Rachana Singh Thakur: January 2011

Sunday, January 09, 2011

Michelson Morely Experment

Michelson had a solution to the problem of how to construct a device sufficiently accurate to detect aether flow. The device he designed, later known as an interferometer, sent a single source of white light through a half-silvered mirror that was used to split it into two beams travelling at right angles to one another. After leaving the splitter, the beams travelled out to the ends of long arms where they were reflected back into the middle on small mirrors. They then recombined on the far side of the splitter in an eyepiece, producing a pattern of constructive and destructive interference based on the spent time to transit the arms. If the Earth is traveling through an ether medium, a beam reflecting back and forth parallel to the flow of ether would take longer than a beam reflecting perpendicular to the ether because the time gained from traveling downwind is less than that lost traveling upwind. The result would be a delay in one of the light beams that could be detected when the beams were recombined through interference. Any slight change in the spent time would then be observed as a shift in the positions of the interference fringes. If the aether were stationary relative to the sun, then the Earth's motion would produce a fringe shift 4% the size of a single fringe.
In 1881, while he was still in Germany, Michelson had used an experimental device to make several measurements, in which he noticed that the expected shift of 0.04 was not seen, and a smaller shift of (at most) about 0.02 was.^[2] However his apparatus was a prototype, and had experimental errors far too large to say anything about the aether wind. For a measurement of the aether wind, a much more accurate and tightly controlled experiment would have to be carried out. The prototype was, however, successful in demonstrating that the basic method was feasible.

Though using a contemporary laser, this Michelson interferometer is the same in principle as those used in the original experiment.

In 1887 he then combined efforts with Edward Morley and spent a considerable amount of time and money creating an improved version with more than enough accuracy to detect the drift.^[3] In their experiment, the light was repeatedly reflected back and forth along the arms, increasing the path length to 11 m. At this length, the drift would be about 0.4 fringes. To make that easily detectable, the apparatus was located in a closed room in the basement of a stone building, eliminating most thermal and vibrational effects. Vibrations were further reduced by building the apparatus on top of a huge block of marble, which was then floated in a pool of mercury. They calculated that effects of about 1/100th of a fringe would be detectable.
The mercury pool allowed the device to be turned, so that it could be rotated through the entire range of possible angles to the "aether wind". Even over a short period of time some sort of effect would be noticed simply by rotating the device, such that one arm rotated into the direction of the wind and the other away. Over longer periods day/night cycles or yearly cycles would also be easily measurable.
During each full rotation of the device, each arm would be parallel to the wind twice (facing into and away from the wind) and perpendicular to the wind twice. This effect would show readings in a sine wave formation with two peaks and two troughs. Additionally, if the wind were only from Earth's orbit around the sun, the wind would fully change directions east/west during a 12-hour period. In this ideal conceptualization, the sine wave of day/night readings would be of opposing phase.
Because it was assumed that the motion of the earth around the sun would cause an additional component to the wind, the yearly cycles would be detectable as an alteration of the magnitude of the wind. An example of this effect is a helicopter flying forward. While hovering, a helicopter's blades would be measured as travelling around typically at 300 mph at the tips. However, if the helicopter is travelling forward at 150 mph, there are points where the tips of the blades are travelling through the air at 150 mph (downwind) and 450 mph (upwind). The same effect would cause the magnitude of an aether wind to decrease and increase on a yearly basis.

Quotes by Great People

"Look deep into nature, and then you will understand everything better"

Albert Einstein

"Our task must be to free ourselves by widening our circle of compassion to embrace all living creatures and the whole of nature and its beauty."

Albert Einstein

"In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual."

Galileo

"I keep the subject of my inquiry constantly before me, and wait till the first dawning opens gradually, by little and little, into a full and clear light."

Isaac Newton

"If I have seen further it is by standing on the shoulders of giants. "

Isaac Newton

"If I have ever made any valuable discoveries, it has been owing more to patient attention, than to any other talent."

Isaac Newton

"I do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."

Isaac Newton

Cathode Ray Tube

A cathode ray tube is a vacuum tube which consists of one or more electron guns, possibly internal electrostatic deflection plates, and a phosphor target.^[2] In television sets and computer monitors, the entire front area of the tube is scanned repetitively and systematically in a fixed pattern called a raster. An image is produced by controlling the intensity of each of the three electron beams, one for each additive primary color (red, green, and blue) with a video signal as a reference.^[3] In all modern CRT monitors and televisions, the beams are bent by magnetic deflection, a varying magnetic field generated by coils and driven by electronic circuits around the neck of the tube, although electrostatic deflection is commonly used in oscilloscopes, a type of diagnostic instrument.^[3]

Heisenberg uncertainty principle

In quantum mechanics, the Heisenberg uncertainty principle states by precise inequalities that certain pairs of physical properties, such as position and momentum, cannot be simultaneously known to arbitrarily high precision. That is, the more precisely one property is measured, the less precisely the other can be measured.
Published by Werner Heisenberg in 1927, the principle means that it is impossible to determine simultaneously both the position and momentum of an electron or any other particle with any great degree of accuracy or certainty. It should be emphasized that this is not meant to be a statement about a researcher's ability to measure these specific pairs of quantities. Rather, it is a statement about the system itself. That is, a system cannot be defined to have simultaneously singular values of these pairs of quantities. The principle states that a minimum exists for the product of the uncertainties in these properties that is equal to or greater than one half of the reduced Planck constant (ħ = h/2π).
In quantum physics, a particle is described by a wave packet, which gives rise to this phenomenon. Consider the measurement of the position of a particle. It could be anywhere. The particle's wave packet has non-zero amplitude, meaning the position is uncertain – it could be almost anywhere along the wave packet. To obtain an accurate reading of position, this wave packet must be 'compressed' as much as possible, meaning it must be made up of increasing numbers of sine waves added together. The momentum of the particle is proportional to the wavenumber of one of these waves, but it could be any of them. So a more precise position measurement – by adding together more waves – means the momentum measurement becomes less precise (and vice versa).
The only kind of wave with a definite position is concentrated at one point, and such a wave has an indefinite wavelength (and therefore an indefinite momentum). Conversely, the only kind of wave with a definite wavelength is an infinite regular periodic oscillation over all space, which has no definite position. So in quantum mechanics, there can be no states that describe a particle with both a definite position and a definite momentum. The more precise the position, the less precise the momentum.
A mathematical statement of the principle is that every quantum state has the property that the root mean square (RMS) deviation of the position from its mean (the standard deviation of the x-distribution):

$\sigma_x = \sqrt{\langle(x - \langle x\rangle)^2\rangle} \,$

times the RMS deviation of the momentum from its mean (the standard deviation of p):

$\sigma_p = \sqrt{\langle(p - \langle p \rangle)^2\rangle} \,$

can never be smaller than a fixed fraction of Planck's constant:

$\sigma_x \sigma_p \ge \hbar/2.$

The uncertainty principle can be restated in terms of other measurement processes, which involves collapse of the wavefunction. When the position is initially localized by preparation, the wavefunction collapses to a narrow bump in an interval Δx > 0, and the momentum wavefunction becomes spread out. The particle's momentum is left uncertain by an amount inversely proportional to the accuracy of the position measurement:

Mathematical derivations

When linear operators A and B act on a function

ψ(x)

, they don't always commute. A clear example is when operator B multiplies x, while operator A takes the derivative with respect to x. Then, for every wave function

ψ(x)

we can write

$(AB - BA) \psi = \left({d\over dx}x-x{d\over dx}\right)\psi(x) = {d\over dx} ( x \psi(x)) - x {d\over dx} \psi(x)$

$= \psi(x) + x {d \over dx}\psi(x) - x {d \over dx}\psi(x) = \psi(x) ,$

which in operator language means that

${d\over dx} x - x {d\over dx} = 1.$

This example is important, because it is very close to the canonical commutation relation of quantum mechanics. There, in the position basis, the position operator multiplies the value of the wavefunction by x, while the corresponding momentum operator differentiates and multiplies by $\scriptstyle -i\hbar$ , so that:

$[p,x] = p x - x p = -i\hbar \left( {d\over dx} x - x {d\over dx} \right) = - i \hbar.$

It is the nonzero commutator that implies the uncertainty.
For any two operators A and B:

which is a statement of the Cauchy–Schwarz inequality for the inner product of the two vectors $\scriptstyle A|\psi\rangle$ and $\scriptstyle B|\psi\rangle$ . On the other hand, the expectation value of the product AB is always greater than the magnitude of its imaginary part:

$|\langle\psi|AB|\psi\rangle |^2 \ge {\left\vert{1\over 2i} \langle\psi|AB - BA |\psi\rangle\right\vert}^2$

and putting the two inequalities together for Hermitian operators gives the relation:

$\langle A^2 \rangle \langle B^2 \rangle\ge {1\over 4} |\langle [A,B]\rangle|^2$

and the uncertainty principle is a special case.

Energy-time uncertainty principle

One well-known uncertainty relation is not an obvious consequence of the Robertson–Schrödinger relation: the energy-time uncertainty principle.
Since energy bears the same relation to time as momentum does to space in special relativity, it was clear to many early founders, Niels Bohr among them, that the following relation holds:^[2]^[3]

$\Delta E \Delta t \gtrsim h,$

Compton Effect

The Compton effect (also called Compton scattering) is the result of a high-energy photon colliding with a target, which releases loosely bound electrons from the outer shell of the atom or molecule. The scattered radiation experiences a wavelength shift that cannot be explained in terms of classical wave theory, thus lending support to Einstein's photon theory. The effect was first demonstrated in 1923 by Arthur Holly Compton (for which he received a 1927 Nobel Prize).

How's It Work?

The scattering is demonstrated in the picture to the right. A high-energy photon (generally X-ray or gamma-ray) collides with a target, which has loosely-bound electrons on its outer shell. The incident photon has the following energy E and linear momentum p:

E = hc / lambda p = E / c

The photon gives part of its energy to one of the almost-free electrons, in the form of kinetic energy, as expected in a particle collision. We know that total energy and linear momentum must be conserved. Analyzing these energy and momentum relationships for the photon and electron, you end up with three equations:

energy
x-component momentum
y-component momentum

... in four variables:

phi, the scattering angle of the electron
theta, the scattering angle of the photon
E_e, the final energy of the electron
E', the final energy of the photon

If we care only about the energy and direction of the photon, then the electron variables can be treated as constants, meaning that it's possible to solve the system of equations. By combining these equations and using some algebraic tricks to eliminate variables, Compton arrived at the following equations (which are obviously related, since energy and wavelength are related in photons):

1 / E' - 1 / E = 1/(m_e c²) * (1 - cos theta) lambda' - lambda = h/(m_e c) * (1 - cos theta)

The value h/(m_e c) is called the Compton wavelength of the electron and has a value of 0.002426 nm (or 2.426 x 10^-12 m). This isn't, of course, an actual wavelength, but really a proportionality constant for the wavelength shift.

De Broglie wavelength

The de Broglie relations

The de Broglie equations relate the wavelength $~\lambda~$ and frequency $~f~$ to the momentum $~p~$ and energy $~E~$ , respectively, as

$\lambda = \frac{h}{p}$ and $f = \frac{E}{h}$

where $~h~$ is Planck's constant. The two equations are also written as

$p = \hbar k$

$E = \hbar \omega$

where $~\hbar=h/(2\pi)~$ is the reduced Planck's constant (also known as Dirac's constant, pronounced "h-bar"), $~k~$ is the angular wavenumber, and $~\omega~$ is the angular frequency.
Using results from special relativity, the equations can be written as

$\lambda = \frac {h}{\gamma mv} = \frac {h}{mv} \sqrt{1 - \frac{v^2}{c^2}}$

and

$f = \frac{\gamma\,mc^2}{h} = \frac {1}{\sqrt{1 - \frac{v^2}{c^2}}} \cdot \frac{mc^2}{h}$

where $~m~$ is the particle's rest mass, $~v~$ is the particle's velocity, $~\gamma~$ is the Lorentz factor, and $~c~$ is the speed of light in a vacuum.
See the article on group velocity for detail on the argument and derivation of the de Broglie relations. Group velocity (equal to the particle's speed) should not be confused with phase velocity (equal to the product of the particle's frequency and its wavelength).

Releation between mass and momentum

In special relativity, the energy-momentum relation is a relation between the energy, momentum and the mass of a body:

$E^2 = m^2 c^4 + p^2 c^2 , \;$

where c is the speed of light, E is total energy, m is invariant mass, and p is momentum.
For a body in its rest frame, the momentum is zero, so the equation simplifies to

$E = mc^2 \;$

If the object is massless then the energy momentum relation reduces to

$E = pc \;$

as is the case for a photon.
In natural units the energy-momentum relation can be expressed as

$\omega^2 = m^2 + k^2 \;$

where $\omega \;$ is angular frequency, m is rest mass and k is wave number.
In Minkowski space, energy and momentum (the latter multiplied by a factor of c) can be seen as two components of a Minkowski four-vector. The norm of this vector is equal to the square of the rest mass of the body, which is a Lorentz invariant quantity and hence is independent of the frame of reference.
When working in units where c = 1, known as the natural unit system, the energy-momentum equation reduces to

$m^2 = E^2 - p^2 \,\!$

In particle physics, energy is typically given in units of electron volts (eV), momentum in units of eV/c, and mass in units of eV/c². In electromagnetism, and because of relativistic invariance, it is useful to have the electric field E and the magnetic field B in the same unit (gauss), using the cgs (gaussian) system of units, where energy is given in units of erg, momentum in g.cm/s and mass in grams.
Energy may also in theory be expressed in units of grams, though in practice it requires a large amount of energy to be equivalent to masses in this range. For example, the first atomic bomb liberated about 1 gram of heat, and the largest thermonuclear bombs have generated a kilogram or more of heat. Energies of thermonuclear bombs are usually given in tens of kilotons and megatons referring to the energy liberated by exploding that amount of trinitrotoluene (TNT).

Saturday, January 08, 2011

Length Contraction

length contraction – according to Hendrik Lorentz – is the physical phenomenon of a decrease in length detected by an observer in objects that travel at any non-zero velocity relative to that observer. This contraction (more formally called Lorentz contraction or Lorentz–Fitzgerald contraction) is usually only noticeable at a substantial fraction of the speed of light; the contraction is only in the direction parallel to the direction in which the observed body is travelling. This effect is negligible at everyday speeds, and can be ignored for all regular purposes. Only at greater speeds it becomes important. At a speed of 13,400,000 m/s (30 million mph, .0447c), the length is 99.9% of the length at rest; at a speed of 42,300,000 m/s (95 million mph, .141c), the length is still 99%. As the magnitude of the velocity approaches the speed of light, the effect becomes dominant, as can be seen from the formula:

$L' = \frac{L}{\gamma(v)} = L \, \sqrt{1-v^2/c^2}$

where

L is the proper length (the length of the object in its rest frame),

L' is the length observed by an observer in relative motion with respect to the object,

v \, is the relative velocity between the observer and the moving object,

c \, is the speed of light,

and the Lorentz factor is defined as

$\gamma (v) \equiv \frac{1}{\sqrt{1-v^2/c^2}} \$ .

The deviation between the measurements in all inertial frames is given by the Lorentz transformation. As the result of this transformation (see Derivation), the proper length remains unchanged and always denotes the greatest length of an object, yet the length of the same object as measured in another inertial frame is shorter than the proper length. This contraction only occurs in the line of motion, and can be represented by the following relation (where v is the relative velocity and c the speed of light)

$L = L_0 \cdot \sqrt{1-\frac{v^2}{c^2}}.$

For example, a train at rest in S' and a station at rest in S with relative velocity of v = 0.8c are given. In S' a rod with proper length L_0^{'}=30\ \mathrm{cm} is located, so its contracted length L' in S is given by:

$L = L_0^{'} \cdot \sqrt{1-\frac{v^2}{c^2}} = 18\ \mathrm{cm}.$

Then the rod will be thrown out of the train in S' and will come to rest at the station in S. Its length has to be measured again according to the methods given above, and now the proper length L_0 = 30\ \mathrm{cm} will be measured in S (the rod has become larger in that system), while in S' the rod is in motion and therefore its length is contracted (the rod has become smaller in that system):

$L' = L_0 \cdot \sqrt{1-\frac{v^2}{c^2}} = 18\ \mathrm{cm}.$

Thus, as it is required by the principle of relativity (according to which the laws of nature must assume the same form in all inertial reference frames), length contraction is symmetrical: If the rod is at rest in the train, it has its proper length in S' and its length is contracted in S. However, if the rod comes to rest relative to the station, it has its proper length in S and its length is contracted in S'.

Length contraction can simply be derived from the Lorentz transformation as it was shown, among many others, by Max Born:[6]

Length contraction can simply be derived from the Lorentz transformation as it was shown, among many others, by Max Born:^[6]

^Derivation

In an inertial reference frame S', $x_{1}^{'}$ and $x_{2}^{'}$ shall denote the endpoints for an object of length $L_{0}^{'}$ at rest in this system. The coordinates in S' are connected to those in S by the Lorentz transformations as follows:

$x_{1}^{'}=\frac{x_{1}-vt_{1}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$ and $x_{2}^{'}=\frac{x_{2}-vt_{2}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}.$

As this object is moving in S, its length

L

has to be measured according to the above convention by determining the simultaneous positions of its endpoints, so we have to put $t_{1}=t_{2}\,$ . Because $L=x_{2}-x_{1}\,$ and $L_{0}^{'}=x_{2}^{'}-x_{1}^{'}$ , we obtain

(1) $L_{0}^{'}=\frac{L}{\sqrt{1-\frac{v^{2}}{c^{2}}}}.$

Thus the length as measured in S is given by

(2) $L=L_{0}^{'}\cdot\sqrt{1-\frac{v^{2}}{c^{2}}}.$

According to the relativity principle, objects that are at rest in S have to be contracted in S' as well. For this case the Lorentz transformation is as follows: