Pope John Paul II was raised in Krakow and was its archbishop from 1964 until he was elected as Pope in 1978. When it was later announced that the city will host World Youth Day in 2016, it will be a tribute to the late Pope who became a saint in 2014.
The current mayor of Krakow is Jacek Majchrowski.
Kraków is twinned, or maintains close relations with:
Portland, Oregon
Portland is the largest city in the U.S. state of Oregon. It is the county seat of Multnomah County. About 538,544 people live in the city of Portland, and about 1,927,881 people live in the city's metropolitan area. It is found in the north part of the state, where the Willamette River meets the Columbia River. Portland has the second most people of any city in the Northwest United States, after the city of Seattle.
Portland has many nicknames. One of these is "The City of Roses," because roses grow well there. Some other nicknames of Portland are "Stumptown" (from the fact that Portland was built over a forest), and "Puddletown" (from the fact that it often rains in Portland).
Many things are from Portland. Some of these things are Powell's Books, Beverly Cleary (and some of her most popular fictional characters, Henry Huggins and Ramona Quimby, which also have statues in the city's "Beverly Cleary Sculpture Garden for Children"), and the Portland Trail Blazers basketball team.
Frame of reference
In physics, a frame of reference, or reference frame, is a perspective that one uses to determine if an object is moving. A frame of reference consists of an object or environment that is considered to be stationary. Typically the observer is at rest in the reference frame; in this context, the term is "observational frame of reference."
For example, when you see a ball roll down a street, you can tell the ball is moving because the frame of reference is the streets, whatever may be on the side of the roads, or the Earth. All of these are frames of reference.
All measurements of motion will be compared to a frame of reference. Therefore, the most commonly used frame of reference is Earth itself, even though it moves. Stars are used as a frame of reference when discussing the motions of the Earth.
When we walk from one side of a ship to the other, our frame of reference is the ship; whether the ship is moving is not relevant to that frame of reference.
An inertial frame of reference is a frame that is either not moving or moving at a constant velocity. In other words, an inertial frame of reference has no net force acting upon it, and therefore Newton's laws of motion are observed.
Inertial reference frames are often used in general relativity and special relativity to describe relative motion of observers or objects and the effects they will endure.
Inertial frames of reference are useful in everyday life as well. For example, consider a situation where two cars are traveling along a road. See Figure 1. If they are separated by a distance of 100 meters, the car in the back is moving at 10 meters per second and the car in front is moving at 5 meters per second, both relative to the ground.
There are three obvious inertial reference frames that can be used to determine when the cars will be even with each other: one where the back car is not moving but the front car is moving backwards at 5 m/s, one where the front car is stationary and the back car is moving forward at 5 m/s, or one where the ground is not moving and the cars are moving at their given velocities. These three perspectives all describe the situation accurately and are physically identical.
Broadway
Broadway is a -long main street in New York City. It is different from other streets in that it runs diagonally across the street grid of Manhattan, diagonally crossing newer streets. The street also goes through the Bronx. It continues north as U.S. Route 9 to Albany, the capital of New York State.
Times Square, where Broadway meets 7th Avenue and 42nd Street, is a center of entertainment and is famous for its many theaters and stage plays.
One of the New York City Subway lines, the BMT Broadway Line, runs under the street in Manhattan, south of Times Square. The IRT Broadway - Seventh Avenue Line runs under Broadway, north of Times Square.
Schrödinger equation
The Schrödinger equation is a differential equation (a type of equation that involves an unknown function rather than an unknown number) that forms the basis of quantum mechanics, one of the most accurate theories of how subatomic particles behave. It is a mathematical equation that was thought of by Erwin Schrödinger in 1925. It defines a wave function of a particle or system (group of particles) which has a certain value at every point in space for every given time. These values have no physical meaning (in fact, they are mathematically complex), yet the wave function contains all information that can be known about a particle or system. This information can be found by mathematically manipulating the wave function to return real values relating to physical properties such as position, momentum, energy, etc. The wave function can be thought of as a picture of how this particle or system acts with time and describes it as fully as possible.
The wave function can be in a number of different states at once, and so a particle may have many different positions, energies, velocities or other physical properties at the same time (i.e. "be in two places at once"). However, when one of these properties is measured it has only one specific value (which cannot be definitely predicted), and the wave function is therefore in just one specific state. This is called wave function collapse and seems to be caused by the act of observation or measurement. The exact cause and interpretation of wave function collapse is still widely debated in the scientific community.
For one particle that only moves in one direction in space, the Schrödinger equation looks like:
formula_1
where formula_2 is the square root of -1, formula_3 is the reduced Planck's constant, formula_4 is time, formula_5 is a position, formula_6 is the wave function, and formula_7 is the potential energy, an as yet not chosen function of position. The left hand side is equivalent to the Hamiltonian energy operator acting on formula_8.
Assuming that the wave function, formula_9, is separable, i.e. assuming the function of two variables can be written as the product of two different functions of a single variable:
formula_10
then, using standard mathematical techniques of Partial differential equations, it can be shown that the wave equation can be rewritten as two distinct differential equations
formula_11
formula_12
where the first equation is solely dependent on time formula_13, and the second equation depends only on position formula_14, and where formula_15 is just a number. The first equation can be solved immediately to give
formula_16
where formula_17 is Euler's number. Solutions of the second equation depend on the potential energy function, formula_7, and so cannot be solved until this function is given. It can be shown using quantum mechanics that the number formula_15 is actually the energy of the system, so these separable wave functions describe systems of constant energy. Since energy is constant in many important physical systems (for example: an electron in an atom), the second equation of the set of separated differential equations presented above is often used. This equation is known as the Time independent Schrödinger Equation, as it does not involve formula_4.
There are many philosophical interpretations of the wave function, and a few of the leading ideas will be considered here. The main idea, called the Born probability interpretation (named after physicist Max Born) comes from the simple idea that the wave function is square integrable; i.e.
formula_21