Diffraction occurs when waves bend around an obstacle. In the case of a slit placed before an incident light, the light bends around the edges of the slit. A screen placed behind the slit will show a diffraction pattern with minima and maxima. Here is a list of various formulas for diffraction in different situations:

Single Slit Diffraction:

Where n is the order of the minima, a is the width of the slip, and Θ is the angle of diffraction.

Diffracted Intensity:

Where is the diffracted intensity and maximum intensity respectively, and .

Circular Diffraction:

Where d is the diameter of the aperture.

Double Slit Diffraction:

Where d is the distance between the centers of the slits, a is the width of the slits, and

Multiple Slit Diffraction:

Maxima:

Just as used in the previous section.

Angular Width:

Where is the angular width of the maxima, N is the number of slits, and d is the distance between the centers of the slits.

Diffraction Gratings:

Dispersion:

Where D is the dispersion (separation) and n is the order.

Resolving Power:

Where R is the resolving power, are the mean wavelength and the wavelength difference, N is the number of rulings in the grating, and n is the order of the maxima.

In Lesson 33 we briefly discussed constructive and destructive interference. In the double-slit experiment light of a single coherent waveform is incident on two small openings, so that a waveform passes through both openings then strikes a screen behind them. This experiment demonstrates the wave-particle duality of light, the property that light can behave as both a wave and a particle in the same experiment. The wave nature of light causes the light passing through the two slits to interfere, yet the light is absorbed on the screen as discrete particles. We can predict where the interference minima and maxima will occur.

Constructive interference will cause a maxima at integer multiples of the wavelenght:

Where d is the distance between the two slits, Θ is the angle between the midpoint of the sliits and the point on the screen, and n is the order. Destructive interference causes a minima at the midpoints:

A thin lens is a lens where the thickness is small compared to the radius.

Ray tracing is a technique which allows us to gain a quick qualitative idea of how a lens will project an image, and can give reasonably close approximations to numerical calculations if done carefully. For a convex lens:

1. A ray coming horizontally from the object leaves the lens on a path through the far focal point.
2. A ray going through the center of the lens continues straight.
3. A ray coming through the near focal point exits the lens horizontally.

For a concave lens:

1. A ray coming horizontally from the object leaves the lens on a path along the near focal point.
2. A ray going through the center of the lens continues straight.
3. A ray going toward the far focal point leaves the lens horizontally.

If a more precise solution is desired, we use the thin lens equation:

Where is the focal length and are the image distance and object distance. The magnification of the image is given by:

If a ray travels from A to B by path P, then it will travel from B to A by reversing path P.

The law of reflection says that the angle of incidence () will be equal to the angle of reflection (), when the angles are measured as deviations from the normal.

For refraction, we use Snell’s Law:

Where the n’s represent the index of refraction. Total internal refraction (where light moving from a higher index of refraction to lower is totally refracted to remain inside the denser material) occurs at the critical angle. This will be where the angle of refraction () is 90º:

Since the sin of 90º is 1:

Now that we’ve briefly covered special relativity, on to the next subject: optics. This is the last subject in my GRE study guide!

The first thing we’ll talk about is waves. There are two main types of waves. A transverse wave oscillates perpendicular to the direction of motion, or propagation. This is the ‘normal’ wave we think about, that looks like a sine wave. An example is a wave on water, where the surface of the water raises up and down as the wave moves to the side.

The other type of wave is a longitudinal or compression wave, where the wave oscillates parallel to the direction of propagation. Sound waves are this kind of wave. Visually, you would picture a spring where parts of it are compressed and parts stretched.

There are a few quantities we use to describe waves:

The period, , is the amount of time to complete one oscillation.

The frequency, , is the number of oscillations in a unit of time, or the inverse of the period:

The wavelength, , is the length of one oscillation, or the distance ‘peak-to-peak’.

All of these quantities are related to the speed (v) of the wave:

When two waves occupy the same space, the result is simply the two waves addded together. This is called the principle of superposition. Superimposing two waves is called interference. Constructive interference is when the sum of the waves is greater than the individual waves (1 + 1 = 2). Destructive interference is when the sum of the waves is less than the individual waves (1 + -1 = 0).

In the last two lessons we’ve used an expression to transform quantities into different reference frames. But so far probably seems a bit mysterious, so now we’ll see where it comes from.

Say we have coordinates measured from two different reference frames, frame 1: (x, y, z, t), and frame 2: (x’, y’, z’, t’). Say frame 2 is moving in the x direction relative to frame 1. In a classical Galilean transformation between the two you only need to account for the slide of space through frame 2 at the speed measured from frame 1:

Galilean Transform

And to convert velocities from the rest frame to the moving frame, differentiate x’, y’, z’ with respect to time, so that the velocity in the x direction is:

Where is the velocity difference between the frames. You may notice a problem here. If we use the speed of light (expressed as ‘c’) as the velocity we’re looking at, this implies the speed of light in the moving frame is less than the speed of light in the rest frame (by subtracting the difference). But this can’t be, the speed of light should be c in all reference frames. The Galilean transformation is a good approximation only as speeds significantly lower than the speed of light, where the subtracted difference makes little difference.

Rather than , we can propose an alternate transformation: . This transform will reduce to approximately the same as the Galilean transformation in the condition that k is close to 1. Transforming the other way, from x’ to x:

Therefore:

Solved for t’:

And since light has the same speed from all reference frames:

and

Multiplying both sides by c:

Solved for x:

For x to equal ct, then:

Solved for k:

Lo and behold, that’s !

Last time we looked at how time and length vary with velocity. Mass also changes with velocity, following the same kind of relationship:

Where is the same as before. This is called relativistic mass. Relativistic momentum is just regular momentum using relativistic mass:

And the relativistic version of Newton’s 2nd Law:

The energy is given by Einstein’s famous formula:

Where k is kinetic energy. With no kinetic energy (a stationary particle), the rest energy is:

The total energy can also be formulated as:

New topic! Now we’re through quantum, time for special relativity. The basic idea of relativity is that measurement systems – time, distance – are different depending on the frame of reference. Here, different frames of reference arise from velocity differences between two objects. If the speed of light is the same as measured from any reference frame, the only way to make that work is for an adjustment of time or space: as velocity increases, time slows down (called time dilation) and length decreases in the direction of motion (called length contraction). We can calculate exactly how much these quantities are affected for various speeds:

Where is time in the moving frame and time in the rest frame, respectively. The second formula uses the same notation for length. In both cases:

Thinking again about the ‘particle in a box’, when the potential forming the walls is greater than the energy of the particle, the particle cannot penetrate the wall and must bounce back. But one of the principles of quantum mechanics is the Heisenburg Uncertainty Principle:

And if we’re saying that we’re absolutely certain the particle is not outside the box, then outside the box – that cannot be. Therefore, there must be some probability the particle has penetrated the barrier. This is called tunneling. In fact, if we look at the probability distribution of the wave function () we’ll see a small but finite probability extending into infinity past the barrier. Stated otherwise, there is a very small probability of finding the particle anywhere. To approximate the transmission probability, or the probability the particle will be found on the other side of the barrier:

where

The expectation value , or the expected mean value, can be found by:

The expectation value of a function dependent on x can be found similarly:

The differences between the harmonic oscillator in quantum perspective (as opposed to classical) are:

1. Discrete energy levels

2. Lowest energy not zero

3. May go beyond ±A (tunneling)

Schrödinger’s Equation for a harmonic oscillater:

This gives energy levels of:

with

This is a topic I’ll definitely need to write more about later.