealized particle beam: Identical point-shaped particles that are randomly distributed over the beam cross-section and that all move at the same constant speed (high particle density). The film shows the classical physical concept of an idealized particle beam: Identical point-shaped particles that are randomly distributed over the beam cross-section and that all move at the same constant speed (low particle density).A particle beam consists of a certain type of elementary particle, for example electrons, all of which move at roughly the same speed in roughly the same direction.

An electron beam is created by accelerating and focusing the electrons that escape from a glowing wire using electric and magnetic fields.

Other sources of particle beams are nuclear reactors (neutron beams) or elementary particle accelerators.

The film shows how one imagines a particle beam in classical physics.

- At any point in time, the particles have a certain location (and a certain speed.
- The trajectory of each particle can be followed individually.

In quantum physics, this idea has turned out to be wrong. It is true that the particles are still thought of as mass points, but the idea of movement along a certain trajectory belongs to the time before quantum physics.

In practice, it would not be possible to follow the individual particle trajectories precisely anyway. Usually there are billions of particles in a particle beam, which also move at extremely high speed. With detectors one can at least measure the approximate particle locations and count how many particles per second get into a certain spatial area (detector opening). Detectors today are so sensitive that they already react to the arrival of individual elementary particles.

Elementary particles can exert forces on one another (electrons, for example, repel one another because of their negative charge). For the particle beams we are considering, the particles in the beam should be distributed so thinly that their interaction can be neglected (low particle density).

We only consider particle beams in which the particles do not influence each other. Every single particle moves as if it were alone.

Elementary particles are extremely small. Even if there are billions of particles in the beam, the distance between the particles can be large compared to the range of their interaction.

Light particles (photons) do not exert any forces on one another. Light rays can therefore have a high intensity. A normal laser pointer already produces approx. 3 x 10^{15} Photons per second and the beam diameter is only about one millimeter. With high intensity (large number of particles), quantum mechanical effects (such as the wave nature of motion) can of course be observed more easily.

##### Visualization of particle beams

Since real particle beams usually contain billions of particles, it makes no sense to draw individual points. Then we simply represent the particle density using a shade of gray.

Schematic representations (density graphics) of particle beams:

Figure 4: The particle density, visualized by a shade of gray, is greatest in the middle and decreases towards the edge. It is constant along the ray.

Figure 5: The dark gray tone symbolizes a low particle density (thin beam, low intensity). The speed of the particles is not visualized by this form of representation.

In contrast to a wave, we do not see any spatially or temporally periodic structure in the particle density of an ideal particle beam (a wave, on the other hand, has wave peaks and troughs).

When particle beams hit obstacles, they can be scattered in all directions. The particle density is then strongly dependent on location.

##### Particle density

Figure 6: Snapshot of a particle cloud with inhomogeneous particle density: The location of each particle is marked by a white point.

Figure 7: Snapshot of a particle cloud with inhomogeneous particle density: The local distribution of the particles is described by a continuous function, the particle density. The function value is visualized by a gray tone. The regions with a low particle density are dark, the regions where there are many particles are light.

If there are very many particles (typically billions) in an area of space, it makes no sense to want to describe the exact location and movement of each particle. A continuous quantity, the so-called particle density, is then used to describe the local distribution of the particles.

Homogeneous distribution: We first consider the case that the particles are evenly distributed in the spatial area under consideration. That means: If we subdivide the spatial area into nothing but equal parts of size ∆V (whereby each part may still contain a lot of particles), then in each part (within the scope of the counting accuracy) there are the same number ∆N of particles. Then the particle density is defined as the quotient

η = ∆N / ∆V.

Inhomogeneous distribution: If the particles are not evenly distributed, then the particle density changes from place to place. This is described by a position-dependent function η (x) which gives the prevailing particle density at each point x. With the help of η (x) one can calculate the approximate number of particles that are in a small volume ∆V around the point x. This number is approximate

∆N = η (x) ∆V

The description of the distribution of point-like particles by a continuous function η is a mathematical idealization. The above equation is only reasonably accurate if the volume ∆V is so small on the one hand that the function η has approximately the same value at all points of ∆V as at the point x (i.e. the value η (x)). On the other hand, ∆V must be so large that there are still very many particles in this volume piece. If η (x) ∆V is too small, due to the random placement of the particles, the particle number ∆N would also show strong random fluctuations.

In one dimension, the particle density can be represented by a function graph. In the plane we represent the particle density by means of a density graphic.

###### Task:

Exercise:

Particle density in one dimension: This figure shows on the left some examples of particle beams (snapshots) in which the particle density is variable. We want to describe the particle density along the particle beam by a function that only depends on the coordinate along the beam direction. The density distribution transverse to the direction of the beam is not to be described. The function should only indicate how many particles per unit of length are at the respective location in the beam. On the right in the figure there are some examples of such density distributions along particle beams. The task is to find out which of the particle densities 1-6 belong to which images A-F.

Figure 8

Solution:

A - 4

B - 3

C - 1 (compare with A, there the density is much higher)

D - 6

E - 2

F - 5 (the ray widens, but the number per x interval remains constant)

##### Particle flux density

Figure 9: In a particle beam with velocity v, all particles in the drawn cuboid pass within time t a rectangle A, which is set up transversely to the direction of movement. The volume of the cuboid is A v t.

We consider a particle beam in which all particles move in the same direction with the velocity v. We now imagine an area A perpendicular to the direction of movement. Within the time t, each particle covers the distance s = v t. In time t all particles in a volume of size A v t pass through this cross-sectional area A. If the particle density η in the spatial area under consideration is η, the number of these particles is equal to particle density η times the volume A v t, i.e.

n = η A v t.

If you divide that by area and time, you get the size

j = η v.

It describes the number of particles that pass through an area (imagined transversely to the direction of movement) per unit of time and per unit of area. It is called the particle flux density.

The particle flux density j = η v describes the number of particles per unit of time that pass through an (imaginary) unit area perpendicular to the direction of movement.

As a rule one tries to prepare particle beams in such a way that all particles have a certain velocity v.In practice the particles will be distributed around an average value

The particle flux density j = η

If the particle density and the flow velocity of the particles is different from place to place, then the particle flux density also differs from place to place. This situation is idealized by a function j (x), which gives the particle flux density at location x.

The particle flux density j (x) describes the mean number of particles that pass through a unit area per unit of time, which is set up at the location x across the direction of movement of the particles.

###### tasks

Exercise:

What is the physical unit of particle flux density?

Solution:

The physical quantity j = η v has the unit dimension of

Particle density * speed = (number / volume) * (length / time) = number / (length * length * time).

The physical unit is therefore 1 / (m^{2} s).

#### Mass and charge densities and associated current densities

###### Mass density and charge density

We consider a particle beam that consists of particles of the same kind. All particles therefore have the same mass m and the same charge e. If the particle density is η, the number ∆N of particles in a small volume area ∆V is ∆N = η ∆V. The total mass in this volume is therefore

∆M = m ∆N = η m ∆V.

The total electrical charge in the volume is ∆V

∆Q = e ∆N = η e ∆V.

If the particles distributed in space have the particle density η (x), one calls m * η (x) the mass density and e * η (x) the charge density.

###### Streams of masses and charges

With the movement of the particles in the beam, mass is transported, the mass "flows". Let

μ = η m

is called the mass flow density. It indicates the mass that flows through a unit area (transverse to the direction of movement) per unit of time.

A charge current density is also defined

ρ = η e

It indicates the charge that is transported per unit of time through a unit area across the direction of movement.

###### Momentum density

Often the state of motion is described by the momentum, p = mv. The mean velocity

= m

The mass flow density

μ = η m

/ ∆V

is therefore equal to the number of particles times the (mean) momentum of each particle divided by the volume, i.e. the total momentum per unit volume. The mass flow density is therefore also called the pulse density of the jet.

Mass flow density = momentum density

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