IMAGINARY FREQUENCY OR WAVE
NUMBER, TUNNELING AND DAMPING
Ken Cheney
4/23/2006
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):
d 2q
dq q
R  0
2
dt
dt C
(1.1)
d 2x
dx
m 2  R  kx  0
dt
dt
(1.2)
L
Or:
So the solutions will be the same if we just let:
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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:
x  x0 cos(0t )
(1.3)
0  k / m
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:
x  x0e

R
t
2m
cos(t )
(1.4)
With:

k  R 


m  2m 
2
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.
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(1.5)
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:
  i '
k
(1.6)
ei  e  i
2
(1.7)
 R 
2
'  
 
 2m  m
Recall the unlikely identity
cos( ) 
If that isn’t familiar start with:
ei  cos( )  i sin( )
(1.8)
And play a little to get Eq. 1.8.
Now with equations 1.8 and 1.7:
cos(t )  cos i ' t 
eii ' t  e  ii ' t
2
 ' t
e  e ' t

2

(1.9)
(1.10)
(1.11)
Combining Equations (1.11) and (1.4) gives the equation for large R:
x  x0e

R
t
2m
 e  ' t  e ' t 


2


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.
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(1.12)
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:
k  R 


m  2m 
2
(1.13)
We are going to do an electrical experiment so Eq. (1.13) becomes, in
electrical terms:
1
 R 


LC  2 L 
2
(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.
R
L
Oscilloscope
Function
Generator
C
Figure 1
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LRC Circuit for variable damping
4
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.
Figure 2
Square Wave
square wave.
Figure 3
Damped 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.
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Figure 4
Over damped Square Wave
QUANTUM MECHANICAL TUNNELING
Schrodenger’s Equation

2
2m
2  ( x, y , z , t )  V ( x, y , z , t )   i


t
(1.15)
If V is only a function of x, y, and z then  can be separated into space
and time parts as:
 ( x , y , z , t )   ( x , y , z ) ( t )
(1.16)
And:
(1.17)
(t )  eit
And:

2
2m
2 ( x, y, z, t )  V ( x, y, z, t )   E
(1.18)
If V=V(x) then this simplifies to:
2

  V ( x )  E
2m t 2
(1.19)
2
2m
   2 ( E  V )
2
t
(1.20)
2
  k 2
t 2
(1.21)
2
Or:
Or:
Where:
k
2m
2
(E V )
(1.22)
With the solutions:
 ( x)  Aeikx  Beikx
(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:
( x, t )  ( Aeikx  Beikx )eit
(1.24)
i ( kx t )
 i ( kx t )
 Ae
 Be
(1.25)
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Since any function of (kx  t ) or (kx  t ) 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.
V
k real
k imaginary
k real
E
E
V
Figure 5
Potential Barrier for Tunneling
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)
k  ik '
k'
2m
2
(V  E )
(1.27)
Putting Equation (1.26) into Equation (1.24) we see:
( x, t )  ( Aeiik ' x  Be iik ' x )e it
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(1.28)
Or:
( x, t )  ( Ae k ' x  Bek ' x )eit
(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!
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