Wave - Particle Duality |

We have seen that **Compton
effect** is a direct evidence of the **corpuscular** nature of light. On the other hand the phenomena of
interference, diffraction, and polarization reveal the wave nature of light. Louis de
Broglie proposed that this dual character of light may also be a characteristic of the
subatomic particles like electrons. According to Broglie's proposal a material particle of
energy **E** and momentum **p** may exhibit characteristics of
waves of wavelength **l** as

**l= h/p,
Eq. (1)**

where **h** is the Planck's
constant and its frequency is determined by Planck's energy quantization expression:

**E = hf.
Eq.
(2)**

Eqs. (1) and (2) can be rewritten as follows,

**p = h/l = hbar
k, E = hbar w, Eq.
(3)**

where **k=2****p/l** is
the wavenumber and **w=2****pf** is the circular frequency.

The dual nature of matter (wave or
particle) is hard to digest. To understand this we do a series of thought experiments with
'bullets' and 'electron guns' as proposed by the famous physicist R. P. Feynman [Examples
taken from ** Feynman Lectures in
Physics - Volume III**].

** First Thought Experiment with
Indestructible Bullets**:

Our experimental setup is shown in the figure below. We also have machine gun (not AK 47!) that shoots a stream of bullets. In front of the machine gun we have a wall with two holes which are big enough for the bullets to pass through. Beyond the wall we have a screen which will absorb the bullets when they hit it. We then can count the number of bullets that have passed through the wall and have been absorbed in the screen.

We may now ask the following question

"What is the **probability**
that a bullet can be found on the screen at a distance 'x' (from the center of the screen)
after it made its trip through the holes?"

The result of such an experiment is plotted
in the graph in Fig. (c) above. We call the probability **P _{12}** as
the bullets can come through any of the holes

Now we do the same experiment but we close
one of the hole, say hole **2**. The probability of the bullets to pass
through hole **1** and hit the screen is say **P _{1}**
[Fig (a) above ] .

Similarly we close hole **1**
and let the bullets pass only through hole **2** - say the probability
distribution in this case is **P _{2}** [Fig. (b)]. Comparing parts
(b) and (c) of the Fig. above we find the important result that:

**P _{12} = P_{1}
+ P_{2}.**

The probabilities just add together. Thus
we can say that the effect of both holes open is the sum of the effects with each hole
open alone. The important observation in this example is that there is '**no interference**.'

** Young's Double-Slit Experiment**:

A parallel beam of monochromatic light is
incident on the two parallel slits of width '**a**' and separated by a
distance '**b**'. When both the slits are open we see a **interference** pattern as you will see in the simulation below. If any one slit is shut
off, a single **diffraction** pattern will be observed.

Please on the simulations below to observe the interference patterns.

**A simulation on Young's Double-slit Experiment**

Another simulation on the double-slit experiment:

**Another simulation on Young's Double-slit Experiment**

Yet another simulation on the double-slit experiment:

** Second Thought Experiment with
Electrons**:

We now repeat the same experiment but we have electrons instead of bullets. Instead of machine gun we fire the electron with an electron gun. We have a detector (may be a Geiger counter) which can detect as soon as the electrons hit the screen after passing through the holes. We repeat the same three steps as we did with the bullets:

(1) keep both the holes open - each
electron either goes through hole **1** or hole hole **2**. The
probability in this case say is **P _{12} **[Fig. (c) below]

(2) close hole **2** and keep
only hole **1** open - each electron have no other choice but will try to go
through **1**. The probability to find an electron on the screen is **P _{1}**
[Fig. (a) below].

(3) close hole **1** and keep
only hole **2** open - each electron have no other choice but will try to go
through **2**. The probability to find an electron on the screen now is **P _{2}**
[Fig. (b) below].

You may think that similar to the case of bullets [Fig. (b) below]

**P _{12} = P_{1}
+ P_{2}.**

but that's not the case as shown in the graph below!

The graphical pattern when both the holes
are open are very similar to the interference pattern in the Young's double slit
experiment [shown above]. We thus conclude the "**There is interference**" and we find that, **P _{12}** is not just the
sum of

__How is it possible?__

There is no way you can explain the
interference pattern physically if you think electrons as simply particles. Let us
first try to understand the mathematics that relates **P _{1}** and

**P _{1} = | j_{1}
|^{2},
P_{2} = | j_{2} |^{2}**.

Remember that **P _{1}**
gives the effect when hole

**P _{12} = | j_{1}
+ j_{2} |^{2}**.

Now we can see that

**P _{12} = | j_{1}|^{2}
+ | j_{2} |^{2} + 2| j_{1}|
|j_{2}| cosq,**

**= P _{1} + P_{2}
+ 2 (P_{1} P_{2})^{(1/2)} cosq ,**

where **q** is
the phase difference between **j**_{1} and **j**_{2}.
The last term is the "**interference
term**" and the physics is the
same as Young's double slit experiment in optics.

So we conclude with the statement - **The electrons arrive in lumps, like particles, and
the probability of arrival of these lumps is distributed like the distribution of
intensity of a wave**. We can then say
that an electron behaves "**sometimes
like a particle and sometimes like a wave.**"

To see a 'cool' demonostration of this thought experiment (along with understandable text) please on the following link:

**Electron Interference Experiment**

** Third Thought Experiment with
Electrons (Watching the electrons!)**:

To our experiment we add a very strong
light source and we place it behind the wall and between the two holes, as shown in the
Fig. below. The purpose is to '*watch*' the electrons.

When an electron passes through one of the
holes to the detector it will scatter some light to our eyes. We will see a flash of light
coming either from the bottom of the light source (if the electron pass through hole **2**)
or from the top of the light source (if the electron pass through hole **1**).
We will assume that the electron is an indestructible particle - so we will not see light
flashes from the top and the bottom of the light source at the same time. Every time
we hear a 'click' from the electron detector we also see a 'flash' either from the top or
the bottom of the light source. We repeat the steps in our previous experiments with
electrons. Using the light flash we keep count of the electrons that pass through hole **1**
or hole **2** and finally make its way to the detector. Say **P _{1}**
and

We now turn off the light source - the situation is identical to our second thought experiment. In this case we don't look at the electrons as if we don't care about their route to the detector. We are just interested in the results. Now we see interference [Fig. (c) above].

From this experiment we conclude that
"**when we look at the electrons
the distribution of them on the screen is different than when we do not look.**" In other words, if the electrons are not seen, we
have interference.

In our experiment we find that it is impossible to arrange the light in such a way that one can tell which hole the electron went through, and at the same time not disturb the interference pattern. This is a general consequence of Heisenberg's uncertainty principle which we will discuss later.

We now briefly **summarize** our
results.

The **probability**
of an event in an ideal experiment is given by the square of the absolute value of a
complex number **j** which is called the **probability amplitude**:

**P = probability,**

**
j = probability amplitude,**

**
P = | j | ^{2}.**

When an event occur in several alternative
ways, the probability amplitude for the event is the sum of the probability amplitudes for
each way separately. **There is interference**:

**j = j _{1}
+ j_{2} ,**

**
P = | j _{1} + j_{2} |^{2}.**

If an experiment is performed which is
capable of determining whether one or another alternative is actually taken, the
probability of the event is the sum of the probabilities for each alternative. **The
interference is lost**:

**P = P _{1} +
P_{2}.**

__The Uncertainty Principle:__

Considering the experimental limitations of
experiments Heisenberg proposed his famous uncertainty principle which is the basis of
quantum mechanics. If you make a mesurement on any object and you measure the **x**-component
of its momentum with an uncertainty D**p**_{x} (please read **delta p _{x}**),
then you can not at the same time, know its

**Dx = h/Dp _{x}**,

where **h** is the Planck's
constant.

The uncertainties in the position and momentum of a particle at any instant must have their product greater than Planck's constant. Mathematically,

**Dx Dp _{x} >= h**

This realtion is known as **Heisenberg's uncertainty principle**. The above one-dimensional consideration can be extended
to **3**-dimesnions.

**Dx Dp _{x} >=
h, Dy Dp_{y}
>= h, Dz Dp_{z}
>= h.**

The uncertainty principle can also be
expressed in other forms. If **E** is the energy of a quantum system at the
time **t**, then the uncertainties D**E** and
D**t** are related by the equation

**DE Dt >= h.**

__Heisenberg's Gamma Ray
Microscope:__

To examine the uncertainty principle of Heisenberg, N. Bohr proposed this hypothetical experiment which consists of a gamma ray microscope as shown below.

The idea is to detect an electron as exactly as possible. To do that Bohr assumed that very short wavelength radiation (gamma rays) may be used to 'illuminate' the electron and this radiation scattered from the electron may then be observed by means of the gamma-ray microscope. Since ordinary optical parts can not focus gamma-rays, such an experiment is actually hypothetical.

Now according to physical optics, the
smallest distance D**x** between two points in an object
that will produce separated images in a microscope is given by (this is known as the
resolving power of a microscope),

**Dx = l/2sinq,
Eq. (4)**

where **2****q**
is the angle subtended by the objective lens as shown in the figure. Now gamma rays of
energy **hf**, possess momentum **p = hf/c= h/****l**.
We assume that the scattered gamma-ray to have the same momentum **p** which
it has originally when it enters the microscope. To be collected by the lens, the photon
may be scattered through any angle between **-****q** to **+****q**. This
imparts to the electron an **x**- component of the momentum **p _{x}**
having any value between

**Dp _{x} = 2h sinq/l.
Eq. (5)**

Multiplying Eqs. (4) and (5) we get,

**Dp _{x} Dx = h.**

This is *in agreement* with the
uncertainty principle.

__Group and Phase velocity:__

According to de Broglie a particle (or
matter) of energy **E** and momentum **p** also possess
charactersitics of waves of wavelength **l** [Eq. (3) above]. Now classically a wave of definite
wavelength and frequency is of infinite extent in space and is of infinite duration. On
the other hand a particle is localized at a definite point in space at a given instant of
time (with a definite momentum and energy). Thus the two pictures (wave and particle) are
incompatible. What we need is a "**wave
group**" or **pulse** which is localized in a finite region of space. This is shown in the Fig.
below.

To obtain such a localized wave group (or wave packet) we need to superpose waves of different wavelengths upon one another. Let us examine the situation mathematically:

Consider a one-dimensional wave of
wavelength **l** and frequency **f** propagating in the
**+x** direction with a phase velocity **v _{p}** (the
crest and trough of the wave moves with this velocity). Such a wave can be mathematically
described as:

**y = A cos( 2px/l - 2pft),**

where **A** is the amplitude
of the wave and **y** is the displacement along the **+y** axis.
We can rewrite the same equation in a slightly different fashion as:

**y = A cos( kx - wt
), **

where, **k = (2**p**/**l**)**
is the wave-number and **w = (2**p**f )**
is the circular frequency. The **phase velocity** of the wave is given by, **w/k
= 2** **p** **f** **l**/**(2****p) =** **f**
**l** = **v _{p}**. Thus we have the phase velocity in terms
of the frequency

**v _{p}= w/k.**

Now imagine two such moving waves of
slightly different wave numbers **k _{1}** and

**y _{1} = A
cos(k_{1}x - w_{1}t),
y_{2}
= Acos(k_{2}x - w_{2}t).**

Using the principle of superposition the resultant wave is,

**y = y _{1} +
y_{2} = A[ cos(k_{1}x - w_{1}t)+cos(k_{2}x - w_{2}t)
].
Eq. (6)**

We now use an identity from trigonometry:

**cosB + cosC = 2 cos
[(B-C)/2]cos [(B+C)/2].**

Eq. (6) now becomes

**y = A' cos [(k _{1}+k_{2})x/2
- (w_{1}+w_{2})t/2 ],
Eq. (7)**

where

**A' = 2A cos[(k _{1}-k_{2})x/2 - (w_{1}-w_{2})t/2
]**

**
= 2A cos [ (**D**k/2) x - (**D**w/2)
t].
Eq. (8)**

and D **k = |k _{1}-k_{2}|**,
D

**v _{g} = **D

**v _{g}** is known as
the

**v _{p} = (w_{1}
+ w_{2})/(k_{1} + k_{2}).
Eq. (10)**

For two waves with nearby frequencies, **w _{1}**
and

**v _{p} = w/k.
Eq. (11)**

Thus the high-frequency wave moves at the
phase velocity **v _{p}**. Using Eqs. (9) and (11) we get a
relationship between the

**v _{g} =
dw/dk = d(k v_{p})/dk = v_{p} + k dv_{p}/dk.
Eq. (12) **

Eq. (12) can be rewritten in terms of the
wavelength **l**

**
v _{g}= v_{p} - l (dv_{p}/dl).
Eq. (13)**

The phase velocity in general depends on
the wavelength **l**. Materials in which the phase velocity varies with
wavelength are said to exhibit **dispersion** (example the different colors of light travel at different
speed inside a glass which is a dispersive medium). For light waves in vacuum, there is no
dispersion. So **dv _{p}/d**

To see a nice demonostration of the group and the phase velocities of a wave please on the simulation below:

**A simulation on Group and Phase velocities of a
travelling wave**

From Eq. (8) we can see that the envelope
term **A**'=**2A cos[(**D**k/2)x]** (at
t=0) has minima (which means **A**'**=0**) when **(**D**k/2) x
= (n+1/2)p** where **n=0,1,2,...**. The distance between two successive
minima D**x** satisfies the condition,

**(Dk/2) Dx = p => Dk Dx = 2p.
Eq. (14)**

Recall that D**k
= |k1 - k2|** is the range of wavenumbers present. Similarly, if **x**
is held constant (say **x = 0** ) and time **t** is allowed to
vary in the envelope portion, the result we get is **(****Dw/2)Dt=p **or

**Dw Dt = 2p.
Eq. (15)**

A general characteristic of wave groups is
that they are both of limited spatial diatnce D**x** and
limited time duration D**t**. The smaller the spatial width of
the pulse, D**x**, the larger the range of
wavelengths (or wavenumbers), D**k**, needed to form a pulse.
Mathematically this mean

**Dk Dx = 1.
Eq. (16)**

Similarly if the time duration, D**t**,
of the pulse is small, we need a wide spread of frequencies, D**w**,
to form the group. Mathematically,

**Dt Dw = 1.
Eq. (17)**

In electronics, this condition is known as
the "**response time-bandwidth
formula**." The above expression
shows that in order to amplify a voltage pulse of time width D**t**
without distortion, a pulse amplifier must equally amplify all frequencies in a frequency
band of width D**w**.

We will now do an example to show how to calculate the group and the phase velocities.

**An Example:**

**Phase and Group velocities of de
Broglie waves -**

The phase velocity of the de Broglie waves is,

**v _{p} = f l = w/k.**

Now from the special theory of relativity,
we have for the electrons the total energy **E** and the momentum **p**
as (**m _{0}** is the rest mass or the proper mass)

**E = gm _{0}
c^{2} = hbar w,
p = gm_{0} v = hbar k,**

where g **=
1/[1-v ^{2}/c^{2}]^{½}** is the

**E/p = c ^{2}/v
= w/k.**

The phase velocity is then, **v _{p}
= w/k = c^{2}/v**, where

**v _{p} = c^{2}/v_{g}
=>
v_{p} v_{g}
= c^{2}.**

On the other hand, the phase velocity can be obtained from the relativistic expression for the energy

**E ^{2} = p^{2}c^{2}
+ m_{0}^{2}c^{4}
=>
hbar^{2} w^{2} = hbar^{2} c^{2} k^{2} + m_{0}^{2}c^{4}.**

Substituing **w = v _{p}k**,
and solving for

**v _{p} = c [1
+ (m_{0} c/hbar k)^{2}]^{½} = c [ 1+ (m_{0} c l/h)^{2}]^{½}
.**

Thus we have an expression for the phase
velocity of the de Broglie waves. We can clearly see that **v _{p} > c**
- but we know from the postulates of special relativity that nothing (no information) can
travel faster than the speed of light '

**v _{g} = c^{2}/v_{p}
= c/[1+ (m_{0} c l/h)^{2}]^{½} < c.**

The answer to the puzzle is that
information is only carried by the wave group that moves with a group velocity **v _{g}**
(not by the individual frequencies which move with a phase velocity

**Reference:**

*Feynman Lectures in Physics - Volume III.*- Link to a cool Site: http://www.colorado.edu/physics/2000/schroedinger/

© *Kingshuk Majumdar (2000)*