Wave-centric interpretation of quantum mechanic

All postings by author

previous: Emitter arrays

up: Three types of waves

next: next blog

Wave-centric interpretation of quantum mechanics

by Peter H. Ceperley

Keywords: interpretation of quantum mechanics, wave-centric, real waves

Alternate location: Mason Archival Repository Service

Table of contents

1. Summary
2. Introduction
3. Absorption
4. Required distribution of noise amplitude
5. Distribution resulting from random phases incident on atoms
6. Questions and brief answers concerning specific applications of the wave-centric interpretation
7. Conclusion

1. Summary

This paper explores one interpretation of quantum mechanics: that perhaps the waves of quantum mechanics are real and the particles, i.e. the quanta, are merely properties of atoms, the atoms which are responsible for launching and absorbing the waves. In this interpretation, the waves of quantum mechanics are emitted in discrete quanta, but after emission they spread out as waves do. A detector's atoms are continuously bathed in a sea of zero point noise which is in equilibrium with these atoms. An outside wave, even if faint, will throw off this equilibrium which results in one or more atoms in a particle detector absorbing a quantum partially from the wave and partially from the zero point noise field. In this article we show that the number of atoms absorbing a quantum will be proportional to the energy density of the external wave.

2. Introduction

All things in quantum mechanics propagate as waves but are emitted and absorbed as particles.

In quantum mechanics all entities are considered to have both particle-like and wave-like attributes. This is often summed up as: entities propagate as waves but are emitted and absorbed as particles. The standard interpretation of quantum mechanics, i.e. the Copenhagen interpretation, puts more emphasis on the particle side, implying that these tiny entities are basically particles but somehow magically propagate as waves. The current paper explores the obvious other interpretation, that elementary "particles" really are waves, not particles, but these waves are emitted and absorbed in such a way as to mimic particle emission and absorption. For other, non-conventional interpretations see wikipedia.

The particles: In this paper the "particles" that we wish to consider as waves are those truly elementary particles that are not composites of smaller entities. Possibilities are electrons, quarks, photons, phonons, etc. Composites such as atoms are localized entities, i.e. particles, that are composed of waves internally. We will put off dealing with composites to a future paper. While we will not be concerned with atoms as particles (as in interfering beams of atoms), we will be very interested in them playing another very important role as wave emitters and absorbers as discussed in Section 3 below.

The waves: We envision these waves to be fairly complicated, similar to the waves that can exist in some solid state materials so as to allow the waves to carry properties that are conventionally associated with elementary particles. That is to say that electron waves will carry charge and spin. The spin aspect would be similar to spin waves that exist in magnetic materials. Quark waves would be even more complex with more properties in the waves. Viewing elementary particles as waves amounts to assuming that space is a medium that supports these various complicated waves. The inclusion of charge as a wave property makes electron and quark waves non-linear; that is, the wave trajectories, wavelength, etc. can depend on the amplitude of the waves.

The atom: In this interpretation an atom is considered to have a central positive core which traps electron waves in various excitation modes, i.e. orbitals. Thus atoms are solitons of a sort. The non-linearity of the waves opens up the possibility that stable electron waves in an atom only exist at one select excitation level for each mode. If we assume such discrete excitation levels then whenever a mode is to be excited, an atom will absorb one quantum of energy and when it radiates energy, it will radiate one quantum of energy. This makes absorption and emission of radiation occur in quanta. We deal more with absorption in the next section.

Emission: If an atom emits a quantum of electromagnetic energy or an electron, that wave will spread and flow outwards away from the atom in a normal wave fashion as dictated by the Maxwell or Schroedinger equations modified to carry spin and charge as appropriate.

3. Absorption

In quantum mechanics "measurements" are the only way we know what is happening. That is to say we envision quantum mechanical waves propagating here and there but until we make a "measurement" we can be sure of nothing. But a measurement is a drastic step because in the conventional interpretation when a measurement is performed, the waves, spread out as they may be, instantaneously collapse onto the site of the measurement and propagate no more. The instantaneous collapse violates relativistic causality, but is accepted as one of the mysteries of quantum mechanics.

Examples of “measurements”

An atom on a photographic film absorbing a quantum of energy. In the process, the atom changes state and changes the chemical bonds with nearby atoms and thus becomes “exposed”. An atom in a solid state particle detector becoming excited and ejecting a quantum of charge into the detector’s conduction band where it will be registered as an electronic event. An atom in a bubble chamber absorbing a quantum of energy, part of which it releases to the surrounding atoms and causes nucleation of a bubble. A neutron in a cold neutron experiment is absorbed by a 3He nucleus of a neutron detector which results in the production of two hydrogen atoms and 0.764 MeV of energy. The 0.764 MeV of energy creates a charge cloud which is picked up by the Geiger-Muller tube which contains the 3He.

What constitutes a measurement? Our view is that a “measurement” consists of absorption of the wave by an atom. (See examples listed at right.) Whenever an atom absorbs a quantum mechanical wave and undergoes a change as a result, we can, if we choose, deduce something about what the quantum mechanical wave was doing before the absorption. The atomic absorption scrambles the quantum mechanical wave and is usually considered irreversible, that is, the initial wave cannot be reconstructed.

Just how does the spread out wave field concentrate itself to allow one quantum of energy or a whole electron to be absorbed by an atom? As we stated above, the conventional interpretation resorts to magic to explain this effect. In the wave-centric interpretation we hope to get rid of the magic and restore causality to the understanding of quantum mechanics.

In the wave-centric interpretation, absorption can be explained in terms of a background noise, most likely the zero-point energy. In this interpretation we assume there is an undetectable noise field filling space in equilibrium with the atoms. This is in addition to any thermal (temperature related) noise. This noise field is undetectable because we interpret zero point effects as due to quantum mechanics.

In the next section we will show that a noise field which provides energy around and available to every atom can cause the quantum mechanical particle-like energy absorption. This does not mean every particle will see the same level of noise energy at all times. Being chaotic as noise fields are, at any instant around some atoms there will be more noise energy and around other atoms there will be less noise energy. It is also true that at any instant the noise around some atom will be phased just right to best add to whatever external wave is arriving. If no external radiation is present, this noise will be in equilibrium with the atoms and will result in very few if any actual transitions.

When there is an external wave present and this wave combines with the noise field, some lucky atom or atoms will be in just the right place and have just the right phase to use the noise energy plus the external radiation to absorb a quantum and make a transition. During the transition, the atom will, as all wave receivers do, emit a canceling wave which will cancel the nearby combined noise and external wave fields and thereby deny other nearby atoms the opportunity to make a transition. This emission of a canceling wave allows the atom to have an effective cross section for absorption much, much bigger than its physical size. This is the same effect that also allows small AM and FM radios to similarly have a very enhanced absorption cross section, again much larger than their physical sizes.

The larger the external wave is, the more atoms will happen to have sufficient surrounding noise fields to allow them to effect a transition. If we count each transition as a "particle" then we will "detect" (via transitions) more "particles" in regions of stronger external wave fields. In the next section we do the math to derive the distribution of noise amplitude required to make the number of transitions be proportional to the energy in the external wave.

To summarize absorption: the wave-centric interpretation hypothesizes that atoms are surrounded by a noise field and that this noise field allows even weak external waves to cause an occasional atomic transition as expected in quantum mechanics. We might view the atoms as popcorn kernels on a hot frying pan just waiting for an extra bit of heat to make them pop; the stronger this "external heat" the more kernels will pop.

4. Required distribution of noise amplitudes:

In this section we assume a triangular distribution of noise amplitudes. We then go on to show that this results in the probability of a detection being proportional to the external wave energy. We also show that random phases in the various waves making up the noise result in a total amplitude having a distribution that is nearly triangular.

Fig. 1.Triangular noise distribution. This curve shows the probability (per amplitude) that noise will be able to impart a given amplitude to a resonator (such as an atom). Later we will explore possible sources of this noise distribution. On the horizontal axis is the component of noise amplitude that is in phase with a particular external wave. Negative noise amplitudes indicate that the noise is phased to subtract energy from the atom. We will be primarily interested in the right half of the distribution which has has a total probability of one (area under the right side of the distribution).

Suppose we pick a point at which to examine the noise and repeatedly measure the noise amplitude at this point over a period of time. Because of the ever changing nature of noise fields we expect to see a variation in amplitudes of our measurements. This variation is due to the mix of waves in the noise field having various frequencies and beating with each other. The resulting time-varying field will, at times, add to any particular external wave and at other times will subtract from it.

Triangular noise distribution: Fig. 1 shows a triangular noise amplitude distribution. This represents one possible distribution of amplitudes. It shows the relative fraction of time a certain amount of noise amplitude will be available to an atom for the purpose of making a transition.

For convenience, we assume amplitude units such that an amplitude of +1 (plotted on the horizontal axis of the graph) is the amplitude required to cause a transition without any external wave. According to the graph in Fig. 1, there is zero probability of such a +1 amplitude.

At this point in our explanation we are only interested in positive amplitudes, those which will potentially add to a particular external wave's amplitude and be involved with a quantum absorption. These positive amplitudes are on the right side of Fig. 1. We can express the linear region on the positive side of Fig. 1 as:

        .     (1)

For larger noise amplitudes where A_noise > 1 ,  the probability P₁ = 0  .

Resulting transitions:

We need to determine the probability that the noise as specified in Fig. 1 will be available to participate in a transition. That is, given an external wave of amplitude A_ext , we need to calculate the probability that the sum of the noise amplitude and the external wave amplitude will be greater or equal to the amplitude for a transition.

For this we need the probability P₂ that the noise will be greater or equal to a given minimum value A_min . This can be calculated from (1):

.

Doing the final evaluation, we get:

        ,     (2)

Fig. 2. Parabolic noise distribution. This is the shape of the noise amplitude distribution when the horizontal axis represents the minimum amplitude.

which we plot in Fig. 2.

In Fig. 3 we have changed the horizontal axis from the minimum noise amplitude into the external wave amplitude required to make the total amplitude available to the atom to be equal to or greater than +1  (i.e. one quantum). That is to say such that    A_min + A_ext = +1   .

We can turn this last equation around and solve for the amplitude of the external wave, i.e.  A_ext = 1 − A_min . Looking at Eqn. (2) we see this makes the probability

P₂ = (A_ext)²      .     (3)

Thus we now have the probability of transition as a function of the amplitude of the external wave, as plotted in Fig. 3 below.

Equation (3) indicates that the transition probability equals the external wave amplitude squared. We know that the wave energy E_ext is also proportional to the wave amplitude squared. We can now write:

P₂ ∝ E_ext     .     (4)

Thus the probability of detection is proportional to the energy of the external wave as shown in Fig. 4.

This is consistent with conventional quantum mechanics. Note again that this is for a background noise with a triangular amplitude distribution as shown in Fig. 1 . As a final cross check we note that Fig. 3 shows that an external wave with an amplitude of +1 which we defined earlier to be equivalent to one quantum (available energy to an atom) which will have 100% chance of causing a transition.

Fig. 3. Probability of transition versus amplitude of external wave adding to the noise amplitude.	Fig. 4. Probability of transition versus external wave energy.

5. Distribution resulting from random phases incident on atoms:

Fig. 5. Typical histogram showing the frequency of occurrence of various total amplitudes resulting from summing many waveforms with random phases, frequencies, and amplitudes. The waveform amplitudes were multiplied by a resonator response function. Also shown is a fitted Gaussian distribution.

We used Octave to calculate the distribution of amplitudes of a particular phase (to match the phase of a specific external wave) that might be expected in a linear resonator (our atom). We summed waves with random phases, amplitudes and frequencies. Fig. 5 shows the resulting distribution with a fitted Gaussian. We see the fit is good.

Fig. 6 shows a Gaussian as compared with a triangular distribution. The two curves differ at the peak around A_noise = 0 and also around the edges, i.e. around A_noise = ±1. If an atom did experience an amplitude with an absolute value greater than 1, this amplitude would cause a transition without any extra outside wave. This means that the atom would instantaneously absorb all the amplitudes with an absolute value in excess of 1 and leave small amplitudes remaining. The small amplitude "clippings" would add to the center part of the distribution curve making the Gaussian more triangular as shown in Fig. 7. As we can see the results are a nearly triangular amplitude distribution. A better modeling of this process would require modeling of the atom as a non-linear resonator surrounded by a noise field which is in equilibrium with other non-linear atom/resonators. Hopefully with more complete modeling a truly triangular noise distribution would be achieved.

Fig. 6. Comparison of triangular and Gaussian distributions.	Fig. 7. Result of clipping amplitudes outside the ±1 quantum region to crudely simulate atoms quickly absorbing one quantum of these amplitudes. The remains of these clippings are added to the distribution increasing the center peak region. Also shown is a fitted triangular distribution (in black) which now is a moderately good fit.

6. Questions and brief answers concerning specific applications of the wave-centric interpretation

1. What about particle tracks such as those in a cloud chamber? Are these not very compelling evidence that there really are subatomic particles? Answer: these are just as much compelling evidence that there really are waves. As explained many years ago [see Schiff, Quantum Mechanics, 2nd Ed. p.209-210, McGraw-Hill 1955] each spot on a cloud chamber or bubble chamber, etc is due to an atomic absorption or excitation and results in the collapse of the wave function (or atomic transition in our interpretation), a re-concentrating of the energy and a re-emission of most of the energy in the direction of momentum. This re-concentration makes it extremely likely that another measurement event will happen just down stream from the first, and that will result in another re-concentration and re-emission, and so on, resulting in a trail of events, i.e. a track. The tracks are due to successive ``measurement" events of the waves and not necessarily due to the presence of a localized particle. In the wave-centric interpretation zero point noise is integrally involved in each measurement.


2. Which-way experiments have been popular in the last several decades. In these a non-linear element splits a very dilute beam of photons and sends the two halves on different paths which eventually recombine at a detector which sees an interference pattern of one type or another. The beam is made dilute enough so that there is only one photon in the setup at a time. If we add a detector along one path to try to determine which path a particular photon is traveling, the interference pattern goes away. Since conventional quantum mechanics considers a single photon unsplittable it is not clear how a single photon can sample both routes and interfere with itself. It is also unclear how a single photon knows when a detector is placed in the path it doesn't take.

   Answer: In the wave-centric interpretation, there are actual waves traveling both routes simultaneously and they recombine and interfere at the final measuring point as waves would. If we block the waves in one path with an additional measurement device, the interference is destroyed just as it would be in classical wave experiments. If we partially block a path, the interference is reduced.



3. What about quantum entanglement experiments? In these experiments twin pairs of "particles" are produced and separate. Before measurement, these particles can have a wide range of properties. Measurement of one particle seems to fix the properties of the other particle instantaneously even if the two are separated by very large distances. These experiments are cited as proof that quantum mechanics is strange and cannot be explained by a deterministic interpretation.

   Wave-centric answer: Most likely any particular pair of wave packets sent out were a lot more limited in space, time and/or other properties than many physicists assume, even though the ensemble of all the particles (i.e. all the wave packets) under consideration involve greatly varying properties. For example, consider the decay of positronium which often results in two oppositely traveling gamma rays. Probably the wave packets of a specific pair of "photons" were directed oppositely along a specific axis during creation due to the internal dynamics of the decay process. It is then no mystery that later detection of the two would reveal them to be traveling in opposite directions.



4. Bell's hidden variable theorem. This applies to particles trying to mimic waves. We are proposing that only waves carry all the measurable properties. This purely wave approach is not in conflict with Bell's theorem.


5. Electron-electron collisions. Our model would set up the calculation with two uniform electron waves colliding with each other. With no tiny particles of very localized strong forces how can we have large angle scattering? This is also a question in conventional quantum mechanics. In conventional quantum mechanics we convert to a center of mass reference frame then proceed to calculate the results of two quantum mechanical plane waves passing through each other. This wave math would be the same as in the wave-centric approach. In both, the specifics of the quantum mechanical wave equation lead to a certain amount of large angle scattering.

7. Conclusion

In this article we have briefly examined the possibility that the waves in the wave-particle duality of quantum mechanics be elevated to "real" and the "particles" be considered as artifacts of the way the waves behave. More specifically we show that waves in the presence of zero point noise could be responsible for the particle-like aspects of electrons, photons, etc. Unlike the conventional Copenhagen interpretation, the interpretation presented here has the advantage of preserving causality. At the same time, this interpretation needs a lots of additional refinement, extension and detailed modeling.

All postings by author

previous: Emitter arrays

up: Three types of waves

next: next blog

Wave Equations

All postings by author

three types of waves

last posting: emitter games

Wave Equations

- what is required for an equation to be a wave equation? -

Mathematically we can define waves in several ways:

1. As a shifting sinusoid

   mathematically expressed in real form as:    where A and B are real constants,     i.e. amplitudes,	(1)
   or in complex form as:     where A is a complex amplitude,     i.e. an amplitude and phase shift.	(2)
   In both the above forms κ is the wavenumber in units of radians per meter and ω is the angular frequency in units of radians per second. The independent variables x and t are position and time. In the complex form only the real part of the complex expression is considered to describe actual physical attributes of a wave. Because of this we are free to put a minus sign in front of the i in (2) without changing the physical attributes of the wave represented as long as we stay consistent in our use of a minus sign or not throughout the calculation at hand. Using a negative κ in either of the above equations represents waves propagating in the negative x direction.

   
   
   
2. A wave can also be an arbitrary function that keeps its shape and moves along as time progresses. You might notice that the sinusoids in (1) and (2) are functions of κx−ωt. Alternatively they could be written as functions of x−ct where c=ω/κ is the wave velocity. Note that κx−ωt can be factored as κx−ωt = κ×(x−ωt/κ) = κ×(x−ct) where κ is the wavenumber for the sinusoids of (1) and (2).
   
   This method of using a function of x−ct can work for any arbitrary function. As an example at the right we see a Gaussian bell shaped function converted into a function that moves with velocity c by replacing the x with an x−ct.
   
   

   A stationary Gaussian as a function of x is	.   (3)
   The moving Gaussian is	(4)
   where σ is a measure of the width of the bell shaped curve. Note that we could have started with any function of x and made a moving version of that function by replacing all occurrences of x in that function with x−ct.

   
   
   
3. A different way to mathematically define a wave is with a differential equation which has as its solution functions such as those above. The most cited example of such a defining differential equation is "the wave equation":
   
      .    (5)
   
   The advantage of differential equations is that they allow the calculation of waves in different coordinate systems and varied boundary conditions such as for the waves inside a spherical resonator. Differential equations can also be modified to accommodate changing properties of the wave media such as varying wave velocities as a function of position.
   
   Differential equations other than (5) may also serve as wave equations, even though (5) has become known as the official "wave equation".
   
   

In this posting we shall explore various differential equations and take a stab at what is required to be a general differential wave equation.

The wave equation - non-dispersive waves

Equation (5) has solutions which are non-dispersive, that is, waves whose velocity of propagation does not depend on the frequency of the wave. Examples of such waves are a acoustical waves of small amplitude and waves on a string, also of small amplitude. Another wave type which is non-dispersive is electromagnetic waves such as light, radio waves and x-rays. Such electromagnetic waves have a wave variable ( the ψ in (5) ) that is a vector in the general case; however the wave variable can be a scaler as in (5) in certain cases, the cases that are often treated in electromagnetic textbooks.

One property of a medium governed by a non-dispersive wave equation is that waves of arbitrary shape keep their shape as they propagate. One way to understand this is that any arbitrary function of x−ct is a solution to the wave equation (5). That is to say direct substitution of f(x−ct) into (5) maintains the equality shown in (5). This type of solution of "the wave equation" was first proposed by Jean le Rond d'Alembert in 1746.

To understand this we use the chain rule of differentiation: so that substituting ψ(x−ct) = ψ(u) into (5) yields:

    (6)

where u=x−ct.

Another way to understand that waves of arbitrary shape propagate and maintain their shape is to understand that any function can be broken down into a sum of sinusoids of varying frequencies. Because these frequency components all travel at the same velocity, the shape will maintain itself.

In contrast in a medium with a dispersive wave equation, the various frequency components of an arbitrary shape will travel at different speeds and so add up differently as time progresses. This results in a wave that changes its shape as time progresses.

Another interesting exercise is to factor (5) which yields:

   .    (7)

One solution of this equation occurs when the right most factor operating on ψ is zero:

   .    (8)

Putting ψ = x−ct into the left side of (8) yields zero and thus satisfies (8), (7) and (5). Inverting the order of the two factors in (7) results in an equation like (8) but with a minus sign. That equation is satisfied by ψ = x+ct which is a sinusoidal wave traveling in the negative x direction.

We see that (5) is a second order differential wave equation with two solution: a positive going and a negative going wave. Equation (8) is a first order differential wave equation with only one solution, a positive going wave.

Water waves - dispersive

A common type of waves are those we see on the surface of water, i.e. water waves. These are dispersive. To better understand their dispersive nature we look at one of the defining differential equations, Equation (8) of this link.

   .    (9)

We have replaced Δy of that link (the vertical displacement of the water's surface due to the wave) with δ_y. The quantity δ_x is the displacement of water in the x direction due to the passing wave.

Looking at Equations (3a) and (3b) of the the link we see that the ratio of δ_y/δ_x = −i tanh κy or solving for δ_y this is δ_y = −i δ_xtanh κy. Substituting this into (9) yields

    (10)

where y is the depth of the water.

In (10) we have a wave equation expressing the horizontal displacement of the water parcels in response to a passing wave as a function of x and t.

To find out the constraint on these waves we substitute the complex traveling wave expression δ_x = A e^i(κx−ωt) into (10) to get: gκ tanh κy = ω² or:

   .    (11)

We next solve for the phase velocity, i.e. the velocity of the wave crests. The standard equation for the phase velocity is v_p = ω/κ which when used with (11) gives us:

   .    (12)

Thus the phase velocity is not constant with frequency. Water waves are therefore dispersive, meaning that different frequencies travel at different velocities.

The group velocity, i.e. velocity of energy and pulses such as the Gaussian shown above, has the standard equation v_g = dω/dκ. Using (11) this yields:

   .    (13)

Incidently, for waves which are without dispersion, i.e. all frequencies travel at the same speed as discussed in the previous section, the phase velocity equals the group velocity and both are a constant.

Waves in a waveguide

Microwaves (electromagnetic waves used for microwave ovens, cell phones and radar) are often channeled down metal pipes called waveguides. These have a rectangular cross section (width a and height b). While there are many modes of propagation, each with its own electric and magnetic field configuration, these modes can be lumped into two classes. We will focus on the TE modes which are the commercially most used. We will further restrict our study of TE modes with non-zero electric fields in the y direction, E_y.

The electromagnetic wave equation written for the E_y component of the electric field in free space is:

   .    (13a)

When confined to a metal waveguide the y electric field parallel to the walls at the walls must be zero. It is also true that E_y at the top and bottom of the waveguide must have ∂E_y/∂y = 0. Due to these two constraints the electric field must assume a solution in the form:

    (14)

where the x and y wavenumbers are given by:

   .    (15)

The modal indices m and n are positive integers. For the modes we are treating m must be greater than zero while n can be zero or greater. (The TE_0,1 mode, i.e. m = 0, n = 1 mode, has all its electric field in the x direction and is zero in the y direction while the TE_0,0 mode has all zero electromagnetic fields and is not a wave mode at all.)

Substituting (14) into (13a) yields:

   .    (16)

The first two terms in (16) are set by the waveguide dimensions, i.e. by (15), and can be replaced by the cutoff frequence ω_c given by:

   .    (17)

Equation (17) calculates the lowest frequency that a particular mode will propagate in the z direction. We can use (17) with (16) to get:

   .    (18)

Equation (18) is the defining differential wave equation for these fields propagating in the z direction. It is similar to (5) except that it has an additional term involving E_y times a constant (the first term in (18) ).

We next replace f(x,t) in (14) with the standard complex propagating sinusoid Ae^{i(κ_zz−ωt)} and then substitute the resulting E_y into (18). We finally solve for ω and get:

   .    (19)

From this we can calculate the phase and group velocities:

    (20)

   .    (21)

Similar wave media: Electromagnetic waves in a plasma

A plasma consists of ionized gas molecules and free electrons and highly interacts with electromagnetic waves trying to propagate through it. There are a number of types of electromagnetic waves possible (with static magnetic fields or without and other criteria). See this link for details. One of the simpler cases is without a static magnetic field and is governed by this equation:

    (22)

where ω_p is the plasma frequency and is the lowest frequency at which this type of wave will propagate. It is related to the density of free electrons in the plasma. Alternate source: D.K.Cheng, 2nd Ed. Field and Wave Electromagnetics p.377 Eq.8-73. Equation (22) is basically the same as (19).

Similar wave media: the Klein Gordon Equation in quantum mechanics

The Klein Gordon equation governs the motion of spinless, chargeless particles which are moving at relativistic speeds (within an order of magnitude of the velocity of light):

    (23)

where c is the velocity of light, m is the mass of the particles under study, and ℏ is a quantum mechanical constant.

With the standard complex propagating sinusoid Ae^{i(κ_zz−ωt)} substituted in for ψ this equation becomes:

   .    (24)

Again we see the same dependence for the frequency ω on the wavenumber κ as we see in (19). All three waves have differential equations that have second order derivatives in space and time plus an extra term that is a constant times the wave parameter. All three display a minimum frequency of wave propagation, i.e. a cutoff frequency. All three approach dispersionless propagation at very high frequencies but show dispersion at frequencies just above the cutoff frequency.

Equation (24) is the relativistic equivalent of the Schrodinger equation which we will discuss next.

The Schrodinger equation

The Schrodinger equation governs the motion of spinless particles which are moving at non-relativistic speeds (much slower than the velocity of light):

    (25)

where V is the potential the particle experiences and can vary with position.

With the standard complex propagating sinusoid Ae^{i(κ_zz−ωt)} substituted in for ψ this equation becomes:

   .    (26)

In (26) we see a very different dispersion relation than we saw in the previous wave media. If potential V is zero, the frequency depends on the square of the wavenumber. If V is not zero but positive then there is an offset, and there will be a minimum propagating frequency; this occurs when a particle has sufficient energy (frequency and energy are proportional in quantum mechanics) similar to the cutoff frequencies in the previously discussed media.

We can calculate the phase and group velocitites:

    (27)

   .    (28)

Equation (28) is consistent with the momentum p = mv = h/λ remembering that κ = 2π/λ and that ℏ = h/2π. The constant h is Planck's constant.

A comparison of the discussed wave types

Dispersion and wave velocities
Fig. 1. Comparison of the dispersion curves of the four wave types discussed above. We see that the "wave equation" or dispersionless curve is linear, that of the water wave shows its square root dependence, that of the wave guide approaches being linear at large wavenumbers, and that of the Schrodinger equation is parabolic.	Fig. 2. Comparison of the phase velocities versus wavenumber of the four wave types discussed above. We see that the dispersionless curve is constant as expected, that of the water waves is largest at low wavenumbers (highest frequencies) and slowly decreases with increasing wavenumbers. The phase velocities of the wave guide waves and waves of the Schrodinger equation are infinite at low wavenumbers and become smaller at larger wavenumbers.
	Fig. 3. Group velocities for the four types of waves discussed above versus wavenumber. The "no dispersion" medium shows a group velocity equal to the phase velocity. The water waves have group velocities closely following the phase velocities. The wave guide shows group velocities related as the inverse of the phase velocities. The Schrodinger equation waves have group velocities linearly proportional to the wavenumber, that follows the quantum mechanics idea that momentum is proportional to the wavenumber.

Table 1. Differential wave equation types
No dispersion - the "wave equation"	(30)
Water waves - waves on a liquid surface	(31) for large depth, i.e. κy >> 1 .
Waveguide	(32)
The Schrodinger Equation	(33)

The a's in the above equations are real constants. The differential equations (30) through (33) are taken from the preceding text with the constants replaced by a's.

Looking at the above differential wave equations in Table 1 we see that all the equations relate derivatives with respect to position x to derivatives with respect to time t. All the derivatives are operating on a wave parameter labelled ψ for the purposes of comparison.

What differs between the four equations are the derivative orders. That is, in (30) we see second derivatives with respect to both x and t whereas in (31) we see a first derivative with repect to x and second derivative with respect to t. Eqs. (32) and (33) have a third term which has no derivative.

Eqs. (31) and (33) have the complex constant i in them which signals that they are written for complex solutions. The i is also necessary so that the standard complex sinusoidal wave, i.e. Ae^i(κx−ωt), be a solution to the equation.

Requirements for a general differential wave equation

It would seem that we could write differential wave equations containing derivatives of any order, even fractional orders, as long as we included the required complex constant to insure that Ae^i(κx−ωt) be a solution.

Could we write a differential wave equation that did not have Ae^i(κx−ωt) as a solution but instead had some other solution that moved along in time? If that solution had a fixed shape as it moved along then this shape could be decomposed into sinusoidal Fourier components which would need to move along in unison so that the shape be preserved in time. This would indicate a non-dispersive differential equation of the type of (30) which of course has sinusoidal solutions.

So for a non-sinusoidal wave equation we are left with solutions which are non-sinusoidal and that do not maintain their shape as they moved along. Such "wave" solutions might be addressed by a differential equation which lacks the required complex constant to make Ae^i(κx−ωt) a solution. It is not clear that these would be classified as waves.

Example of creating a diferential wave equation

Suppose we want to create a differential wave equation for a media that supports sinusoidal waves and has a dispersion relation of . To avoid fractional order derivatives we square both sides to have integer powers of κ and ω: where a is the proportionality constant. We now write a differential equation that includes a second derivative of ψ with respect to time t and a third derivative with respect to position x: where b is a complex constant to be determined. We substitute our standard wave into that and arrive at: . This means or . Substituting b into our differential equation we have our final wave equation: .

Non-linear wave equation?

All the above work assumes that we are dealing with a linear wave medium. That is to say if ψ₁ and ψ₂ are both solutions of the wave equation for the medium in question then the sum of these, i.e. ψ₃ = ψ₁ + ψ₂, will also be a solution. Note for example that our derived wave equation just above is first order in ψ even though it involves higher order derivatives of ψ.

In reality, many wave media are not completely linear, especially for large amplitude waves. We will not address such waves here, except to say they are more complicated to deal with.

All postings by author

three types of waves

last posting: emitter games