Imaginary Frequency Or Wave Number, Tunneling and Damping

Imaginary frequencIES AND wave numberS; DAMPING AND TUNNELING

Ken Cheney

4/27/2006

ABSTRACT

The physical results of imaginary frequency or wave number will be investigated three ways.

The phenomena of tunneling through areas where traveling waves are impossible will be investigated with microwaves in wave-guides and for total internal reflection.

The same phenomena occur in Quantum Mechanical tunneling (e.g. alpha decay and tunnel diodes) but we won’t explore that in this lab.

The result of a frequency changing from real to imaginary will be investigated by varying the resistance of a damped LRC circuit.

ABOUT IMAGINARY FREQUENCIES AND OSCILLATIONS

Identical differential equations occur in mechanics (damped harmonic motion) and in an LRC circuit (damped oscillations):

(1.1)

Or:

(1.2)

So the solutions will be the same if we just let:

q becomes x

L becomes m

1/C becomes k

I’ll discuss the mechanical version since it is easier to visualize but we will do the electrical version in the lab because variable electrical resistors are easier to implement than mechanical variable damping.

To see what the solutions must be visualize a simple mass on a spring. If R=0 then the mass oscillates with the same amplitude forever. For simplicity we will assume we start with some displacement but no velocity. It’s not hard to generalize but beside the point for our purposes now.

In this case we can solve the equation to find a solution:

(1.3)

Physically adducing damping can give two rather different results. A small amount of damping (due to air resistance say) leads to a steadily decreasing amplitude of oscillation.

But a large damping (say putting the mass and spring in a vat of honey) produces no oscillations at all, just a slow return to equilibrium.

Mathematically a finite R gives (with a good deal more effort) a solution:

(1.4)

With:

(1.5)

This looks ok for damped oscillations, even a smaller frequency is reasonable since the mass is slowing down all the time.

But, what about the “no oscillation” result with large damping.

Worse, equation 1.6 gives an imaginary frequency for sufficiently big R!!!

Perhaps there is no physical way to get omega imaginary. No hope, we can have R as big as we want and k as small as we want.

Let’s bit the bullet and try an imaginary omega. To make it clear where the imaginaries are let’s define a real omega prime for the conditions where omega is imaginary:

(1.6)

Recall the unlikely identity

(1.7)

If that isn’t familiar start with:

(1.8)

And play a little to get Eq. 1.8.

Now with equations 1.8 and 1.7:

(1.9)

(1.10)

(1.11)

Combining Equations (1.11) and (1.4) gives the equation for large R:

(1.12)

No oscillations, just decay.

You might wonder about the term with the positive exponent but it is multiplied by the negative term that is always larger, see Eq. 1.7.

Oscillations Experiment

What to look for

All the interesting stuff above starts when omega becomes imaginary. From Eq. (1.5) we see that happens when:

(1.13)

We are going to do an electrical experiment so Eq. (1.13) becomes, in electrical terms:

(1.14)

We will make a series LRC circuit and observe the voltage across the capacitor. Of course the charge on the capacitor is proportional to the voltage so the trace on the scope will show q versus time.

Figure 1 LRC Circuit for variable damping

We will charge and discharge the capacitor by using a function generator set to produce square waves in series with the LRC circuit. The period of the function generator should be about five times the period of the LC oscillator in order to see several oscillations.

When R is large however the oscillations will stop. At “critical damping” the output will be almost a perfect square wave with just a little rounding of the leading edges. As R gets larger the rise and fall time of the wave increases giving a much more rounded leading edge.

QUANTUM MECHANICAL TUNNELING

QUANTUM TUNNELING

Schrodenger’s Equation

(1.15)

If V is only a function of x, y, and z then can be separated into space and time parts as:

(1.16)

And:

(1.17)

And:

(1.18)

If V=V(x) then this simplifies to:

(1.19)

Or:

(1.20)

Or:

(1.21)

Where:

(1.22)

With the solutions:

(1.23)

Either part is a solution, since Schrodenger’s Equation is linear any linear combination of solutions is also a solution. Even complex coefficients A and B will work!

To see what this is trying to tell us we can reconstruct the complete solution including time:

(1.24)

(1.25)

Since any function of is a traveling wave (going “right” or “left” respectively) these solutions must be traveling waves of some sort.

This all makes as much sense as can be expected of Quantum Mechanics. If the particle has enough energy to overcome the local potential the particle is free to travel. Sounds quite classical .

But what if we are looking at a space where the local potential is greater than the energy of the particle. Classically we don’t bother to consider this situation for skateboarders rolling up a hill. They never go over a hill if their kinetic energy is less than the potential (mgh) energy needed to reach the top of the hill.

What does the math tell us if E<V? Math tells us a tale much like the one we saw above for oscillations. k is imaginary. To separate out imaginary parts let’s define a k’ to be real when k is imaginary:

(1.26)

(1.27)

Putting Equation (1.26) into Equation (1.24) we see:

(1.28)

Or:

(1.29)

Definitely not a traveling wave!

One might worry about the B term going to infinity as x goes to infinity, not a pleasant prospect! Or the A term would go to infinity as x goes to minus infinity. Happily it works out that the A term applies going in the plus x direction and the B term applies going in the minus x direction.

The moral is that there is a probability of finding the particle at a spot where E<V but that probability decreases with distance.

If this region of E<V changes to a region with V>E then the traveling wave solution reappears and the particle moves with constant probability as a function of x. The particle has “tunneled” through a region where it would be impossible classically for the particle to exist.

The probability of this tunneling can be worked out by matching boundary conditions before and after the barrier (E<V) region. The calculus is fairly simple but the algebra is quite ugly so I’ll refer you to a Quantum Mechanics book for the full solution.

The point here is, again, that a complex number (wave number k in this case) can lead to real, observable results, tunneling!

TUNNELING IN A WAVE GUIDE

Sending electromagnetic waves (microwaves in this case) down a channel made of parallel conducting plates (a wave guide) gives another example of the real, observable use of an imaginary wave number. For a simple geometry we can show that a traveling wave is a solution of the wave equation for a “wide” wave-guide and that the solution changes to a decaying amplitude with an imaginary wave number when the wave-guide is “narrow”.

Electromagnetic energy or signals are transmitted by wires, coax cables, wage guides, and in free space. In the microwave region over relatively short distances wave-guides work very well. In the visible region fiber optics work as wave guides for intercontinental distances, with a few repeaters.

A wave-guide for microwaves is a hollow conducting tube, it looks like a piece of plumbing. We can analyze a simple version consisting of just two parallel conducting plates.

Let the wave travel in the x direction. The electric field will be in the y direction and the plates making up the wave guide will be in an xy plane at z=0 and z=a. Sorry, we seem to have a left handed coordinate system, no matter here.

Since there can be no tangential electric fields at conducting surfaces the electric field must go to zero at the plates.

A traveling wave that will do this is:

(1.30)

With n=1, 2, 3, . . .

There are different modes for different values of n. We don’t care here but for optimum focusing (width) or data transmission (different velocities) you want n=1.

Let’s see what restrictions the wave equation places on this wave:

(1.31)

Since there are no y terms in E:

(1.32)

Combining Eq. (1.30). (1.31), and (1.32) we find that almost everything cancels out and we are left with:

(1.33)

Then

(1.34)

This is what we are interested in here but if you solve for velocity as omega over k you find it is greater than the velocity of light! If you can’t stand that investigate wave and phase velocity.

Is there some physical reason why we can’t set it up to get an imaginary k? No, all we need is a small separation of the plates a, easily done!

To see what happens with an imaginary wave number let’s (for cases where k is imaginary) define:

(1.35)

Then, as before, is real when k is imaginary.

Looking at the part of Eq. (1.30) involving k:

(1.36)

(1.37)

Inserting Eq. (1.37) into Eq. (1.30) we get:

(1.38)

which is not a traveling wave but an electric field decaying exponentially in the x direction.

But if the wave-guide becomes wider than the critical a, cut off width, then a traveling wave will appear again, with a smaller amplitude Ey. This critical a is the minimum a for real k. This is smallest for n =1. Setting k=0 in Eq. (1.34) gives this a.

Wave Guide Experiment

The experiment will add three parts in succession:

Part I has just the source, a section of wave guide wider than the cut off width, and the detector.

Part II adds a section of wave guide narrower than the cut off width between the wave guide of section I and the detector.

Part III adds a section of wave guide wider than the cut off width between the narrow section and the detector.

To start:

Calculate the wave length of the 10GHz microwave.

Calculate the critical, cut off width for a. k=0, n=1in Eq. (1.34)

Set the voltmeter to DC, the detector is a diode and will change the signal to DC.

Be sure the microwave source and detector are oriented to give you the maximum signal. Put them face to face and try rotating the detector.

Be sure the source is emitting waves polarized correctly for the wave guide. With a short section of wide wave guide try the source and detector rotated (together) both ways.

To check:

Is there a traveling wave in section I?

Measure the strength of the signal at the detector as you make section I longer and longer. Plot Voltage verses the length of the wave guide.

Does the intensity decrease suddenly when a reaches the cut off width?

Push the sides of I closer and closer, plot the intensity versus width.

Does the intensity decrease as part II becomes longer?

Restore part I to its original width (wider than the cut off width), add part II (a wave guide narrower than the cut off width).

Measure the intensity as the length of part II increases, add to your plot from 1.

Do you get a traveling wave if the wave guide becomes wider than the cut off width?

Add a wider wave guide (part III) to the end of part II.

Measure the intensity as part III is made longer. Add to your plot.

Tunneling and Total Internal Reflection

If waves go from a medium (lucite block) to a medium (air) where the wave goes faster the angle of refraction is larger than the angle of incidence. Clearly for some angle of incidence the angle of refraction is ninety degrees. Instead of skittering along the surface the waves are totally internally reflected, for this angle and all larger angles of incidence.

It is easy to predict the critical angle using Snell’s law:

(1.39)

If you know the indexes of refraction just set r to ninety degrees and solve for i. If you don’t know the ns you can use Snell’s law to find them.

What is interesting here is whether there is any electric field outside the block when the beam is undergoing total internal reflection, and whether a wave can tunnel through the gap if another lucite block is placed across a small gap from the first block.

This experiment can be done with semicircular Lucite blocks as sketched first or with right triangular lucite blocks as shown second.

Experiment for Tunneling and Total Internal Reflection

Convince yourself that total internal Reflection does occur. Explain clearly what you did!

Plot, on the same graph, the reflected and transmitted intensities as a function of the gap width. Start with zero width.

We might expect an exponential decay in amplitude but life is not that simple in general. Most of us (Like Feynman) will be satisfied to observe tunneling at all, and a semi exponential decay but for those determined to see the theoretical result the transmissivity is:

(1.40)

with:

(1.41)

where I denotes the angle of incidence, n the refraction index of the medium, Z the gap thickness, and lambda the vacuum wavelength.

(1.42)

for the case where E has a component transverse to z. I can’t tell what is divided by what either, I just copied this literally! Or:

(1.43)

A. Kodre and J. Strnad, American Journal of Physics , Vol. 44, No. 2, February 1976, p 181

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